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0 for each h(.) and clearly by linearity, we have
ftl (OL
/.h(t)dt
=0 I siiy 8L),, is continuous, it must be for each h(.). Since the mapping t '- (at - dtd S1identically zero along y(.) and the Euler-Lagrange equation (2.2) is satisfied. o
ax
TX_
0
2.1.2 Hamiltonian Equations The Hamiltonian formalism, which is the natural formalism to deal with the maximum principle, appears in the classical calculus of variations via the
Legendre transformation. Definition 8. The Legendre transformation is defined by
p=
si (t, x, i)
(2.3)
and if the mapping cp : (x, i) - (x, p) is a diffeomorphism we can introduce the Hamiltonian:
H : (t,x,p)'-' p.x - L(t,x,P).
(2.4)
30
2 Optimal Control for Nonlinear Systems
Proposition 10. The formula (2.1) takes the form t, 8L (_. 8x o
d 8L l
QC t
dt 8X I
t, 1.,,'h(t)dt + [ & - Hot1 to
(2.5)
and if -y(.) is a minimizer it satisfies the Euler-Lagrange equation in the Hamiltonian form
±(t) =
Xt)
(t, x(t), p(t)),
e (t, x(t), p(t))
(2.6)
2.1.3 Hamilton-Jacobi-Bellman Equation Definition 9. A solution of the Euler-Lagrange equation is called an extremal. Let Po = (to, xo) and P1 = (tl, xl ). The Hamilton-Jacobi-Bellman (HJB) function is the multivalued function defined by t,
S(Po, Pi) = f L(t, 7(t),1'(t))dt to
where 'y(.) is any extremal with fixed extremities xo, x1. If -y(.) is a minimizer, it is called the value function.
Proposition 11. Assume that for each Po, P1 there exists an unique extremal joining Po to Pl and suppose that the HJB function is C1. Let P0 be fixed and let S : P - S(Po, P). Then S is a solution of the Hamilton-JacobiBellman equation 69S
8t
49S
(Po,
P) + H(t, x, 8x)
(2.7)
0.
Proof. Let P = (t, x) and P + bP = (t + dt, x + bx). Denote by y(.) the extremal joining Po = (to,xo) to P and by ry(.) the extremal joining Po to P+BP. We have
aS = S(t + dt, x + dx) - S(t, x) = C(y) - C(y) and from (2.5) it follows that:
A3=AC,
t(8L-
10 \ 8x
d8LI ,ti'h(t)dt + dt x I
[pbx -
L
H8t ] to,
where h(.) = ry(.)-y(.). Since y(.) is solution of the Euler-Lagrange equation, the integral is zero and
AS = 'dc , In other words we have
[P,6x -
HUI
to
2.1 A Short Visit into the Classical Calculus of Variations
31
dS = pdx - Hdt. Identifying, we obtain 63-
at
= - H,
Hence we get the HJB equation. Moreover p is the gradient to the level sets {x E R°; S(t, x) = c}. 0
2.1.4 Euler-Lagrange Equations and Characteristics of the HJB Equation Under some extra regularity conditions, the extremals are the characteristics of HJB equation. Indeed, let u(.) be a solution of the HJB equation. Hence we can write (2.7) as
F(t'x'
as as as at' ax) = at
as
+H(t'x' ax) = 0
and let us assume the map F to be C2. Introduce p = as, T = as and z = S(t, x). Then, according to (63] the characteristic curves parametrized by s are solutions of. dx ds
OF
OH
ap
ap
dpOF_OF _ _ax p Ts
dz
OH
az
ax
' aH
OF
T =pap + T =pap - H
dtOF1 _
dT Ts
OF at
ds OF az
OT
aH p
at
In particular since 7, = 1, we deduce that dx
OH
dt - ap'
dp dt
OH
ax
which is the Hamiltonian form of the Euler-Lagrange equation.
2.1.5 Second Order Conditions The Euler-Lagrange equation has been derived using the linear terms in the Taylor expansion of W. Using the quadratic terms we can get a necessary and sufficient second order condition. For the sake of simplicity, from now
2 Optimal Control for Nonlinear Systems
32
on we assume that the curves t - x(t) belongs to R, and we consider the problem with fixed extremities: x(to) = xo, x(tl) = xl. If the map L is taken C3, the second derivative is computed as follows:
LC = f", L(t, y(t) + h(t),7(t) + ih(t)) - L(t, y(t),7(t)))dt ti 8L d8L hZ (t) +28'L J to Jtp e2L 8x Wt 8i I7' h(t)dt+12 (t' ( azft h(t)h(t) &2
+ o(h, h)2.
If y(.) is an extremal, the first integral is zero and the second integral corre-
sponds to the intrinsic second-order derivative 62C, that is: 62C
1
-2
f1,
o
a2 L
\ axe I, h2+
2a.92 L
xax IY
h(t)h(t)
a2 L
+ ax
h2dt.
(2.10)
Using h(to) = h(ti) = 0, it can be written after an integration by parts as 62C
where
=f'
P1822
2ax 6
(th2ct) + Q(t)h2(t))dt
(2.11)
d 82L Q = 2 ( axe - dt axax 1
82L
Using the fact that in the integral (2.11) the term Ph2 is dominating (see (471), we get the following proposition.
Proposition 12. If y(.) is a minimizing curve for the fixed extremities problem then it must satisfy the Legendre condition: 82L
(2.12)
2.1.6 The Accessory Problem and the Jacobi Equation The intrinsic second-order derivative is given by b2C =
f ' (P(t)h2(t) + Q(t)h2 (t)) dt,
h(to) = h(ti) = 0,
where P, Q are as above. We write 62C
= /o ((P(t)h(t))h(t) + (Q(t)h(t))h(t))dt
and integrating by parts using h(to) = h(ti) = 0, we obtain
2.1 A Short Visit into the Classical Calculus of Variations
b2C = Jol (Q(t)h(t)
-
33
dt(P(t)h(t)))h(t)dt.
Let us introduce the linear operator D : h
Qh - dt (Ph). Hence, we can
write
52C = (Dh, h)
(2.13)
where (,) is the usual scalar product on L2([to, t1]). The linear operator D is
called the Euler-Lagrange operator. Definition 10. From (2.13), 62C is a quadratic operator on the set Co of C2-
curves h : [to,tl] - R satisfying h(to) = h(tl) = 0, h # 0. Rather to study 62C > 0 for each h(.) E Co we can study the so-called accessory problem: min b2C. hECo
Definition 11. The Euler-Lagrange equation corresponding to the accessory problem is called the Jacobi equation and is given by
Dh = 0
(2.14)
where D is the Euler-Lagrange operator: Dh = Qh order linear differential operator.
d(Ph).
It is a second-
Definition 12. The strong Legendre condition is: P > 0 where P = z elry. If this condition is satisfied, the operator D is said to be nonsingular.
2.1.7 Conjugate Point and Local Morse Theory See [52, 85].
Definition 13. Let y(.) be an extremal. A solution J(.) E Co of DJ = 0 on [to, tll is called a Jacobi curve. If there exists a Jacobi curve along y(.) on [to,tll the point -y(tl) is said to be conjugate to -y(to).
Theorem 4 (Local Morse theory [851). Let to be fixed and let us consider the Euler-Lagrange operator (indexed by t > to) Dt defined on the set Co of curves on [to, t] satisfying h(to) = h(t) = 0. By definition, a Jacobi curve on [to, t] corresponds to an eigenvector Jt associated to an eigenvalue At = 0 of Dt. If the strong Legendre condition is satisfied along an extremal y : [to, T] - R", we have a precise description of the spectrum of Dt as follows.
There exists to < ti < . . . < to < T such that each y(ti) is conjugate to -y(to). If ni corresponds to the dimension of the set of the Jacobi curves Jt{
associated to the conjugate point y(ti), then for any T such that to < tl < < tk < T < tk+I < . < T we have the idendity -
k k
n , =
ni
(2.15)
34
2 Optimal Control for Nonlinear Systems
where nT = dim{linear space of eigenvectors of DT corresponding to strictly negative eigenvalues}. In particular if t > tj we have min
hECo
ft. (Q(t)h2(t) + P(t)h2(t))dt = -oo.
(2.16)
2.1.8 Scalar Riccati Equation Definition 14. The quadratic differential equation
P(t)(Q(t) + tb(t)) = w2(t)
2.17)
is called the scalar Riccati equation.
Its connection with the problem is the following. Assume P > 0 on [to, ti] and assume that there exists a solution u(.) of the Jacobi equation such that this solution does not vanish on the interval [to, t,]. Let h(.) be any C2-function such that h(to) = h(ti) = 0. Then t,
d(w(t)h2(t))dt = 0
fto and
b2C =
t ((Ph2 + Qh2) + d(wh2))dt
Jto
= f (Ph+ 2whh + (Q + w)h2)dt.
If w(.) is a solution of the Riccati equation, the previous expression can be written as 62C h(t)) 2dt. P(t) (h(t) + r_
to
1
Hence 62C
=
tl
P(t)cp2(t)dt
to
where W(t) = h(t) + "'tit . Now observe that if we set w(t) = -u t P(t) where u(.) is nonvanishing on [to,ti], then u(.) is a solution of the Jacobi equation:
Q(t)u(t) and
h(t) +
d
(P(t)u(t)) = 0
w(t)h(t)
h(t)u(t) - h(t)u(t)
P(t)
U(t)
Hence cp =- 0 is equivalent to
h(t)u(t) - h(t)u(t) = 0.
2.1 A Short Visit into the Classical Calculus of Variations
35
This is possible if and only if h(.) = Cu(.) where C is a constant. It contradicts
the fact that u(.) does not vanish on [to, t1] and that h(to) = h(ti) = 0 if h 0 0. Hence ca 0 unless h- 0 and 62C
t,
=
J to
is nonzero for each h(.) E Co and 62C > 0.
2.1.9 Local CO Minimizer - Extremal Field - Hilbert Invariant Integral Definition 15. Consider the time-minimal problem with fixed extremities: Let 7(.) be a reference trajectory. It is called a C°(to, xo), (ti, xi) E minimizer if it is a local minimum for the C°-topology: Rn+i.
d(x, x*) = ma`el IIx(t) - x (t)II tE
To obtain C°-sufficient optimality conditions we use the concept of ex-
tremal (or Mayer field). Definition 16. Let 7 : [to, t1] - W1 be a reference extremal issued from xo at t = to: 7(to) = xo. An extremal field is a mapping 0 : (a, t) -- Rn+1, a E D = parameter space C R" such that: 1. 4)(ao, t) = (t, 7(t)) is the reference extremal and 10(a,.); a E D} is a family .T of extremals;
2. the image of 0 denoted T is a tubular neighborhood of 7(.) and through each point (t, x) of T there passes an unique extremal of F whose derivative is denoted by u(t, x). We assume that t '- u(t, x) is Cl and we use the following notations: P
aL
L=Ll,(
(2.18) =),
H=HIa=p
where H = p.x - L is the Hamiltonian. We can easily prove the following lemma.
Lemma 6. The following relations hold: Oj3
8Pk
8xk = 8xi '
OH
Opi
8xi
at
In particular the one form w = -Ifdt + Pdx, called Hilbert-Cartan form, is closed.
36
2 Optimal Control for Nonlinear Systems
Theorem 5 (Hilbert invariant integral theorem). The integral
Jr
-Hdt + pdx
is independent of the curve r(.) on T. Moreover if n) is an extremal of F it is given by fr Ldt. Proof. The first assertion is a consequence of the fact that the form w is = u(t, x) and -Hdt+pdx =
closed. Moreover if T(.) is an extremal we have (L - 3.u(t, x))dt + pdx. 0
Remark 3. 1. The resolution of c:i = dS on the domain where S : (t, x) F-+ R is a smooth function, is equivalent to solve the Hamilton-Jacobi equation. 2. In practice we use a central Mayer field, i.e. all the extremals are starting at to - E from a single point xo, for e > 0 small enough.
Corollary 6. Let -y(.) be the reference extremal with extremities (to,xo), (t1,x1) and let rd be any curve of T with same extremities. We define by E the excess Weierstrass mapping:
E(t, x, x, w) = L(t, x, w) - L(t, x, z) - (w - z) 8x (t, x, z)
z = u(t, x), (t, x) E T. Then, if E > 0 we have that y(.) is a Co-minimizer on T. Proof. We have
L(t, x, i)dt =
J(L - p.u(t, x))dt +
J7
zC = fr L(t, x, i)dt - fy L(t, x, i)dt = fr ((L - L) - (i - u(t, x).p) dt = fr E(t, x, u(t, x), i)dt. This proves the assertion.
2.2 Optimal Control and the Calculus of Variations 2.2.1 Problem Statement We consider the autonomous control system
2.2 Optimal Control and the Calculus of Variations
i(t) = f (x(t), u(t)),
x(t) E R", U(t) E .R
37
(2.19)
where f is a C'-mapping. Let the initial and target sets Mo, M1 be given. We assume M0, M, to be C'-submanifolds of R. The control domain is a given
subset Sl C Rm. The class of admissible controls U is the set of bounded measurable mappings u : [0,T(u)] -i 1. Let u(.) E U and xo E R" be fixed. Then, by the Caratheodory theorem, see [77], there exists a unique trajectory of (2.19) denoted by x(., x0, u) (in short x(.)) such that x(0) = xo. This trajectory is defined on a nonempty sub-interval J of [0, T(u)) on which t F- x(t, xo, u) is an absolutely continuous function and is solution of (2.19) almost everywhere. To each u(.) E U defined on [0, T] with response x(., xo, u) issued from x(0) = x° E Mo defined on [0, T], we assign a cost T
C(u) = J f°(x(t),u(t))dt
(2.20)
0
where f ° is a C'-mapping. An admissible control u*(.) with corresponding trajectory z ' (., x' (0), u') defined on 10,T*] such that x' (0) E M0 and x'(T') E M1 is optimal if for each admissible control u(.) with response x(., x(0), u) on [0, T], X(0) E M0 and x(T) E M1, then
C(u*) < C(u).
2.2.2 The Augmented System The following remark is straightforward but is geometrically very important
to understand the maximum principle. Let us consider f = (f, fo) and the corresponding system on R"+' defined by the equations I = f (±(t), u(t)), i. e:
±(t) = f (x(t), u(t)), x°(t) = f°(x(t), u(t)).
(2.21)
(2.22)
This system is called the augmented system. Since f is C1, according to the Caratheodory theorem, to each admissible control u(.) E U there exists an admissible trajectory &(t, $o, u) such that io = (x0i x°(0)), x°(0) = 0 where the added coordinate x°(.) satisfies x°(T) = fu f°(x(t),u(t))dt. Let us denote by AMo the accesibility set Uu(.)Eu (T, xo, u) from Mo =
(Mo, 0) and let if, = M1 x R. Then, we observe that an optimal control u'(.) corresponds to a trajectory !'(.) such that :to* E Mo and intersecting M1 at a point &* (T*) where x° is minimal. In particular &* (T) belongs to the boundary of the accessibility set AMo , see Fig. 2.1.
38
2 Optimal Control for Nonlinear Systems
Fig. 2.1.
2.2.3 Related Problems Our framework is a general setting to deal with a large class of problems. Examples are the following:
1. Nonautonomous systems:
i(t) = f (t, x(t), u(t)). We add the variable t to the state space by setting as = 1, t(so) = so. 2. Fixed time problem. If the time domain [0, T(u)J is fixed (T(u) = T for all u(.)) we add the variable t to the state space by setting d8 = 1, t(so) = so
and we impose the following state constraints on t: t = 0 at s = 0 and t = T at the free terminal time s. Some specific optimal problems important for applications are the following.
1. If f ° __ 1, then min fo f °(x(t), u(t))dt = min T and we minimize the time of global transfer. 2. If the system is of the form: ±(t) = f (t, x(t), u(t)), f (t, x, u) = A(t)x(t) +
B(t)u(t), where A(t), B(t) are matrices and C(u) = fo L(t, x(t), u(t))dt where L(x, u,.) is a quadratic form for each t, T being fixed, the problem is called a linear quadratic problem (LQ-problem).
2.2.4 Optimal Control and the Classical Calculus of Variations Classical problems of calculus of variations can be easily stated as optimal control problems as follows.
2.2 Optimal Control and the Calculus of Variations
39
1. Holonomic problems: min f L(t, x(t), x(t)dt. We introduce the control
system by setting x(t) = u(t) and we must minimize a cost C(u) _ f L(t, x(t), u(t))dt. In particular the accessory problem min h(.)
J
(P(t)h2(t) + Q(t)h2(t))dt
is transformed into the LQ-problem h(t) = u(t),
min u(.)
J
(P(t)h2(t) + Q(t)u2(t))dt.
2. Nonholonomic problems: more generally we consider the problem min x(.)
J
L(t, x(t), x(t)dt)
among a set of curves satisfying the constraints x(t) E D(x(t)) can be reformulated into an optimal control problem when the differential inclusion can be restated as a system x(t) = f (t, x(t), u(t)). An important example in our study is the sub-Riemannian problem: T
min fa (z(t), x(t))g dt with x(t) E D(x(t)) and
D(x) = Span{F1(x), ... , F,(x)}
where the distribution D generated by the vector fields F's is of constant rank and (,)g is the scalar product associated to a Riemannian metric g.
2.2.5 Singular Trajectories and the Weak Maximum Principle Definition 17. Consider a system of R": ±(t) = f (x(t), u(t)) where f is a Coo -mapping from R" x R"' into R" . Fix xo E R" and T > 0. The end-point mapping (for fixed xo, T) is the mapping Exo.T : u(.) E U -. x(T, xo, u). If u(.) is a control defined on [0, T] such that the corresponding trajectory x(., x0, u) is defined on [0, T), then E--°,T is defined on a neighborhood V of u(.) for the L°°([0,TJ) norm.
2.2.6 First and Second Variations of E,O,T It is a standard result, see for instance [99], that the end-point mapping is a C°°-mapping defined on a domain of the Banach space L°D ([0, T] ). The formal computation of the successive derivatives uses the concept of Gateaux
derivative. Let us explain in details the process to compute the first and second variations.
40
2 Optimal Control for Nonlinear Systems
Let v(.) E L°°([O,T]) be a variation of the reference control u(.) and let us denote by x(.) + f (.) the response corresponding to u(.) + v(.) issued from x0. Since f is C°O, it admits a Taylor expansion for each fixed t:
(x, u)f + L (x, u)v + lku (x, u)(6, v)
f (x + l;, u + v) = f (x, u) +
e8
2
+2
2
(x,u)(C1C)+28 (x,u)(v,v)+
Using the differential equation we get
k(t) + fi(t) = f (x(t) + fi(t), u(t) + v(t)) Hence we can write 1; on the form: 61x + 52x + where 51x is linear in v, Sex is quadratic, etc and are solutions of the following differential equations: 61x
a2x
+ 8f
8x
(x, u)51 x + au (x, u)v
(x, u) a2x +
02
(2.23)
(x, u ) (a lx, v)
-
+2 a (x'u)(6 1x, 61x)+ 2au
(x,u)(v,v).
( 2 . 24 )
Using 1;(O) = 0, these differential equations have to be integrated with the initial conditions 61x(0) = 52x(0) = 0. (2.25) Let us introduce the following notations:
A(t) = Lf (x(t), u(t)), ax Definition 18. The system
B(t) =
(x(t), u(t)) 49U
6x(t) = A(t)6x(t) + B(t)5u(t) is called the linearized system along (x(.), u(.)).
Let M(t) be the fundamental matrix on [0, T] solution a.e. of
Af(t) = A(t)M(t),
M(O) = identity.
Integrating (2.23) with 61x(0) = 0 (see Lemma 1) we get the following expression for 61x: T
81x(T) = M(T) fo M -1(t)B(t)v(t)dt This implies the following lemma.
(2.26)
2.2 Optimal Control and the Calculus of Variations
41
Lemma 7. The Frechet derivative of Ex0,T at u(.) is given by rT Eu O.T(v) = &,x(T) = M(T) J M-1(t)B(t)v(t)dt. 0
Definition 19. The admissible control u(.) and its corresponding trajectory x(., x0i u) defined both on [0, T] are said to be regular if the Frechet derivative E1x0,T is surjective. Otherwise they are called singular.
Proposition 13. Let A(xo, T) = Uu(.)EUx(T, x0, u) be the accessibillity set at time T from x0. If u(.) is a regular control on [0, T] then there exists a neighborhood U of the end-point x(T, x0i u) contained in A(xo, T).
Proof. Since E'xO,T is surjective at u(.), we have using the open mapping Theorem [31] that ExO,T is an open map. Theorem 6. Assume that the admissible control u(.) and its corresponding trajectory x(.) are singular on [0, T]. Then there exists a vector p(.) E R"\{0} absolutely continuous on [0, T) such that (x, p, u) are solutions a.e. on [0, T] of the following equations: (x (t),
dt (t) = an
p(t), u(t)),
dt (t)
_ - eH (x(t), p(t), u(t)) (2.27) a (x(t), p(t), u(t)) = 0 (2.28)
where H(x, p, u) = (p, f (x, u)) is the pseudo-Hamiltonian, (,) being the standard inner product. Proof. We observe that the Frechet derivative is solution of the linear system
81x(t) = A(t)6,x(t) + B(t)v(t).
Hence, if the pair (x(.), u(.)) is singular this system is not controllable on [0,T]. We use the proof of Proposition 1 to get a geometric characterization of this property. The proof which is the heuristic basis of the maximum principle is given in details. By definition, since u(.) is a singular control on [0, T] the dimension of the linear space
{jT M (T)M1(t)B(t)v(t)dt; v(.) E L°°([O,T])} is less than n. Therefore there exists a row vector p E R"\{0} such that
EM(T)M-1(t)B(t) = 0 for a.e. t E [0, T]. We set
42
2 Optimal Control for Nonlinear Systems
p(t) = pM(T)M-1(t) By construction p(.) is solution of the adjoint system
P(t) = -p(t) Lf (x(t),u(t)) Moreover, it satisfies almost everywhere the following equality:
p(t) Lf (x(t), u(t)) = 0. Hence we get the equations (2.27) and (2.28) if H(x, p, u) denotes the scalar product (p, f (x, u)). 0
2.2.7 Geometric Interpretation of the Adjoint Vector In the proof of Theorem 6 we introduced a vector p(.). This vector is called an adjoint vector. We observe that if u(.) is singular on [0, T], then for each 0 < t < T, ul10,, is singular and p(t) is orthogonal to the image denoted K(t) of E'x0,T evaluated at u11011. If for each t, K(t) is a linear space of codimension
one, then p(t) is unique up to a factor.
2.2.8 The Weak Maximum Principle Theorem 7. Let u(.) be a control and x(., xo, u) the corresponding trajectory, both defined on [0, T]. If x(T, xo, u) belongs to the boundary of the accessibility set A(xo,T), then the control u(.) and the trajectory x(.,xo, u) are singular.
Proof. According to Proposition 13, if u(.) is a regular control on [0, T] then x(T) belongs to the interior of the accessibility set.
Corollary 7. Consider the problem of maximizing the transfer time for system i(t) = f (x(t), u(t)), u(.) E U = L°°, with fixed extremities xo, xl. If u`(.) and the corresponding trajectory are optimal on 10,,r*), then u`(.) is singular. Proof. If u* (.) is maximizing then x' (T) must belongs to the boundary of the accessibility set A(xo, T) otherwise there exists E > 0 such that x` (T - e) E A(xo, T) and hence can be reached by a solution x(.) in time T: x* (T - E) =
I(T). It follows that the point X*(T) can be joined in a time t > T. This contradicts the maximality assumption.
Corollary 8. Consider the system +(t) = f (x(t), u(t)) where u(.) E U = L01([0, T)) and the minimization problem: min
rT
J
u(.)EL/ p
L(x(t),u(t))dt, where
the extremities xo, xI are fixed as well as the transfer time T. If u'(.) and
2.2 Optimal Control and the Calculus of Variations
43
its corresponding trajectory are optimal on [0, T[ then u' (.) is singular on [0,T] for the augmented system: x(t) = f (x(t), u(t)), to (t) = L(x(t), u(t)). Therefore there exists P* (t) = (p(t),po) E Rn+1\{0} such that (i*j5*,u*) satisfies
X(t) _ 4V (1(t),P(t),u(t)),
p(t)
a (x(t),*t),u(t)), (2.29)
aH u
WO, p(t), u(t)) = 0
where 1 = (z, x°) and H(x, P, u) = (p, f (x, u)) + poL(x, u).
Proof. We have that x' (T) belongs to the boundary of the accessibility set A(:io,T). Applying (2.27),(2.28) we get the equations (2.29) where Po = = 0 since ft is independent of x°. Hence po is a constant. 0
-
2.2.9 Abnormality In the previous corollary, P*(.) is defined up to a factor. Hence we can normalize po to 0 or -1 and we have two cases:
1. Case 1: u(.) is regular for the system x(t) = f (x(t), u(t)). Then po 0 0 and can be normalized to -1. This case is called the normal case (in calculus of variations), see [18].
2. Case 2: u(.) is singular for the system x(t) = f (x(t), u(t)). Then we can choose po = 0 and the Hamiltonian H evaluated along (x(.), p(.), u(.)) doesn't depend on the cost L(x, u). This case is called the abnormal case.
2.2.10 The Weak Maximum Principle and Euler-Lagrange Equation We can deduce from Corollary 8 the standard Euler-Lagrange equation. Indeed consider the problem of minimizing fo L(t, x(t), x(t))dt, where L R2n+1 -- IR is a smooth map, among the set of absolutely continuous curves
t t-- z(t) in R", with bounded derivative and satisfying the boundary conditions: x(0) = xo, x(T) = xl. We introduce for almost every t, the linear system +(t) = u(t), u(t) E R", with u(.) a measurable bounded function. We have H(i,1P, u) = p.u + poL(x, u)
where p is a row vector in R". Since the linear system is controllable, any = 0, we get optimal control is normal and we can set po = -1. Using
44
2 Optimal Control for Nonlinear Systems
Pi (t) = -Po au W* u(t)) = 8u; (x(t), u(t)),
i = 1, ... , n.
Moreover
AM
eH (x(t), P(t), u(t)) = -P0 8x (x(t), u(t)) = gxi (x(t), u(t)).
Integrating this last equation with respect of t, we get Pi (t) = Pi (to) + f
tax (x(s), u(s))ds
and we write the Euler-Lagrange equation in the integral form (satisfied a.e.):
rt
8xi W* u(t)) = Pi (to) +
8L (x(s), u(s))ds,
(2.30)
Moreover, if the curve t'- x(t) is C2 we obtain by differentiating at a (x(t), u(t)) = ax WO, u(t))
everywhere on [0, TI.
2.2.11 Comparison with the Calculus of Variations Although we can recover the Euler-Lagrange equation from the weak maximum principle, the two viewpoints are radically different and the point of view of the maximum principle is much superior to the one of the calculus of variations in several ways: 1. we impose minimal regularity assumptions on the set of curves; 2. we use the concept of the augmented system where the derivative of the
cost is a state variable and the adjoint vector have a clear geometric explanation; 3. we obtain a set of equations in the Hamiltonian form without using the Legendre transformation which is not in general well-defined.
2.2.12 LQ-Control and the Weak Maximum Principle We can apply the maximum principle to get optimality necessary conditions in the LQ-problem. For the sake of simplicity we analyze only the autonomous case. We consider the problem of minimizing the cost
IT( C(u) =
tx(t)Rx(t) +t u(t)Uu(t)dt
2.3 Pontryagin's Maximum Principle (PMP)
45
among the set of curves satisfying
zb(t) = Ax(t) + Bu(t), X(t) E Rn, U(t) E Rm where A, B, R, U are constant matrices, and R, U are symmetric. We assume to have fixed boundary conditions: x(O) = xo, x(T) = x1. Moreover we impose the (strong Legendre) regularity condition: U > 0
and we assume that the linear system i(t) = Ax(t) + Bu(t) is controllable, that is the rank(R) = n where of 7Z = [B, , An-1B]. From this last assumption, any minimizer is normal and according to Corollary 8, a minimizer is solution of the following constraint Hamiltonian system:
x(t) =
8p
(x(t), p(t), u(t)),
p(t)
aH (x(t), p(t), u(t) ),
a WO, p(t), u(t)) = 0 where FI (x, p, u) = (p, Ax + Bu) - 1( txRx + tuUu). Solving the linear = 0, we get with U > 0, that an optimal control is defined by equation the (dynamic) feedback
u(p) = U-' tB tp
where p is written as a row vector. Introducing the Hamiltonian function
H(x,p) = H(x,p,u(p))
.,
-
= 0, we get _ . Hence and using the constraint an optimal trajectory is the projection on the x-space of a solution of the following Hamilton system:
i(t) =
8
(x(t),p(t)),
P(t)
aft
a (x(t),p(t))
2.3 Pontryagin's Maximum Principle (PMP) In this section we state the Pontryagin maximum principle and we outline the proof. We adopt the presentation from Lee and Markus [77] where the result is presented into two theorems. The complete proof is complicated but rather standard, see the original book from the authors [91].
Theorem 8. We consider a system of R': i(t) = f (x(t), u(t)), where f Rn++n -- R' is a Cl-mapping. The family U of admissible controls is the set of bounded measurable mappings u(.), defined on [0, T] with values in a control domain Il C Rm such that the response x(., xo, u) is defined on [0, T].
Let U(.) E U be a control and let 7(.) be the associated trajectory such that
46
2 Optimal Control for Nonlinear Systems
T(T) belongs to the boundary of the accesibility set A(xo, T). Then there exists p(.) E 1111\{0), an absolutely continuous function defined on [0,21 solution almost everywhere of the adjoint system:
At) = -p(t) of (T (t), u(t))
(2.31)
such that for almost every t E (0, T] we have
H(T(t), p(t), u(t)) = M(Y(t), p(t))
(2.32)
where
H(x, p, u) = (p, f (x, u)) and
M(x, p) = max H(x, p, u). UEf1
Moreover t F-- M(T(t), p(t)) is constant on [0,T].
Proof. The accessibility set is not in general convex and it must be approximated along the reference trajectory T(.) by a convex cone. The approximation is obtained by using needle type variations of the control u(.) which are closed for the L1-topology. (we do not use L°° perturbations and the Frechet-derivative of the end-point mapping computed in this Banach-space) 0
Needle type approximation. We say that 0 < t1 < T is a regular time for the reference trajectory if d
dt a1
J
f (7(7-), u(r))dr = f (Y(tl), u(tl))
and from measure theory we have that almost every point of [0, T] is regular. At a regular time t1, we define the following L'-perturbation U,(.) of the reference control: we fix 1, e > 0 small enough and we set
ue(t) =
u1 E .R constant on [tI - le,tl] -u(t) otherwise on [0, T]
We denote by 7,(.) the associated trajectory starting at TE(0) = x0, see Fig. 2.2. We denote by a -+ at(e) the curve defined by at(e) = Te(t) for t > ti. We have
xf(tl) = (t1 - lE)+
t1
t1 -IE
f (Ye(t),uf(t))dt
where uE = ul on [t1 - lE, t1]. Moreover
Y(ti) = 7(t1 - lE) +
1
t1
!e
f (T (t), u(t))dt
2.3 Pontryagin's Maximum Principle (PMP)
47
xo
Fig. 2.2.
and since ti is a regular time for a(.) we have
Ye(ti) -Y(ti) = 1E(f (x(ti),ui) - f (x(tl), u(ti)) + o(E). In particular if we consider the curve E '- at, (E), it is a curve with origin T(ti) and whose tangent vector is given by
v = 1(f(T(ti),u1) -f(x(ti),'u(ti)).
(2.33)
For t > ti, consider the local diffeomorphism: ¢t(y) = x(t,ti,y,u) where x(., ti, y, u) is the solution corresponding to u(.) and starting at t = ti from y. By construction we have at(e) = cbt(at, (e)) for e small enough and moreover for t > ti, vt = W16-0 t t(E) is the image of v by the Jacobian V.L. In other words vt is the solution at time t of the variational equation dV
Of
dt = ax
(x (t), u(t))V
(2.34)
with condition vt = v for t = ti. We can extend vt on the whole interval [0,T). The construction can be done for an arbitrary choice of ti, I and ul. Let 17 = {t1,1, ui } be fixed, we denote by v[j (t) the corresponding vector vt.
Additivity property. Let t1, t2 be two regular points of u(.) with ti < t2 and 11, 12 > 0 small enough. We define the following perturbation ii (t) =
I
ui on [ti - lie,ti] t2] u2 on [t2 u(t); otherwise on [0, T]
48
2 Optimal Control for Nonlinear Systems
where u1, u2 are constant values of (1 and let 7E(.) be the corresponding trajectory. Using the composition of the two elementary perturbations III = {t1i l1, u1 } and 172 = {t2, 12, u2} we define a new perturbation
H : {t1i t2, ll)l2, ui, u2}. If we denote by vn, (t), vn2 (t) and v,7 (t) the respective tangent vectors, a computation similar to the previous one gives us:
vn (t) = vn, (t) + vn, (t),
fort > t2.
We can deduce the following lemma.
Lemma 8. Let 1 7 = It,,
, t87 A111 i , Asia, ul, , us} be a perturbation at regular times ti, ti < < t8i li > 0, Ai > 0, Fi=1 Ai = 1 and corresponding to elementary perturbations Hi = {ti, li, ui} with tangent vectors vn: (t). Let 7E(.) be the associated response with perturbation H. Then we have
xE(t) = x(t) +
EAivrfi (t) + o(e)
(2.35)
i=1
where
0, uniformly for 0 < t < T and 0 < Al < 1.
Definition 20. Let u(.) be an admissible control and a(.) its associated trajectory defined for 0 < t < T. The first Pontryagin's cone K(t), 0 < t < T is the smallest convex cone at x(t) containing all elementary perturbation vectors for all regular times ti.
Definition 21. Let vi,
, v,, be linearly independent vectors of K(t), each vi being formed as convex combination of elementary perturbation vectors at distinct times. An elementary simplex cone C is the convex hull of the vectors vi.
Lemma 9. Let v be a vector interior to K(t). Then there exists an elementary simplex cone C containing v in its interior. Proof. In the construction of the interior of K(t), we use convex combination of elementary perturbation vectors at regular time not necessary distincts. Clearly by continuity, we can replace such a combination by a cone C in the interior with n distinct times.
Approximation lemma. An important technical lemma is the following topological result whose proof uses the Brouwer fixed point theorem.
Lemma 10. Let v be a nonzero vector interior to K(T), then there exists A > 0 and a conic neighborhood N of Av such that N is contained in the accessibility set A(xo,T). Proof. See [77].
2.3 Pontryagin's Maximum Principle (PMP)
49
The meaning of this lemma is the following. Since v is interior to K(T), there exists an elementary simplex cone C such that v is interior to C. Hence for each w E C there exists L() a perturbation of u(.) such that its corresponding trajectory 7e(.) satisfies
7,(T) = Y(T) + ew + o(w). In particular there exists a control U,(.) such that we have
Y,(T) = x(T) + ev + o(e). This geometric situation is illustrated on Fig. 2.3, by construction 7e(T) E K(T). In other words K(T) is a closed convex approximation of A(xo, T).
Ev
Fig. 2.3.
Separation step. To finish the proof, we use the geometric Hahn-Banach theorem. Indeed if x(T) E OA(xo, T) there exists a sequence x V A(xo, T) such that x,, -, Y(T) when n - +oo and the unit vectors x^-x T have
a limit w when n - oo. The vector w is not interior to K(t) otherwise from Lemma 10 there would exist A > 0 and a conic neighborhood of Aw in
A(xo, T) and this contradicts the fact that x,, 0 A(xo, T) for any n. Let a be an hyperplane at a(T) separating K(T) from w and let p' be the exterior unit normal to 7r at !(T), see Fig. 2.4. Let us define p(.) as the solution of the adjoint equation P(t) = -p(t) 8x (x(t), u(t)) satisfying p(T) = p. By construction, we have
p(T)v(T) < 0 for each elementary perturbation vector v(T) E K(T) and since for t E [0, T] the following equations hold:
50
2 Optimal Control for Nonlinear Systems
A(x0,T)
Fig. 2.4.
P(t) = -p(t) ax (x, u),
v(t)
ax
(Y, u)v
we have
p(t)v(t) = 0.
Hence p(t)v(t) = p(T)v(T) < 0, Vt. Assume that the maximization condition (2.32) is not satisfied on some subset S of 0 < t < T with positive measure. Let t1 E S be a regular time, then there exists ul E S7 such that p(t1)f(7(t1), 9(t1)) < p(t1)f(T(t1), ul) Let us consider the elementary perturbation 171 = {t1,1, ul } and its tangent vector
VIII (ti) = 1[f(x(tl),ul) -f(x(t1),u(t1)}. Then, using the above inequality we have that P(tl )vn, (t1) > 0
which contradicts p(t)vn, (t) < 0, for all t. Therefore the equality
H(T(t),p(t),u(t)) = M(Y(t),p(t)) is satisfied a.e. on 0 < t < T. Using a standard reasoning we can prove that t - + M(x(t), p(t)) is absolutely continuous and has zero derivative almost everywhere on 0 < t < T, see [77].
2.3 Pontryagin's Maximum Principle (PMP)
51
Theorem 9 (Maximum Principle). Let us consider a general control system: i(t) = f (x(t), u(t)) where f is a continuously differentiable function and let Mo, M1 be two Cl submanifolds of Rn. We assume the set U of admissible controls to be the set of bounded measurable mappings u : [0, T(u)] .0 E Rm, where !l is a given subset of Rm. Consider the following minimization problem: minUEU C(u), C(u) = fo f °(x(t), u(t))dt where f ° E C', x(O) E Mo, x(T) E Ml and T is not fixed. We introduce the augmented system:
i°(t) = f°(x(t),u(t)),
x°(0) = 0
(2.36)
i(t) = f (x(t), u(t)),
(2.37)
x(t) = (x°(t),x(t)) E R"+1, f = (f°, f). If (x*(.),u*(.)) is optimal on [0,T'1, then there exists P'(.) = (p°, p(.)) : [0, T*] -+ R"+1\{0} absolutely continuous, such that (i' (.), P' (.), u' (.)) satisfies the following equations almost everywhere on 0 < t < T*: x (t) = 8p WO , P(t) , u (t)) , p- (t) = - aH (x (t) , P(t) , u (t)) fI(x(t), P(t), u(t)) = )cf(x(t), P(t))
(2 . 38)
(2.39)
where
H(x,0,u) _
Ax, u)), M(x,P) = En H(x,P,u)
Moreover, we have
M(x(t),P(t)) = 0,dt, p° < 0
2.40)
and the boundary conditions (transversality conditions):
x'(0) E Mo, x*(T*) E Ml, P (0)1T .(o)Mo, p'(T')1T=.(T)Ml.
(2.41)
(2.42)
Proof (for the complete proof, see [771 or [91]). Since (x'(.),U*(.)) is optimal
on [0,T*] the augmented trajectory t -4 x' (t) is such that x' (T) belongs to the boundary of the accessibility set A(x' (0), T*). Hence by applying Theorem 8 to the augmented system, one gets the conditions (2.38),(2.39) and k constant. To show that M = 0, we construct an approximated cone K'(T) containing K(T) but also the two vectors ± f (x' (T ), u' (T)) using time variations (the transfer time is not fixed). To prove the transversality conditions, we use a standard separation lemma as in the proof of Theorem 8. Definition 22. A triplet (x(.), p(.), u(.)) solution of the maximum principle is called an extremal.
2 Optimal Control for Nonlinear Systems
52
2.4 Filippov Existence Theorem In order to solve optimal control problems, we need an existence theorem about optimal trajectories. The following existence theorem can be found in [77] with a complete proof (in a more general setting than the one stated here).
Theorem 10. Consider a control system in 1R": ±(t) = f (x(t), u(t)), where f is C1, with the following data: - The initial and target sets M0, Ml are nonempty compact sets of R. - The control domain 42 is a nonempty compact set in Rm. - The state constraints are of the form hi (x) > 0, i = 1, , p where the hi are Co functions on R". - The set U of admissible controls is the set of measurable mappings u(.) [0, T] -> Si, such that each u(.) has a response x(.), 0 _< t < T steering
x° E Mo to x(T) = x1 E Ml, and t i-- x(t) is entirely contains in the restraint set: h; (x) > 0. - The cost to be minimized is for each u(.) E U of the form:
C(u) =
o 0
f°(x(t),u(t))dt
where f ° is a C1 function. We assume the following.
1. The family U is not empty, that is there exists u(.) steering xo E M° to
x1EM1. 2. For each response x(.) defined on [0,T] corresponding to u(.) E U, there exists an uniform bound:
Ix(t)I
as (a)
fo.(x,u) = 1
(2.43)
and the maximum is reached for the optimal control. Let us now make the following additional assumption:
Assumption (H2) The function x " T(x) is C2. We can derive the equations of the maximum principle as follows. Consider the function n 9(x,u) _
YX-
Let u(.) be an optimal control and x(.) be the corresponding trajectory. Fix t and consider the mapping
x " g(x, u(t)) Then for all x, we have g(x, u(t)) < 1 and for x = x(t) the maximum is reached. Hence differentiating with respect to x we get 09(x, u(t)) = 0 at x = x(t). T
Simple computations lead to the following relation at x = x(t): n
n (a
a2 axa
(X)f. (x, u) +
a (x) axa (x, U)) = 0
for i = 1, ... , n.
2.5 Dynamic Programming and the Maximum Principle
55
We can write n
2
E as
ax;(x(t))fa(x(t),u(t)) =
E a- (ax) (x(t)) ac (t) a=1
a=1
d (0,
d ax (x(t))). Therefore along an optimal pair (x(.),u(.)) we have d
dt
(
n
(t))
E 8x. (x(t)) a (x(t),u(t))
aax;
, pa (t))
Introducing the adjoint vector p(t) = (pl (t), level set w(x) = constant: P=(t)
as the gradient to the
= ax (x(t)),
we get the equation a
8i (x(t), u(t))P.(t) and the maximization relation n
pa(t)fa (x(t), u(t)) = m
n
P. (t)fe(x(t), v).
Moreover the maximum is constant and equal to 1. These are the relations of the maximum principle.
Notes and Sources The results about the calculus of variations are standard. See the book by Bolza for his historical interest and the books by Caratheodory [37), GelfandFomin [47] for a modern presentation. The reference [30] makes a concise and
nice introduction of the subject and contains interesting examples and exercises. The book by Bliss [18) is a classic and rather complete reference of the discipline. For the Pontryagin maximum principle standard references are Pontryagin and al. [91], Lee-Markus [77] and Fleming-Rishel [44]. A maximum principle with minimum regularity assumptions was obtained recently by Sussmann [101] but is rather technical; see also Clarke [41]. For existence theorems, see [77) or [38].
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2 Optimal Control for Nonlinear Systems
Exercises 2.1 We consider the following functional in R2: rep
J(x, y) =
f (t, x, y)
1 + ±2(t) + 0(t)dt
Jto
defined on the set of smooth curves whose extremities belong respectively to surfaces parametrized by y = w(t, x), y = ii(t, x).
Use the fundamental formula of the classical calculus of variations to writ necessary optimality conditions (transversality conditions). Give a geometric interpretation.
2.2 We consider the minimization problem in R, with fixed extremities: min fto' L(t, x(t), i (t))dt. Let t'--, x(t) be a piecewise smooth minimizer, with an angular point at c E)to, tl [. Use the fundamental formula to get necessary
optimality conditions at c (Erdmann and Weierstrass conditions (1877)). Present those conditions using Hamiltonian formalism. Compare with the maximum principle.
2.3 Let J(f) = fo (f'2(x) - b f 2(x))dx) where a, b > 0, evaluated on the set of smooth functions f : [0, a) R, such that f (0) = f (a) = 0. Find a relation between a, b for which
1. J(f) > 0 for f 0 0. 2. J(f) > 0. Deduce the following inequality: f07r
f 2(x)dx > f 2.4 Consider the linear system: i(t) = A(t)x(t) + B(t)u(t), x(t) E 1R", u(t) E 1R' and the quadratic cost T
C(u) = tx(T)Qx(T) + f (tS(t)W(t)x(t) +t u(t)Imu(t))dt o
with x(to) fixed and x(T) free. We assume t --, A(t), B(t), W (t), U(t) be continuous matrices and Q, W (t), U(t) being symmetric matrices with Q, W(t) > 0. The control u(.) is taken in L2([to,TJ). We call Riccati equation associated to the LQ- problem the equation
k(t) = -tA(t)K(t) - K(t)A(t) + K(t)B(t)tB(t)K(t) - W(t)
(2.44)
and we denote by K(t, K1, t1) the maximal solution defined on a subinterval J of IR", with K(tl) = K1.
2.5 Dynamic Programming and the Maximum Principle
57
1. Consider the scalar equation: k(t) = k2(t) + 1 and k(O) = 0. Show that the solution is not defined on R. 2. Assume there exists a symmetric matrix K(.) solution of the equation (2.44) on [to, TJ and let x(.) be a curve solution of ±(t) = A(t)x(t) + B(t)u(t). Show the relation T (t Jeo
t BK
r 0
tAK+KA][ux])dt-[
l [uxJlKB K+
T xKxJto
-- 0.
3. Assume that the solution 17 = K(t, Q, to) is defined on the whole interval to < t < T. Show that there exists a control u' (.) minimizing C(u) among
all the solutions satisfying i(t) = Ax(t) + Bu(t), x(to) = xo fixed, x(T) free, given by the feedback
u'(t,x) = - LB(t)17(t)x*(t) and that the minimal cost is
C(u*) = tx'(to)17(to)x'(to). 4. Consider a general Riccati equation
W(t) = A(t) + W(t)B(t) + C(t)i4'(t) + W (t)D(t)W(t)
(2.45)
where W, A, B, C, D- are respectively n x k, n x k, k x k, n x n, k x n matrices
and t i-- A(t), B(t), C(t), D(t) are continuous. Show that (2.44) is a particular case of (2.45) with: k = n, A = -W, B = -A, C = -'A, ,b = BtB. 5. Consider the matrix equation
.k(t) X (t) C(t) A(t) (1'(t)) _- ( -D(t) -B(t)) ( Y(t) )
(2,46)
where X (t) is a n x k matrix and Y(t) an invertible k x k matrix. For to < t < T, set W(t) = X(t)Y-1(t) and show that W(.) is solution of the Riccati equation. 6. Show that the PMP equations associated to the LQ-problem with A, B, W constants are of the form
(i) -H(x)'
H- (BtB A)
Lifting this equation to
CY) =H(Y)
X, Y: n x n matrices
write the corresponding Riccati equation. Compare with equation (2.44). 7. Show that for each t, H(t) satisfies JH +t HJ = 0 where J is the 2n x 2n
matrix:
In
I0 ).
58
2 Optimal Control for Nonlinear Systems
2.5 Consider the following system on Rn: b(t) = X (x(t)) + u(t)Y(x(t)), where X, Y are analytic vector fields.
1. Show that a singular trajectory x(.) corresponding to u(.) satisfies the following equations for some p(.), p(t) E R'\{0}:
= X(x(t)) + u(t)Y(x(t)),
x(t)
(2.47)
At) _ -At)(g x(t) + u(t)
x(t)) (p(t), Y(x(t))) = 0, Vt.
(2.48)
2. Let [Z1, Z21 = s Z2 - s Zi be the Lie bracket for Z,, Z2 : Rri - R" and adZi (Z2) = [Zl, Z2]. Show that if (p(t), ad2Y(X)(x(t))) t then the singular control is given by (p, ad2X (Y)(x))
u (x, p) _ + (p, ad2Y(X)(x)
0 for each
(2 . 49)
and the trajectory t ' x(t) is analytic and solution of
i(t) =
aH(x (t) ,p (t))
(2 . 50)
(p(t), Y(x(t))) = (p(t), [X, Y](x(t))) = 0
(2.51)
ap
( x (t) ,p (t))
, P(t)
with H(x, p) = (p, X (x) + uY(x)). 3. Assume that t i- x(t) is a trajectory on [0, T] corresponding to the (analytic) control u(t) = 0. Show that t i- x(t) is singular if and only if there exists p(t) E R"\{0} such that (p(t), adkX (Y)(x(t))) = 0
(2.52)
for each k E N, and each t E [0, T]. Deduce that for 0 < t < T, the image of L- ([0, t]) by Frechet derivative of the end-point mapping evaluated at u =_ 0, x(0) = xo is K(t) = Span{adkX (Y)(x(t)); k E N).
4. Consider now the system +(t) = X(x(t)) + u(t)Y(x(t)) defined on the plane. Show that the singular trajectories are contained in the set S : {x E R2; det(Y(x), [X,Y](x)) = 0}. Assume X and Y linearly independent on
a nonempty open set U C R2. Define the one-form on U. a = aldxl + a2dx2 by the relations (a, x) = 1 and (a, X) = 0, where (a, Z) = a(Z) for a given vector field Z on R2. Show that S n U is the set of points where dot is not a volume form i.e. {x E R2; da(x) = 0}. 2.6 The objective of this exercise is to solve the Dido problem: among all closed curves in the plane whose length is fixed to L find the one enclosing a domain D with maximal area. 1. Show that D is convex and that we may divide D into two domains D+ and D_ with same area and equal length 2 .
2.5 Dynamic Programming and the Maximum Principle
59
2. Deduce that the problem is equivalent to maximize A = J
1 - y'2(s)ds
y(s)
0
where s represents the length of the curve, y(O) = y(Z) = 0 and y(s) > 0 for s E [0, ]. a 3. Apply the Euler-Lagrange equation to prove that the solution is given by the half-circle with radius r = 2 . 4. Consider the following optimal control problem: th(t) = u(t)
W) = v(t) z(t) = 2 WOO) - i(t)y(t)) T
min
J0
22(t) + y2(t)dt.
a) Show that z(T) - z(0) represents the area swept by the curve t '(x(t), y(t)), t E [0, T]. b) Apply the maximum principle and compute the extremals. What is conclusion do you reach? 2.7 [Bernouilli problem] We consider the following problem: determine for an heavy particle the curve of quickest descent in a vertical plane between two
given points. We take the positive y-axis vertically downward and denote by g the gravitational constant, by yo the initial y-coordinate and by vo the initial velocity, which we suppose to be nonzero.
1. Show that the problem can be set as:
t, min
to
x2(t) + y2(t) dt
y(t --yo + k
9
where k = 2g 2g
2. Assume that we restrict the problem to curves given by a graph of the form (x, y(x)). Show that then the problem can be restated as: 1 -TO (x)
min Z'01'
--yo y(x)
+ k
dx.
Write the corresponding Euler-Lagrange equation.
3. Let L :
(y, y)
R be a smooth mapping. Prove that H =
y-L
is constant along the solutions of the Euler-Lagrange equation. LYse this property to deduce the smooth graphs (x, y(x)) solutions of the Bernouilli problem.
60
2 Optimal Control for Nonlinear Systems
4. Let a E R and consider the following minimization problem: min
xl y°`
Jxo
1 + y2(x)dx
among the set of smooth curves y e -+ y(x) with y(xo) = yo, y(xl) = yi. Discuss the solutions according to a.
2.8 [Weierstrass 1879] Let F : (x, y, i, y) -+ R be a smooth mapping and assume that for all k > 0 the following is satisfied: F(x, y, ki, ky) = kF(x, y, i, y).
Consider the minimization problem given by: min ft. t F(x(t), y(t), i(t), y(t))dt.
(2.53)
to
1. Prove the following relations:
F. = ±F±1 + iFyx
FU =iFiy+yFyv iFyz + yFyy = 0
iFxy+yFii,=0 and deduce that for (i, that:
# 0, there exists a smooth mapping F1 such F00 = i2F1
Fyi-y2F1 F:jo _ -iyF1. Compute F1 for F = y\,/±2 + y2. (0, 0). Prove 2. Let T = Fxy - F11i + F1 (1+9 - #) and assume that (i, that the Euler-Lagrange equation is equivalent to the Weierstrass equation: T = 0. 3. Consider the problem: min f F, among the set of piecewise smooth curves with fixed extremities and contained in a domain D with a boundary. Let
(i(t), y(t)) be a curve contained in the boundary and joining 2 to 3 as represented in Fig. 2.5. By taking variations (7, y) with fixed extremities
i = i + , y = y + n, prove that if (a, y) is a minimizer, then
T 0 in the domain and let ,1-, be the curvature
2.5 Dynamic Programming and the Maximum Principle
61
(2.55)
Take P a point in the boundary and let be the curvature of (1, y) at P and let .1 be the curvature at the same point P of the extremal tangent to the boundary. Prove that (2.55) can be written: 1
1
- > r
Assume F =
(2.56)
r
x2 -+0, find a geometric interpretation of the above
condition.
4. Let 023 be a minimizing curve meeting the boundary at the point 2. By taking variations indicated on Fig. 2.5, prove the following necessary condition: E(x2, y2) ±2, /2, x2, y2) = 0
(2.57)
where E is the excess Weierstrass function: E(x, y, x, 1/l, X-, Y-) = F(x, y, x, y) -
(F(x,y,±,) + yF'F (x, y, x, y)) (2.58)
and (d2, y2), (x2, y2) are the respective tangents at point 2 of the arcs 02 and 23.
Assume Fl to be nonzero in the domain. Prove that a minimizer must hit the boundary tangentially. 5. Let 2 = 3 in Fig 2.5 and let 021 be a minimizing curve with a single point 2 in common in the boundary. Obtain the necessar o timality conditions and find a geometric interpretation if F = i2 + y2. 2.9 Consider the following quadratic functional: 6
J(h) = j ( P(x)h 2(x) + Q(x)h2(x))dx
(2.59)
2 Optimal Control for Nonlinear Systems
62
where P(x) > 0 for a < x < b and h(a) = h(b) = 0. We divide the interval [a, b] into (n + 1) equal parts of length dx:
dx = xi+1 - xi = (n + 1)'
2 = 0,1,...,n
(2.60)
and let Pi, Qi, hi be the value of the functions P(x), Q(x), h(x) at the point
x=xi.
1. Show that the functional J can be approximated by a quadratic form R in n variables h1, , hn defined by the symmetric n x n matrix: 0
0
0
0
0
0
0 b2 a3 0 0
0
f a1 bl 0
...
b1 a2 b2
0 0 0 ... bn-2 an-1 bn-1 000 0 bn-1 an where
az
ns
1.
2. If Di are the descending principal minors, prove the following recurrence relation:
Di=a1Di_1_b?_1Di,
i=3,...,n
(2.61)
and set be convention Do = 1, D_1 = 0. Deduce the relation:
Di = (Qidx +
Pi-1 + Pi
dx
)D,-1
_ P? Di-2, i = 1, ... , n. (1x)2
(2.62)
3. Making the following change of variables: __
D`
P1...Pi Zi+11 i = 1,
,n
(AX)i+1
D-1=Zo=0, prove for i = 1,
, n the following relation:
1 (PZi+1-Zi
QiZi - dx
AX
`
-Pi-1Zi+1-A
Ax
= 0.
(2.63)
Passing formaly to the limit Ax - 0, obtain the following differential equation:
-
(PZ') + QZ = 0.
Compare with the Jacobi equation.
(2.64)
2.5 Dynamic Programming and the Maximum Principle
63
4. assume R > 0 and use the Sylvester criterion to prove that Z; > 0 for i = 1, , n + 1 and deduce formally that the solution of (2.64) with initial condition Z(O) = Za = 0, Z'(0) = limax.o Z1-Z0 = 1 does not vanishe for a < x < b.
3 Geometric Optimal Control
Geometric optimal control is the computation of the optimal control using geometric methods. It is mainly based on the geometric analysis of the equar tions of the maximum principle. The objective of this chapter is to give an introduction to this analysis. We use three articles for this presentation. One is the article by Ekeland [42], which is historically the first one using singularity theory to locally classify the extremals under generic assumptions. The second one is from Kupka [72] and is a basic article to deal with the time optimal control problem for single-input affine systems. Also [711 gives a model for the Fuller phenomenon. Finally, we present the results from Sussmann [102] for planar systems. It is an introduction to finite closed loop parametrization to compute the synthesis for time minimum problem of affine control systems.
First of all we give an introduction to symplectic geometry which plays an important role in geometric optimal control.
3.1 Introduction to Symplectic Geometry In the sequel, all objects are assumed to be COO.
Definition 23. Let V be a It-linear space of dimension 2n. This space is said to be symplectic if there exists an application w : V x V -> R which is bilinear, skew-symmetric and nondegenerate, that is if w(x, y) = 0 for all x E V then y = 0. Let W be a linear subspace of V. We denote by W- the set
Wl = {x E V; w(x, y) = 0 dy E W}.
The space W is isotropic if W C W -L. An isotropic space is said to be Lagrangian if dimW =dim a . Let (Vi, wi ), (V2, w2) be two symplectic lin-
ear spaces. A linear mapping f
:
V1 - V2 is symplectic if wl (x, y) _
W2 (f (x), f (y)) for each x, y E V1.
Proposition 14. Let (V, w) be a linear symplectic space. Then there exists a basis fell , en, fi, ',fn} called canonical defined by w(ei, ej) = w(fi, fj) = 0 for/1 for( 1 < i,, j < n and w(ei, fJ) = 8ij (Kronecker symbol). If
J is the matrix
U
)
where I is the identity matrix of order n, then
66
3 Geometric Optimal Control
we can write w(x, y) _ (Jx, y) where (,) is the scalar product (in the basis (e,, f;)). In the canonical basis, the set of all linear symplectic transformations is represented as the symplectic group defined by Sp(n, R) = IS E
GL(2n, R); tSJS = J}. Definition 24. Let M be a C°°-manifold with dimension 2n. A symplectic structure on M is defined by a 2-form w such that dw = 0 and w is regular, that is Vx E M, wz is nondegenerate.
Proposition 15. For any C°°-manifold M with dimension n, the cotangent bundle T*M admits a canonical symplectic structure defined by w = da where
a is the Liouville form. If x = (x1, .. , x") is a coordinate system on M and (x, p) with (pi, , pn) the associated coordinates on T'M, the Liouville form is written locally as a = p;dx' and w = doe = Enj=1 dp; A dx'. 1
Proposition 16 (Darboux). Let (M, w) be a symplectic manifold. Then given any point of M, there exists a local system of coordinates called Darboux-coordinates, (xl, , xn, p1i , pn) such that w is given locally by dpi A dx'. (Hence the symplectic geometry is a geometry with no local F n1 invariant)
Proof (sketch of the proof). A standard proof uses a recurrence argument on the dimension of the space, but we use here the homotopy method which
is a basic technique to compute normal forms in geometric problems. Since the result is local we work in a neighborhood of U C R2n of the origin. We consider the 2-form wo = dp A dq and we want to construct a germ of diffeomorphism cp such that yo*wo = w1, where w1 = w (notation). We proceed as follows:
- Step 1. Using a linear change of coordinates, we may assume w(0) _ (dp n dq)I0.
- Step 2. Let us consider the following deformation: wt = (1 - t)wo + tw, where t E [0,1]. The idea of the proof is to construct a family of diffeomorphisms Pt, 0 < t < 1 such that Vt * wt = wo. This family is obtained by integrating a time depending vector field: n
x(t, x) _
+ E x;(t, x) 8xi i=1
Since 'Pt * wt = wo for all t, we have 0
=
d wt-(Wt * Wt)-
- Step S. To solve the previous equation, we use the Cartan-Baker-CampbellHausdorff formula: Lx 12 = d(ix 12) + ixdfl
3.1 Introduction to Symplectic Geometry
67
exp tx.11-tt where 0 is a smooth form, X a vector field, Lx fl = (Lie derivative) and ix is the interior product (see Definition 25). Hence
we get dwt
0 = Sot * (dt + ixdwt + d(ixwt)) and since wt is closed, we deduce that: dwt = 0. Therefore, we must solve the equation: d(ixwt) att.
Hence, we must find a 1-form at = ixwt such that dat = - dt = WO - w1. Since d(wl - wo) = 0 we can apply the Poincare lemma i.e., there exists a one form ,Q = at such that dQ = wo - wl. To conclude the proof, one needs to compute the vector field X such that at = ixwt. This equation is locally solvable since wt is of full rank at 0: wtlo = (1 - t)wolo + twilo = wolo
Definition 25. Let (M, w) be a symplectic manifold and let X be a vector field on M. We note ixw the interior product defined by ixw(Y) = w(X, Y) -, for any vector field Y on M. Let H : M -+ R. The vector field denoted by H and defined by ig(w) = -dH is the Hamiltonian vector field associated to H. If (x, p) is a Darboux-coordinates system, then the Hamiltonian vector field is expressed in these coordinates by:
H=
"8H8 -aHa i=1
api ax=
axi
a i
Definition 26. Let F, G : M - R be two mappings. We denote by {F, G} H the Poisson-bracket of F,G defined by IF, G} = dF(G).
Proposition 17 (Properties of the Poisson-bracket). 1. The mapping (F, G) " IF, G} is bilinear and skew-symmetric. 2. The Leibnitz identity holds: {FG, H} = G{F, G} + F{G, H} S. In a Darbouz-coordinates system, we have
IF, G}
n 8G OF =
api axi
8G 8F axi api
4. If the Lie bracket is defined by [F, G] =G o F - F o G, then its relation with the Poisson bracket is given by: [F, G] ={F,G}. 5. The Jacobi identity is satisfied: {{F, G}, H} + {{G, H}, F} + {{H, F}, G} = 0
68
3 Geometric Optimal Control
Definition 27. Let H be a Hamiltonian vector field on (M, w) and F : M R. We say that F is a first integral for H if F is constant along any trajectory of H, that is dF(H) _ IF, H} = 0. Definition 28. Let (x, p) be a Darboux-coordinates system and H : M - R. The coordinate x1 is said to be cyclic if = 0. Hence F : (x, p) H p1 is a first integral. Definition 29. Let M be a n-dimensional manifold and let (x, p) be Darbouxcoordinates on T *M. For any vector field X on M we can define a Hamiltonian vector field Hx onT'M by H(x, p) = (p, X (X)); Hx is called the Hamiltonian lift and Hx= X 8 - t ox p 8 . Each difeomorphism V on M can be lifted into a symplectic diffeomorphism co on T'M defined in a local system of coordinates as follows. I f x = co(y), then cp: (y, q) --, (x, p) _ (cp(y), t !q) .
Theorem 11 (Noether). Let (x, p) be Darboux-coordinates on T*M, X a vector field on M and Hx its Hamiltonian lift. We assume Hx to be a complete vector field and we denote by Wt the associated one parameter group.
Let F : T*M - R and let us assume that F o cot = F for all t E R. Then Hx is a first integral for F.
Definition 30. Let M be a manifold of dimension 2n + 1 and let w be a 2 -form on M. Then for all x E M, wz is bilinear, skew-symmetric and its rank is 0, such that: 1. z is contained in .(l = {z; (p, [Y, [X,Y]J(x)) 0 01; 2. x is contained in the set
12' = {x; X (x) and Y(x) are linearly independent}.
3.5 Geometric Classification of Extremals near E;
79
Then t ,-> x(t) is a C°°-curve, not reduced to a single point and moreover the mapping t h-' x(t) is an immersion.
Proof. According to Proposition 21, t '-- z(t) = (x(t),p(t)) is a COO-curve and the control t N u(t) is COO. Since i(t) = X(x(t)) + u(t)Y(x(t)), if X, Y are linearly independent then t 1--+ x(t) cannot be reduced to a point and this mapping is of rank 1.
3.5 Geometric Classification of Extremals near Ei In this section we describe the geometry of regular extremals obtained by Kupka [71].
Definition 37. Let (z, u), with z = (x, p) be an extremal defined on [0,T]. A time s E [0, T] is called a switching time if s belongs to the closure
of the set of t E [0, T] where z(.) is not C'. The set z(s), z(.) being any extremal is called the switching set. Observe that this set is a subset
of E, = {(x, p); (p,Y(x)) = 01. Let z = (x, p) be any C°°-solution of i(t) = W (x(t), p(t), u(t)), p(t) = - aH (x(t), p(t), u(t)) defined on [0,T], H(x, p, u) = (p, X (x) + uY(x)), corresponding to a C°°-control u(.). The switching function fi is the mapping t - (p(t), Y(x(t))) evaluated along z(.).
If u = +1 (reap. -1), we set z = z+ and tA = P+ (resp. z = z- and 5 = 0-). Lemma 13. The mapping t - 0(t) is COO and its first two derivatives are given by:
fi(t) = (p(t), [Y, X] (x(t)))
(3.10)
gj(t) = (p(t),[[Y,X],X](x(t)))+u(t)(p(t),[[Y,X],Y](x(t)))
(3.11)
3.5.1 Normal Switching Points Let zo = (xo,po) E El and let us assume that Y(xo) # 0 and zo E E1\E2i where E2 = {(x, p); (p,Y(x)) = (p, [Y, X](x)) = 0}. The point zo is then called normal. The behavior of regular extremals near zo is given by one of the two situations of Fig. 3.6 where E+ (reap. E-)= {(x, p); (p,Y(x)) > 01 (resp. < 0). F1som relations (3.10),(3.11) we have the following lemma.
Lemma 14. Let to be the switching time given by z+(to) = z-(to) = zo. Then, the following equalities hold:
0+ (to) =
(to) = (po,Y(xo)).
Moreover, let z = (x, p) be an extremal passing through zo. We have:
1. if (po, [Y,X](xo)) < 0, then x = -y+7-; 2. if (po, [Y,X](xo)) > 0, then x = 7-7+
(3.12)
3 Geometric Optimal Control
80
Fig. 3.6.
where 'y+ (resp. y_) is an arc solution of the system and corresponding to u = +1 (resp. u = -1) and -y+-t- represents the trajectory corresponding to the concatenation of u = +1 with u = -1, i.e. an arc ry+ is followed by an
arc -f-
3.5.2 The Fold Case Let zo = (xo,po) E E2. We assume that Y(xo) 96 0 and we suppose that E2 is a smooth surface of codimension 2. If we denote H+ (x, p) = (p, X (x) + Y(x)) and H_ (x, p) = (p, X (x) - Y(x)), then El is locally a C°°-manifold of codimension one and is defined by: {z; H+ (x, p) - H_ (x, p) = 0}. Moreover E2 : {z; H+ (x, p) - H_ (x, p) = {H+, H_ } (x, p) = 0}. According to (3.10) -.
-4
and (3.11), at zo E E2 both vector fields H+ and H- are tangent to E1 at zo.
If we set
At = (po, [[Y,X],Xj(xo) ± (po, [[Y,X],Y](xo))
(3.13)
then if both A* 0 0, the contact of the trajectories of H+ and H- with El is of order 2. Such a point is called a fold. According to [73] we have the three distinc cases:
a) A+A_ > 0: parabolic case b) A+ > 0 and A_ < 0: hyperbolic case c) A+ > 0 and A_ < 0: elliptic case The respective behaviors of regular extremals are represented on Fig. 3.7 and we have the following. In the parabolic there exists a neighborhood V
of zo such that each regular extremal in V has at most two switchings and in the hyperbolic case at most one switching. In the elliptic case the situation is more complex: there exists a neighborhood V of zo such that each
regular extremal in V has a finite number of switchings, although there
3.5 Geometric Classification of Extremals near E,
81
exists no uniform bound for the number of switching for all the extremals.
(a)
(b)
(C)
Fig. 3.7.
Another important geometric remark is the following. The surface E2 is the set containing all the singular extremals. In the hyperbolic and the elliptic
case, then (po, [[Y,X],Y](xo)) 0 0 and through zo there passes locally an unique singular control satisfying I u 1< 1 and hence is admissible for the bound I u I< 1. Moreover according to Fig. 3.7, in case (b) we can connect at zo a regular extremal with a singular arc to form an extremal but not in case (c). In the parabolic situation, if zo E fl there passes through zo a singular extremal, but the singular control does not satisfied the bound I u 1:5 1 and hence the singular arc is not an admissible one. Hence from our analysis we deduce the classification of all extremals
near a fold point in fl. Proposition 23. Let zo be a fold point in fl. Then there exists a neighborhood V of zo such that: 1. In the hyperbolic case, each extremal trajectory has at most two switchings
and is of the form ryfry,yf (by convention each arc of the sequence can be empty).
2. In the parabolic case, each arc is bang-bang with at most two switchings and has one of the two form: ry+7-7+ or 8. In the elliptic case, each arc is bang-bang but with no uniform bound on the number of switchings.
82
3 Geometric Optimal Control
3.6 Comparaison Between the Calculus of Variations and the Time Minimum Problem for Affine Systems. The Role of Singular Extremals It is striking to notice the similarity between the classification of extremals in the calculus of variations and the one of regular extremals for the time minimum problems for affine systems. Ideas and techniques involved for the proofs are even the same, but the structure of optimal trajectories is not the same. In the calculus of variations we are dealing with non convex problem and we must convexify the problem to get optimal trajectories. Convexifying the problem means exactly to add admissible optimal trajectories in the set El. For the time minimum problem for affine systems, the problem is already convex and if we convexify the problem with only regular extremals where u E { -1, +1 }, we get the original problem with J u J.-5 1. Here the role of
the extremals in E1 for the convexifled problem is precisely played by the singular trajectories.
3.7 The Fuller Phenomenon In the elliptic case, the main problem when analysing the extremals is to prove that each extremal defined on [0, T] has a finite number of switchings. One of the main contribution of geometric optimal control was to prove that it is not a generic situation.
Definition 38. An extremal (z, u) defined on [0, T] is called a Falter extremal if the switching times form a sequence: 0 < t1 < t2 < . . . < T such
that t - T when n - +oo and if there exists k > 1 with the property: tn+1 - tn - k^ as n - +00. 3.7.1 Fuller Example In 1963 was published an article [45] given an example in which optimal control problem has an optimal trajectory which is a Fuller extremal. This example is the following:
x(t) = y(t), y(t) = u(t), min u(.)
r
I U(t) JS 1
+00
x2(t)dt.
This problem is a LQ-problem where u(.) is not penalized in the cost. Such a LQ-problem is called a cheap LQ-control problem. The system has no singular trajectory and the Hamiltonian is given by:
H(x, p, u) = -x2 + Ply + P2u,
P = (Pi, P2)
3.7 The Fuller Phenomenon
83
Since there is no boundary condition at t = +oo, it can be shown, see (77], that the adjoint vector must satisfy the transversality conditions
pi(+oo) = P2(+00) = 0. Introducing the augmented system
i(t) = y(t), y(t) = u(t), z(t) = x2(t) the adjoint system is then
Pi(t) = 2x(t),
P2(t) = -pi(t)
and an extremal control is defined by
u(t) = sign p2(t). It turns out that the optimal synthesis is characterized by a switching locus S in the (x, y) space given by the equation
x+hyIyI=0 where h 0, 4446, and each optimal non trivial solution exhibits a Fuller phenomenon as represented on Fig. 3.8. Here k = 11-24 > 1, where is the
positive root of x4 + i2
- is =
0. Such optimal trajectories provide Fuller
Y
Fig. 3.8. extremals for the time minimum problem where the system is the augmented system. Hence we have Fuller extremals for systems in R3. This example is not
stable, for instance observe that the switching locus is a curve symmetric with respect to 0. Hence the following theorem is important. Theorem 12. If the dimension of the state space is large enough there exists a stable model (X, Y) exhibiting Fuller extremals.
84
3 Geometric Optimal Control
The (complicated) proof of this result can be found in [71] and semi-algebraic
conditions F are given for describing the model. They involve the Poisson brackets of H± = X ± Y at zo up to order 5 and all the Poisson brackets up to order 4 must be 0. The Fuller example satisfies those conditions at the point x0 = (0, 0, 0) and po = (0, 0, -1) (see the transversality conditions). In the Fuller example, there exist optimal trajectories x"(t),0 < t < T with an infinite number of switchings. Such phenomenon is not admissible to construct optimal synthesis using the tools of the analytic geometry like sub-analytic sets. This makes the problem very difficult. Nevertheless, the Fuller phenomenon is not a stable phenomenon for systems in dimension 2 or 3. This leads to the research program initialized by Sussmann and completed by Schattler to describe the stucture of optimal synthesis for systems in small dimension. We shall here present some results initially obtained by Sussmann, see [102] for all the details.
3.8 Geometry of the Time Optimal Control in the Plane 3.8.1 Preliminaries If Z is a C°°-vector field we denote by {exp tZ} the local one parameter group of Z, i.e. (exptZ)(zo) = z(t,zo) is the unique solution to the Cauchy problem z(t) = Z(z(t)), z(0) = zo. Hence we consider a COO-single affine control system in the plane: z(t) = X(z(t)) + u(t)Y(z(t)), I u(t) I< 1 with z = (x, y). We take zo E R2 and the program is to investigate the local structure of the time optimal control law in a small neighborhood U of zo. Using Filippov existence theorem such an optimal control exists if U is taken small enough. Indeed, the problem is convex and moreover since the controls are uniformly bounded, there exists T > 0 such that every solution z(t, xo) is defined on [0, TJ and stays in a compact set K. Hence using a cut-off we can restrict the system to K and apply Theorem 10. According to the maximum principle an optimal trajectory is solution of the equations:
z(t) = M (z(t), p(t), u(t)), p(t)
aH(z(t), p(t), u(t))
(3.14)
H(z(t), p(t), u(t)) = max(p(t), X (z(t)) + vY(z(t)))
(3.15)
X(z(t)) + vY(z(t))) > 0
(3.16)
w1 0. Prove that each trajectory that lies in the boundary of the small time accessibility set from xo is bang-bang with at most two switchings.
3.7 Consider the following system in R4:
zo(t) = 1
i1(t) = u(t) ±2(t) = x1(t)
x3(t) = 2xi(t) with [ u(t) 1< 1 and x(0) = 0. 1. Prove that all Lie brackets with length > 4 are 0. 2. Prove that if u(.) is a control defined on [0, T] and x(.) is the corresponding trajectory, then the accessibility set at time 1 can be obtained from
the accessibility set at time T by letting: u(t) = u(+), &i(t) = Txi(
fori=1,2,3.
3. Compute the accessibility set at time 1.
)
4 Singular Trajectories and Feedback Classification
The objective of this chapter is to enlight the relationship between feedback classification of systems and classification of singular trajectories. This relation is somehow obvious. Indeed a singular trajectory corresponds to a
singularity of the end-point mapping and this singular point is clearly feedback invariant. It follows that the classification of singular trajectories parametrized by the maximum principle is related to the feedback classification. It is important to remark that the classification of singular trajectories is easiest because we consider only change of coordinates on the state and we are faced with a well studied problem of classifying differential equations up to a set of diffeomorphisms. Less trivial is the assertion that for generic
control systems it will allow to compute a complete set of invariants, that is both classification problems are equivalent. Hence we can use singular extremals to compute feedback invariants. Conversely the feedback group is a basic tool to investigate the properties of singular extremals, like time optimality as it will be shown later. Hence both problems are deeply interconnected and have to be simultaneously studied when dealing with the control of nonlinear systems. The analysis presented in this book is mainly based on the article [20]
which deals with affine control systems and is completed by results from [58] on general systems. Both problems are not independent and the rela-
tion is given by the Goh transformation: the system i(t) = f (x(t), u(t)) can be seen as a very specific affine system in the extended space a(t) = f (x(t), u(t)), ii(t) = v(t). Hence the classification of general control systems is a subcase of the classification of affine systems i(t) = Fo(z(t)) + ,=I v;(t)F;(z(t)), where the distribution Span{Fi(z), tive and with constant rank. Definition 40. Consider control systems of the form
a(t) = f (x(t), u(t)),
,
F,,,(z)} is involu-
x(t) E R", u(t) E IIt""
(4.1)
where f is C°° or C' (real analytic) and we denote by S = {f (x, u)} the set of such systems. Two systems f (x, u) and f'(y, v) are said to be feedback equivalent if there exists a C°° or CW -diffeomorphism of the form q : (x, u) -' (y, v), y = cp(x),v = z,1i(x, u) which transforms f into f i.e. d.iof (x, u) = f'(y, v)
98
4 Singular 'trajectories and Feedback Classification
and f is called the image of f by W. We use the notation: f = O * f. We give a global definition of feedback equivalence, but they are various useful local concepts which have to be used in various situations:
- local feedback equivalence at a point xo E R" of the state space; - local feedback equivalence at a point (xo, uo) E R" x Rm; - local feedback equivalence near a given trajectory: x(t) or (x(t), u(t)) of
the system (4.1).
Also we can restrict the feedback group G = {0;
to a given
subgroup. In particular an important subclass of systems is the class of affine systems. This leads to the two following definitions.
Definition 41. Consider the class of affine control systems
i(t) = Fo(x(t)) + F(x(t))u(t), where F = (Fl,
X(t) E R",u(t) E Rm
(4.2)
F,,,.). We identify such systems to the set A of (m + 1)uplet of vector fields: {Fo, F}. The vector field Fo is called the drift and let D be the distribution defined by D(x) = Span{Fj(x), , Fm(x)}. The system ,
is called regular if for each x E R', rank D(x) = m. For affine systems we restrict the feedback transformation to the subgroup G' = {(cp, a,#) } (the product law is well defined by the action on A) of the form -O(x, u) = (sc(x), O(x, u)),
where O(x, u) = a(x) + p(x)u. We observe the following: if we take (Fo, F) E
A and 0 = (sp, a, p) E G', then the image of (Fo, F) is the affine system (FO', F') given by
i) FO' = cp * (Fo + Fa)
ii) F =cp*Ff3 In particular the second action corresponds to the equivalence of the two dis-
tributions D and D' associated to F and P.
Definition 42. Consider a general control system: x(t) = f (x(t), u(t)), x(t) E R", u(t) E Rm. The control can be viewed as a state variable: y = u and let us introduce a new control v(.) by setting u(t) = v(t). Such a transformation is called a Goh or an I - D transformation and we can canonically associate to the original system the following affine control system: Wi(t) = f (x(t), u(t))
u(t) = v(t) where (x, u) E R"i"m is the state space and v E Rm the control.
(4.3)
4.1 Classification of Affine Systems
99
Such a transformation is not only a trick but has many applications in control engineering. A practical application will be given in Chap 7 where we deal with the study of the time minimal control problem for chemical batch reactors. For the feedback classification problem we have the following.
Proposition 28 (Global statement). Two general systems (4.1) are feedback equivalent if and only if the corresponding affine systems (4.3) are feedback equivalent (up to feedback affine transformation)
Proof. The proof is straightforward and is given in [59]. Hence solving the feedback classification problem for general control sys-
tems is a subcase of solving the feedback classification problem for affine systems. In fact it is much simpler because the distribution associated to (4.3) is the flat distribution and in this subclass we avoid the problem of classifying the distribution. We shall now present the results concerning affine systems.
4.1 Classification of Affine Systems 4.1.1 Computations of Singular Controls In this section, we will consider affine control systems:
i(t) = Fo(x(t)) + F(x(t))u(t),
x(t) E Rn, u(t) E R"'
, F,,, are assumed to , F,,,) and the vector fields FO, F1i be C°'. According to Chap. 2, the singular extremals are the solutions of the
where F = (Fl,
following equations:
x(t) = a WO, p(t), u(t)),
OH
At) OH
WO, p(t), u(t)) = 0
where H(x, p, u) = (p, f (x, u)) and f (x, u) = Fo(x) + Zcn l u;F;(x). The constraintH = 0 has to be satisfied almost everywhere, it is equivalent to (p(t), Fo(x(t))) = 0, Vi = 1,
m
(4.4)
Since (x(.), p(.)) is continuous it is satisfied for all t. The constraint (4.4) define a subset El of the space Rn x R"\{0}. We shall now generalize the algorithm of Sect. 3.4 to compute generic singular trajectories. Differentiating relation (4.4) with respect to t, we get the equation
L(x(t), p(t)) + O(x(t), p(t))u(t) = 0
(4.5)
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4 Singular Trajectories and Feedback Classification
where L(x, p) is them x 1 matrix (Li) with Li = (p, [Fi, Fo] (x)) and O(x, p) is the m x m matrix Oij with Oi, = (p, [Fi, FJ}(x)). Let s be the max(,,,p) rank O(x, p) and observe that O(x, p) is a skewsymmetric matrix. Let Al be the set of affine systems (FO, F) such that: if m is even then s = m and if m is odd then s = m - 1. To compute the control we must distinguish two cases: Case 1: The number of inputs is even. Take (Fo, F) E Al and let u be defined on an open dense subset of 1R" x R's by
u(x,p) = -0-1(x,p)L(x,p)
(4.6)
Now let us choose (Fo, F) such that det 0 is not identically zero on El Define the singular control is defined a.e. on El by u(t) = u(x(t), p(t)).
Case 2: The number of inputs is odd. The computation is similar to the previous one, the only complication being the existence of a kernel for O(x, p).
Take (Fo, F) E A1, then for a.e. z = (x, p) the dimension of kernel O(z) is one. Let zo be such a point. Then from equation (4.5) we have L(zo) = 0 for each u E ker O(zo). Near zo (our computations are local) there exists an orthogonal matrix P(z) such that P-1(z)O(z)P(z) = (01, 0)
where 01 is a skew-symmetric matrix of dimension m - 1. Moreover P(z) can be chosen analytic since the eigendirection corresponding to the zero eigenvalue of O(z) depends analytically on the coefficients of O(z). P-1L can be written in two blocks (L1, L2) where L1 is a (m-1) column vector and L2 is a scalar. Let us write :9 = P-1u as 1i +u2, where ul E and U2 E R. Equation (4.6) is then equivalent to 1Rm-1
11 (Z) + 01(z)ul = 0 L2(z) = 0.
(4.7) (4.8)
Let ul(.) be the map defined a.e. by
ul (z) = -01 1(z)L1(z). To compute the component of u(.) in the kernel of 0 we differentiate (4.8) along a solution. We must observe that L2 (z) is near zo analytic in the coefficients of O(z) and L(z). These coefficients are of the form (p, Z(x)) where Z belongs to the Lie algebra generated by { FO, F1, , Fm }. Therefore by differentiating we get a relation of the form
f(z(t)) + gl(z(t))ul(t) + g2(z(t))u2(t) = 0.
(4.10)
Let (FO, F) be chosen such that 92 is not identically zero and let u2(.) be the map defined a.e. on 1R2n by
4.1 Classification of Affine Systems u2(z) = -(f (z) - 9i(z)ui(z))g2 1(z).
101 (4.11)
A control u(t) = Pi(t) defined by the analytic restriction of ul(z(t)),'2(z(t)) to El fl L2(z) = 0 is a singular control.
Definition 43. Let V be the analytic set defined by: if m is even, V = El and if m is odd V = El n {L2 = 0}. Observe that in each case the number of equations defining V is even. Let A. be the subset of good systems in Al given by:
(i) if m is even, det O is not identically 0 on El; (ii) if m is odd, 0 is a.e. of rank (m - 1) on El and 92 is a.e. invertible on V.
Take (Fo, F) E Ag and let u be the map defined a.e. on 1W1 x R'1 by:
(i) if m is even, u is given by (4.6); (ii) if m is odd, ii = Pu with t- given by (4.9) and (4.10). By construction u is meromorphic and let S be the set of points of V where u is not analytic. From our computation, on V\S the singular extremal is said of minimal order. Let us denote by H the meromorphic function defined a.e. on R2s by (4.12) H(x, p) = (p, Fo(x) + F(x)u(x, p)).
ti
We note H the associated constrained Hamiltonian vector field where a solution is by definition an analytic curve: t i--+ z(t) which stays in El\S and
solution a.e. of the vector field H. Observe that z(t) is contained in V\S and at the regular points of V it is solution of an analytic vector field denoted Hreg.
We have the following proposition.
Proposition 29. Let (Fo, F) be a system of Ag. Then the singular extremals of minimal order are solutions of the constrained meromorphic differential equation :k(t)
=ap8H
(x(t), P(t)),
P(t)
8H
a x (x(t), P(t)),
(x, P) E El.
(4.13)
Conversely through each point zo = (xo, po) of V\S there passes an unique singular extremal of minimal order solution of the above equation.
Remark 5. The previous algorithm to compute singular extremals is valid for generic systems (FO, F) for the Whitney topology. Nevertheless in the nongeneric situation similar computations hold. For instance, if the distribution D(x) = SpanF(x) is involutive, see [20]. In each case we have to
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4 Singular Trajectories and Feedback Classification
solve only linear equations. Even for a system (Fo, F) in A. we have computed only singular extremals of minimal order. See Chap. 8 for a discussion on the existence of higher-order singular extremals in the generic situation.
Also observe that linear controllable systems (A, B) are in the bad set and indeed they have no singular trajectories. In the single-input case: ±(t) = Fo(x(t)) + u(t)F(x(t)), u(t) E R we have the following: V = E2 = {(x, p); (p, F(x)) _ (p, [F, Fo](x)) = 0}
and S is the set of points of V where (p, ad2F(Fo)(x)) = 0. At the regular points of V\S, the symplectic form da = dp A dx is nondegenerate. It follows that singular extremals of minimal order are locally solutions of an Hamiltonian vector field on (V\S,da). This result can be extended to the general case. Observe also that by making the change of parametrization da = dt((p,ad2F(Fo)(x))), we can extend the differential equation to R2n.
4.1.2 Singular Trajectories and Feedback Classification Definition 44. Let E, F be R-linear spaces and let G be a group acting linearly on E and F. A homomorphism X : G -+ IR\{0} is called a character. Let X be a character. A semi-invariant of weight X is a map A : E --+ R such that for all g E G, we have A(gx) = X(g)X(x) for every x E E. If X = 1 it is called an invariant. A map A : E F is a semi-covariant (resp. covariant) of weight X if for all g E G, we have A(gx) = X(g)gA(x) dx E E (resp. X = 1). Given a constrained meromorphic differential equation, we must give a precise definition of equivalence up to diffeomorphisms.
Definition 45. Let (Zi, ci, Si) and (Z2, C2, S2) be two constrained meromorphic differential equations on IIY9 where ci are the constrained sets. Let G be a subgroup of the group of diffeomorphisms over II89. The two constrained differential equations are called G-equivalent if there exists a diffeomorphism i E G such that
i) maps cl onto C2, S1 onto S2; ii) ip maps trajectories of (Zl, cl, Sl) onto trajectories of (Z2i c2, S2). (By convention a trajectory is an analytic solution of Zi which stays in ci\Si)
Definition 46. Let (Fo, F) be a system in A9 and let (H, El, S) be the associated Hamiltonian constrained differential equation. Let A be the map A : (FO, F) H (H, El, S). If (gyp, a, (3) is an element of the affine feedback
group G', we define the action of (cp, a, (3) on (H, El, S) as follows. We can lift cp into a symplectomorphism W defined by
a -1 x = jo(y),
p = q 8x
4.1 Classification of Affine Systems
(Here p, q are row vectors)
103
y
The action of (cp, a, p) on (H, E, cp) is by definition reduced to the action induced by the change of coordinates defined by gyp. In other words the feedback
acts trivially. The following result is straightforward, for the proof see [20].
Theorem 13. The diagram of Fig. 4.1 is commutative: In other words A is
A Ag
G'
A(Ag)
I
Ag
G'
A
A(Ag)
Fig. 4.1.
a covariant.
Less obvious is the assertion that two systems are feedback equivalent if their singular flow are diffeomorphic. Of course, this result is not true in general, for instance both planar systems x(t) = x2(t) + y2(t), y(t) = u(t)
and i(t) = x2(t) - y2(t), y(t) = u(t) have the same singular trajectories ±(t) = x2(t) but they are not feedback equivalent. Hence we must assume that we restrict our analysis to systems with "enough singular trajectories".
Definition 47. Let D(x) = Span F(x) be a distribution and denote by HD the stabilizer of D i.e. HD
C°'-diffeom.with W *Fj
Two vector fields Fo, Fo are called equivalent modulo D if there exists p E HD such that cp * F0 = Fo (mod D), i.e. (cp * Fo - FF)(x) E D(x), Vx E IR".
The following lemma is obvious.
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4 Singular Trajectories and Feedback Classification
Lemma 15. The following assertions are equivalent: i) (Fo, F) and (Fo, F) are feedback equivalent; ii) a) There exists a C"-difeomorphism tp such that 'p * D = D' b) cp * F0 and Fo are equivalent modulo D' Assumption Let .Ar be the set of affine systems (Fo, F) such that
i) (Fo, F) E A9 ii) ir(V\S) contains a non empty open subset U of R", where 7r is the projection on the state space (x, p) '-+ x. Theorem 14. Let (Fo, F) and (Fo, F) be systems in ,A,.. Then the following assertions are equivalent: i) (Fo, F) and (Fo, F') are feedback equivalent; ii) The constrained Hamiltonian vector fields A(Fo, F) and A(FQ, F') are G' equivalent.
Proof. Let q = maxx rank D(x) and q' = maxz rank D'(x). Hence q, q':5 m where m is the dimension of the control space and we may assume q = q' = m.
Assume A(Fo, F) = (H, El, S) and A(FF, F') = (H', Ei, S') be equivalent. Then there exists a diffeomorphism cp with symplectic lifting cp which maps
El onto E'. Observe that by definition
E, = {(x,p); (p,Y(x)) = 0,`dY E D}
and El is D. Similarly E; = (D')-L. Take a non empty open set W on which the rank of D is maximal. Since the restriction of 0 to W maps the restriction of El onto the restriction of Ei, then P maps locally D onto D' and then globally by analycity. Since D and D' are diffeomorphic we may assume D = D'. Since A(Fo, F) and A(FF, F) are equivalent, there exists a diffeomorphism 'P of R" such that 'P preserves D and satisfying V * X =1Y'
on V'\S'. In particular we have on V'\S'
V * Fo+'p*Fiio V= Fo+F'u' where cp E HD. Therefore
cp*Fo=cp*Fo (modD=D') for x E 7r(V'\S'). By assumption, the equation holds a non empty subset of R'. By analycity it is true everywhere. Thus using Lemma 15 we have proved ii) - i). Hence the theorem is proved. Corollary 9. If n is the dimension of the state space and m is the dimension
of the control space, let us assume m < n - 1 if m is even and m < n - 2 if m is odd. Then, for an open dense set of systems (Fo, F) for the Whitney topology, the feedback classification is equivalent to the G'-classification.
4.1 Classification of Affine Systems
105
Proof. The set V almost everywhere foliated by singular extremals is defined by m equations in the even case and (m + 1) equations in the odd case. They
are linear in p. Hence 7r(V\S) contains generically an non empty subset if m < n - 1 in the even case and m < n - 2 in the odd case.
4.1.3 Time Optimality and Feedback Classification The time optimality status of singular trajectories is feedback invariant and can be used to compute feedback invariants. We shall use this remark to
compute feedback invariants for single input systems in 1R2 and 1R3. Both situations are different: for a planar system, there exists no enough singular trajectories to compute all the feedback invariants; in dimension 3 the singular flow is rich enough to generate all the invariants. In both cases it is important to relate feedback invariants to the geometric classification of optimal control problems.
The planar case. We consider a system in 1R2: 1(t) = X(z(t))+u(t)Y(z(t)), u(t) E R, z = (x, y). Along a singular extremal, the following constraints are satisfied: (p,Y(z)) = (p, [X,Y](z)) = 0. Hence the singular extremals are contained in the set Sl defined by {z; det(Y(z), [X,YJ(z)) = 0}. The singular control of minimal order is given by u(z) = - p.nd2JC Y z and is defined outside the set S2 : det(Y, ad2Y(X )) = 0. Another feedback invariant is the set of points C such that det(X, Y) = 0 where X, Y are collinear. The geometry of the time optimal control in the plane was investigated in Sect.
3.8. Outside the set C it can be investigated using the clock one form. We have two situations: - hyperbolic case: a singular extremal of order 2 is time minimizing. - elliptic case: a singular extremal of order 2 is time maximizing. Example 3. Consider both systems
x(t) = x2(t) + y2(t) y(t) = u(t)
i(t) = x2(t) - y2(t) y(t) = u(t)
The singular trajectories are contained in Sl : y = 0 and are solutions of the one dimensional differential equation: i(t) = x2(t). To distinguish between
the two cases we use the optimality status of the singualr arc in y = 0, x > 0. In the first case it is time maximizing and in the second case it is time minimizing.
The local feedback classification when Y is non zero is analyzed in details in [92] in the generic case. We work around 0 and Y can be taken as . It is based on the classification of the clock form. A basic invariant is the first integer k such that adkXY is not collinear to Y at 0. We state the theorem.
106
4 Singular Trajectories and Feedback Classification
Theorem 15. Any generic C°° pair (X, Y) such that Y does not vanish is locally feedback equivalent at 0 E 1R2 to one of the following canonical forms:
Y = y and for X : (M2) yex,
(M1) (y + War; (M3) (y2 + 1)
(1V14) (y2 - 1) 0
,
(NF5) (y2 + Ax) (NF6) (y3 + xy + a(x)) go; where A E R is an invariant and a(x) is a C°°-function such that a(0) # 0 with the following invariance property: two systems (NF6) with a(x) and a(x) respectively are equivalent if and only if a(x) = a(x) for x < 0. The proof is left in exercise to the reader or see [92].
a
Systems in 1183. Consider the following system
i,(t) = X (v(t)) + u(t)Y(v(t)), where v = (x, y, z) E R3 and X, Y are C°°-vector fields. Introduce D1 = det(Y, [Y, X), [f Y, X], Y]), D2 = det(Y, [Y, X], [[Y, X], X]), D3 = det(Y, [Y, X], X).
(The Des are depending on X, Y and we use the notation DX'Y when necessary)
Convention. In the sequel of this section we consider only systems (X, Y) such that D1 is not identically zero. By singular trajectories we mean a singular trajectory contained in 1R3\{D1 = 01. Such a trajectory is of minimal order.
Proposition 30. The singular trajectories are the solutions of
i(t) = X(v(t)) - D2(v(t))Y(u(t)) D1(v(t))
(4.14)
in 1183\{D1 = 0}.
Proof. Computing the singular control we get (p, Y(v)) = (p, [Y, X1(v)) _ (p, [[Y, X1, X](v) + u[[Y, X],Y](v)) = 0
which implies D2 (v) + uD1(v) = 0. Hence the singular control is the feedback u(v) Di °
Definition 48. Let Z be the vector field X - aY on II83\{D1 = 0}. We observe that Z is the restriction of ir* H to R3\{D1 = 0} where zr is the canonical projection (v, p) ,- v. We denote by A, the mapping (X, Y) +-o Z. The feedback group G' = {(cp, 8)} induces the following action on the set of such vector fields in the state space: (,p, a. O)Z = cp * Z.
4.1 Classification of Affine Systems
107
Clearly we have the following proposition.
Proposition 31. The map A,r is a covariant. Moreover in the analytic case the feedback classification of systems (X, Y) with DI non identically zero is equivalent to the G' classification of pairs (Z, RY). In particular the time optimality status of singular trajectories is encoded in (Z, RY). We shall now describe how the small time optimality can be determined. In Chap. 6 we shall give a definition of curvature associated to the concept of conjugate points to deal with optimality on fixed intervals.
Lemma 16. Let (cp, a,13) E G', then: i) ii)
(v) = det (?X) (cp-1(v)) for i = 1, 2,3 i 8V DX'Y Dx+Y«,Y" =13 Dx'Y
iii) D3 +Y°'YQ iv) D3
= f33 (D2 'Y + aDX'Y)
+Ya'Y(3 = 132D3 'Y
Proof. Computations.
Corollary 10. Let f : R3 - 118 be a C°°-mapping. Define the action of ( 0. Take any M > 0 such that the singular control satisfies I u(t) I < M on [0, T]. Then observe that
each point m(t) = (ry(t), p(t)) is an hyperbolic fold point in the hyperbolic situation and an elliptic fold point in the elliptic case, where fold points are defined in Chap. 3. It can be shown that if T is small enough, as for planar systems, the singular arc is time minimizing in the hyperbolic case and time maximizing in the elliptic case. This result is true in a small C°-neighborhood of -y(.) even if u(t) E R. This can be easily seen using a normal form where y(.),Y and [X,Y] form a frame.
Singular trajectories are characteristics. On R3\{D1 = 0}, the vector fields Y, [X, Y] are independent and we can define the one form w = pdv using
the relation (p, X (x)) = 1 and (p, Y(x)) = (p, (X, Y](x)) = 0. Using Cartan's formula it can be checked the following.
Proposition 33. The vector field Z defined by (4.14) is the characteristic field associated to the one form w.
4.2 Singular Trajectories and the Problem of Classification of Distributions 4.2.1 Preliminaries Consider an affine driftless system (F0 = 0): m
±(t) = E ui(t)Fi(x(t))
(4.15)
i=1
where the vector fields Fi are smooth and defined on an open subset U of itt". In this case the feedback classification is equivalent to the classification of
the distribution: D(x) = Span{Fl(x), , Fm (x)). We shall assume that D is of constant rank m on U. One of the main geometric property of a driftless system (4.15) is that any smooth immersion t - x(t) solution of (4.15) can be reparametrized in order to satisfy the constraints ( u(t) 1= 1 where u = (ul, , u,,,) and I . is any norm on the control domain u E R"'. In particular we can locally use the charts of the projective space P(R'") obtained by fixing the parametrization with ui = 1 for some i between 1 and m. Hence to the smooth driftless affine system (4.15) we can associate m I
affine control systems defined by
i(t) = F,(x(t)) + E ui(t)Fi(x(t)), j = 1, ... m. i#i
(4.16)
4.2 Singular Trajectories and the Problem of Classification of Distributions
109
To compute feedback invariants for the distribution D we can use the singular trajectories of (4.15). They are related in an obvious way to the singular trajectories of systems (4.16). Let (x, p, u) be a singular extremal of (4.15) defined on [0, T], then the following relation holds:
(p(t), Fi(x(t))) = 0 f o r i = 1,
(4.17)
, m and observe that the Hamiltonian M
H(x,p,u) = (p,uiFF(x)) i=1
is zero along a singular extremal. We call such singular extremals exceptional. To compute the singular extremals, we differentiate (4.17) and we obtain for i = 1, , m the constraint m
(p, E u, [Fi, Fj] (x)) = 0.
(4.18)
j=1
Hence to compute the singular extremals of minimal order we can proceed like for affine systems. We distinguish between two cases: - if the number of inputs is odd the singular control is computed by solving the linear equation (4.7); - if the number of inputs is even, we must differentiate the additional relation (4.9) with u E ker O(x, p) where O(x, p) = (p, [Fi, Ft] (x)) to compute the singular control. In any case, we observe that since the Hamiltonian is constant along a singular
arc, then any piece of singular extremals such that the control u(.) belongs to the chart uj = 1 for j = 1, , m can be computed using the algorithm for the affine systems. Hence we have the following proposition.
Proposition 34. The singular extremal arcs of minimal order of driftless affine systems (4.15) are singular extremals of one of the affine system (4.16).
They are exceptional, that is H = 0 and they form a subset of codimension one in the set of all extremals of minimal order of (4.16). Having computed such trajectories we shall use the singular trajectories to compute feedback invariants. Of course we have clearly the following.
Proposition 35. The singular extremals of minimal order are feedback invariants.
4.2.2 Local Classification in Dimension 3, with rank D = 2 See [110] for details. Let us consider a driftless affine system:
110
4 Singular Trajectories and Feedback Classification
NO = u1(t)Fi (v(t)) + u2(t)F2(v(t)),
v = (x, y, z) E R3
and with rank D = Span{FI, F2} = 2. Our classification is localized near vo and we consider only the generic cases, which are all the cases with codimension_< 3. We have three situations which are classified according the nature of the singular flow. The singular extremals must satisfy (p(t), F1(v(t))) = 0,
i = 1,2
(p(t), [F1, F2](v(t))) = 0
and hence they are contained in the set M : {v E R3, det(Fl, F2, [FI, F2J) _ 0). The singular controls of minimal order satisfy (p, u1 [[FI, F2], Fi](x) + u2[[Fi, F2], F2](x)) = 0.
(4.19)
We define the singular set S by S = Si n S2 where Si = {v; det(Fi, F2, [[Fi, F2], F;J) = 0}. We have three generic situations.
Case 1: Take a point vo 0 M, such a point is called a contact point and through xo there passes no singular are. In this case, D is C°° (or C") isomorphic to ker a where a is the contact normal form: a = ydx + dz. For this normalization da = dyAdx and 8 is the characteristic direction. Case 2: (Codimension one) We take a point vo E M\S. Such a point is called
a Martinet point. Since one of the Si is non zero, we observe that M is near vo a C°° (or C'') surface. This surface is foliated by the singular trajectories. A normal form is given by D = ker a, where a = dz - Zdx and if the vector fields Fi are analytic then the isomorphism is analytic. In this normal form we have the following identification:
-M:{y=0}
- The singular trajectories are the solutions of Z = j restricted to y = 0
Case 3: (Codimension 3) We take a point vo e M n S and we assume that the point vo is a regular point of M. The situation has been analyzed in [111]. We can proceed geometrically as follows. If Z is the vec-
tor field direction defined locally near vo on M\{vo} whose solutions are singular trajectories, it can be smoothly extended at vo and has a singularity at vo. The extension is obtained as follows. If we introduce Di = det(Fi, F2, [[F1, F2], F1]), the singular control is defined by the re-
lation D1u1 + D2u2 = 0, then u2 = -"D a.e. and the vector field Z is given by Z = D2F1 - DIF21,, and is tangent to M. Introduce A the linearization of Z at vo and let al, a2 be the eigenvalues of A. Then we have al + a2 = 0 and we have two generic situations: - hyperbolic case: v1, o2 E 1R\{0} - elliptic case: a1, a2 E ilR\{0}
4.3 Feedback Classification and Analytic Geometry
111
The computations in [1101 show that there exists in the C°°-category an unique feedback invariant b in the two normal forms corresponding to the two cases.
Hyperbolic case: D = ker a, a = dy + (xy + x2z + bx3z2)dz Elliptic case: D = ker a, a = (dy + xy + 3 + xz2 + bx3z2)dz --1, the The invariant b has the following interpretation. Since singularity is resonant and b is an obstruction to the formal linearization.
Of course in the analytic category we have numerous additional invariants.
4.3 Feedback Classification and Analytic Geometry 4.3.1 Preliminaries Although the classification of general control systems ±(t) = f (x(t), u(t)) is equivalent to the classification of the sub-class of affine systems of the form x(t) = f (x(t), u(t)), fl(t) = v(t), we can make a direct research of feedback invariants for nonlinear systems using mainly Weierstrass preparation theorem. This is the point of view adopted by Jakubczyk in [581. In essence the starting point is the same. Introducing the Hamiltonian H(x, p, u) = (p, f (x, u)), the singular extremals are the solutions of H contained in the set TU- = 0 and in general they are several solutions Hi. As shown by the discussion of the article of Ekeland in Chap. 3, the time optimal synthesis is related to a competition between the Hamiltonian vector fields Hi. Hence the idea is to recover all the
feedback invariants from the Hi s. Of course it has an interpretation with the time minimal synthesis for the associated affine systems.
Here we shall present several points of the article [581, restricting our study for the sake of simplicity to single input control systems. Definition 50. We consider a single input control system ±(t) = f (x(t), u(t)) and our study is localized near a point (xo, uo) E X x U. Assume that f is real analytic and extend locally f to a complex analytic mapping using the Taylor expansion of f at (x0, uo). We assume the following regularity assumption: Assumption 1
©
(xo, uo) # 0.
Assumption 2 Let F(x) = f (x, U) and F = U=EXF(x). Then assume F C TX is a regular submanifold of TX. The Hamiltonian associated to the system is H(x, p, u) = (p, f (x, u)) where
p 0 0 and it is defined on T'X x U, and linear with respect to p. Hence
4 Singular Trajectories and Feedback Classification
112
its admits a complex extension on en X W where W is a complex neighborhood of (xo, uo). The constraint OH = 0 defines locally an analytic set E : {(x, p, u); g(x, p, u) = 0) where g(x, p, u) = OH. Let po 76 0 be a real direction such that (zo, uo) E E where za = (xo, po). We assume the following:
Assumption 3 E is irreductible near (zo, uo) and is of order p, 1 < p < +oo with respect to u.
4.3.2 Critical Hamiltonians and Symbols Let us consider a point (zo, uo) of multiplicity p with respect to u(.). Then, from the Weierstrass preparation theorem the equation g(z, u) = 0 has the same roots than a polynomial equation
Let V be the discriminant of the above polynomial and S be the set {z; D(z) = 0). If we take a point zl S the equation g = 0 has p distinct roots near zl, in fact the solutions form a p-sheeted covering above zj, denoted fil (z), , u, (z). We define the critical Hamiltonians by
Hk(z) = H(z,u:(z)) These Hamiltonians are not in general holomorphic because the singularity set S. To define holomorphic functions near the branching point (zo, uo) we use the symmetrization procedure from Galois. We introduce for z 0 S, the following functions:
Sk = FIi +
+ Hp
called symbols by Jakubczyk and they are extended by continuity on S. We have the following proposition, see [58].
Proposition 36. a) The functions Sk, k > 1, are holomorphic near (zo, uo) and are also real analytic; b) for k > p, the function Sk are polynomials of S1i , SP; c) the functions Sk are homogeneous of order k with respect to p.
Now we can use the symbols to analyze the feedback classification problem. Indeed if we take two Hamiltonian lifts H(x, p, u) = (p, f (x, u)) and H(x, p, u) = (p, j (x, u)) such that assumptions in Definition 50 hold at (zo, uo) and (zo, uo) respectively we have the following theorem, see [58].
4.3 Feedback Classification and Analytic Geometry
113
Theorem 16. a) If the systems t(t) = f (x(t), u(t)) and i(t) = f (x(t), u(t)) are locally feedback equivalent near (xo, uo) and (xo, uo) respectively, then the corresponding local symbols are related up to a symplectic lifting
0, r > 1 and a°(.), , ar_2(.) are germs of analytic functions such that a,(0) = 0 for i = 1, , r - 2. 2. If two systems in the above normal form are locally feedback equivalent near 0, then either a = 0 for both systems or a = ± 1 for both of them and then they both have the same numbers p and r.
4.7 let M be a smooth manifold and let D be a distribution of rank k. We set D1 = D, D2 = D + [D, DJ, , D,,,+1 = D. + [D, D + [D, DI, , D(-+1) = D(-) + [D(-), D(-)]. The small growth vector of D at q E M is the sequence {nl, n2i n3, } of dimensions at q of and the the ascending flag of modules of vector fields D1 C D2 C D3 C big growth vector of D at q the sequences {k1, k2, k3, ...} of dimensions at q of the flag D(O) C Dill C D(3) C . 1. Prove that if two distributions are equivalent up to smooth local diffeomorphisms they have the same small/big growth vector. 2. Let D be a 2-dimensional distribution on a (n+2)-dimensional manifold. We say that D is a Goursat form if D has at every point the big growth vector a) Classify Goursat distributions in dimension 3 and 4.
b) Let D be a Goursat distribution in R5. Prove that at every point q, D is locally equivalent to one of the two distributions around 0: 8 8 8 a s G1: Span{ -,+ql-+q3-+q4-} aql aq2 aqs aq4 aqs
a aql
s
a
ft
a
a
aq4
aqs
G2 : Span{ - , - +q, - + q, q3 - + q, q4 - } 49q3
Show that the small growth vector is (2,3,4,5) in the first case and (2,3,4,4,5) in the second case.
4.8 A quadratic control system is a system in R" of the form: 4(t) = Q(q(t)) + u(t)b
(4.23)
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4 Singular Trajectories and Feedback Classification
where Q is a quadratic mapping and b is a constant n x 1 matrix. Two systems (Q, b) and (Q', b) are called isomorphic if there exists a linear diffeomorphism such that:
Q' = P-1Q o P,
b' = P"1b
(4.24)
and they are called feedback equivalent if there exists such a P as well as a feedback u = a(x) + /3u' where a is quadratic and ,a is an invertible scalar such that: Q' = P-1(Q + ab) o P, b' = P-1(fb). (4.25) Let {e1,
,
en} be a Rn-canonical basis and let n
xi(t) _
ajkxi(t)xk(t) i,k=1
be a quadratic differential equation with ajk = ak,.
ajkei. Prove 1. Endow 1R" with the multiplication defined by: ej.ek = 1 that the associated algebra is commutative but nonassociative in general. 2. Let vi, v2 E R', prove: a) [Q,vlj(v2) = 2v1.v2, b) [[Q, v1], v2] = 2v1.v2.
3. Assume n > 3. Prove that for an open dense set of pairs (Q, b) the system is feedback equivalent to an unique system with the following normalization: i) b = en
ii) E lank+lei=2ek iii) ant =0 forp=
k=1, ,n-1 1
iv) when n is odd all = a or when n is even all = z
v) ask=0 for 4.9 Let 4(t) = Q(q(t)) + u(t)b be a quadratic control system in 1R3. Compute all the singular trajectories when {Q, b}AL(q) =1R3, for all q E 1R3.
5 Controllability - Higher Order Maximum Principle - Legendre-Clebsch and Goh Necessary Optimality Conditions
In the time optimal control problem, one of the main question is to determine the optimality status of the singular trajectories. They are two distinct problems: small time optimality and time optimality over a fixed interval. We
shall discuss the first problem in this chapter, the second one is related to the concept of conjugate point along a singular arc and it will be discussed in the Chap. 6. To solve the small time optimal problem we use an extension of the maximum principle. Indeed, in the singular case the first order Pontryagin cone K introduced in the proof of the maximum principle is a linear space and hence is not a good approximation of the accessibility set. Hence the idea is to extend this cone using higher order directions. This idea formalized by Krener [67] and Hermes [54) provides higher order necessary optimality conditions. The most useful are the Legendre-Clebsch condition and the Goh condition. These are conditions obtained using the intrinsic second order derivative of the end-point mapping but here we present these conditions using the Baker-Campbell-Hausdorff formula. Before explaining these conditions we shall present two geometric results about local and global controllability that we shall need later.
5.1 Some Notations and Formulas from Differential Geometry In this chapter we shall denote by M a smooth (C°° or C") manifold of dimension n, connected and second countable. We denote by TM the fiber bundle and by T'M the cotangent bundle. Let V(M) be the set of smooth vector fields on M and Dif f (M) be the set of smooth diffeomorphisms.
Definition 51. Let X E V (M) and let f be a smooth function on M. The Lie derivative is defined as: LX f = df (X). If X, Y E V(M), the Lie bracket is given by
adX(Y)=[X,Y]=LyoLX-LXoLy.
If x = (x',
,
xn) are a local system of coordinates we have: X (X)
n
_
Xi(x) axi
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5 Controllability and Higher Order Maximum Principle
Lx f (x) = axf X (x)
[X, Y] (x) = 8 (x)Y(x) - 8 (x)X (x) The mapping (X, Y) H [X, Y] is IR-linear and skew-symmetric. Moreover, the Jacobi identity holds:
[X, [Y,ZII+[Y,[Z,XII +[Z,[X,Y1] =0. Definition 52. Let X E V(M). We denote by x(t,xo) the maximal solution of the Cauchy problem: x(t) = X (x(t)), x(0) = xo. This solution is defined on a maximal open interval J containing 0. We denote by exp tX the local one parameter group associated to X, that is: exp tX (xo) = x(t, xo). The vector field X is said to be complete if the trajectories can be extended over R. Definition 53. Let X E V (M) and cp E Di f f (M). The image of X by cp is V * X = dcp(X o cp-1).
We recall the following results.
Proposition 37. If X, Y E V (M) and
E Di f f (M), we have:
1. The one parameter local group of Z = p * X is given by: exp tZ = p o exp tX o
(p-1
2. W * [X, Yl = [gyp * X,
O
small enough in the analytic case. 5. The ad-formula is: k tadkX(Y)
exptX*Y= k>o
k!.
where the series is converging for t small enough.
5.1 Some Notations and Formulas from Differential Geometry
119
Definition 54. A polysystem D is a family {V; i E I} of vector fields. We denote by the same letter the associated distribution, that is the mapping
x '- Span{V(x); V E D}. The distribution D is said to be involutive if
[V,V]CD,VV,VJED. Definition 55. Let D be a polysystem. We design by DAL the Lie algebra generated by D. By construction the associated distribution to DAL is involutive. The Lie algebra DAL is constructed recursively as follows:
Dl = Span{D}, D2 = Span{D1 + [DI, D1]}, , Dk = Span{Dk_I + [Di,Dk_I]} and DAL = Uk>1Dk. If x E M, we associate the following sequence of integers: nk(x) = dim Dk(x).
5.1.1 Controllability with Piecewise Constant Controls Definition 56. Consider a smooth system on M, given in local coordinates by
±(t) = f (x(t), u(t)), x(t) E M, U(t) E U C l[2'n
(5.1)
The set of admissible controls u(.) is the setU of piecewise constant mappings.
If x(t, xo, u) is the solution of (5.1) associated to u(.) starting at x(O) = xo, we denote by A(xo, T) the accessibility set xo, u) in time T and A(xo) the accessibility set UT>oA(xo,T). The system is controllable at time T if for each xo we have A(xo, T) = M and is controllable if for each xo we have A(xo) = M.
Definition 57. Consider a control system (5.1) on M. We can associate to this system the polysystem D = { f (., u); u constant, u E U}. We denote by ST(D) the set k
kEN,tj>Oand
t,=T,Vi E D}
and by S(D) the local semi-group: UT>OST(D). We denote by G(D) the local group generated by S(D), that is G(D) = {exp t1 V1
exp tk Vk; k E N, t; E 1R, V E D}.
Properties. 1. The accessibility set from xo in time T is:
A(xo,T) = ST(D)(xo) 2. The accessibility set from xo is the orbit of the local semi-group:
A(xo) = S(D)(xo). Definition 58. We call orbit of xo the set O(xo) = C(D)(xo). The system is said weakly controllable if for every xo E M, O(xo) = M.
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5 Controllability and Higher Order Maximum Principle
5.2 Integrating Distributions 5.2.1 Preliminaries Let D be a polysystem and DAL the Lie algebra generated by D. We consider
the distribution d : x t-, DAL(X). It is an involutive distribution and the problem of integrating d at a point xo is to find a submanifold N, containing
xo such that for each y E N, TyN = A(y). It is a generalization of the Cauchy problem for integrating a single vector field. Here, we are presenting two results:
- if near xo the rank of d is constant, then we have the Frobenius theorem which is a generalization of theorem of linearization of a smooth vector field
X near a regular point; - if the rank is not constant but if the vector fields of the polysystem are real analytic, then the result is still true. It was proved by Nagano-Sussmann. The proofs of both results are radically different.
5.2.2 Frobenius Theorem Theorem 17. Assume that the rank of the distribution d is constant near the point xo: rankd = p. Then there exists a local coordinate system x = 1. In particular near (x' , . , xn) such that d is generated by { ., , xo the integral manifolds are given by x' = constant, i = p + 1, , n. Proof. The proof is standard and is a reccurence on p.
- If p = 1, then locally d = RX where X E V(M), X(xo) 0 0. We use the linearization theorem for ordinary differential equations. - If p > 2, locally d = Span {Y1, , Yp}. We choose a coordinate system y = (y1, . . . , yn) centered at xo such that Y1 = -. Consider the following p vector fields Z1, , Zp of d defined by Z1 =Y1,
Zk=Yk-(LYky1)Y1, 2 2.
2. For k > 2: first of all we observe that axl (LzkxP+r) = Lz (LzkxP+r) Since Lz, xP+r = 0, we can write
axi (LzkxP+r) = L(zk,Zil (XP+r) and we know by construction that
[Zi, Zk] E Span{Zj; j > 2}. Hence we can write
,(LzkxP+r)Aj(Lz,p+r) j=2
where the Aj are scalar. It is a linear differential equation with respect to x1. For x1 = 0, we have Zk = Vk and by construction LzkxP+r = 0. Since the solution of a linear system with values 0 at x1 = 0 is the identically zero solution, we have Lzk Xp+r = 0. The theorem is proved.
0 5.2.3 Nagano-Sussmann Theorem See [105]. When the rank condition is satisfied (rankL =constant) we get from the Robenius theorem a description of all the integral manifolds near xo. If we only need to construct the leaf passing through xo the rank condition is clearly too strong. Indeed, if D = {X } is generated by a single vector field
X, there exists an integral curve through xo if X is locally Lipschitz. For a family of vector fields this result is still true if the vector fields are analytic.
Theorem 18. Let D be a family of analytic vector fields near xo E M and let p be the rank of A : x H DAL(x) at xO. Then through xo there exists locally an integral manifold of dimension p.
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5 Controllability and Higher Order Maximum Principle
Proof. Let p be the rank of A at xo. Then there exists p vector fields of DAL : X1, , XP such that Span{X l (xo), . the mapping
,
XP(xo) } = A(xo). Consider
a : (t1,...,tp) i--+ exptjX1 ...exptpXp(xo)
It is an immersion for (t1, , t p) = (0, , 0). Hence the image denoted by N is locally a submanifold of dimension p. To prove that N is an integral manifold we must check that for each y E N near xo, we have TAN = A(y). It is a direct consequence of the equalities DAL(exptX;(x)) = dexptXi(DAL (x)),i = 1,
,p
and x near x0, t small enough. To show that the previous equalities hold, let V(x) E DAL(x). Then there exists Y E DAL such that V(x) = Y(x). By analycity and the ad-formula for t small enough we have = 1: k (dexptXi)(Y(x)) k>O
Hence for t small enough, we have
(dexptXi)(DAL(x)) C DAL(exptXi(x)) Changing t into -t we show the second inclusion. O
5.2.4 C°°-Counter Example To prove the previous theorem we use the following geometric property. Let
X, Y be two analytic vector fields and assume X(xo) # 0. From the adformula, if all the vector fields adkX(Y), k > 0 are collinear to X at x0, then for t small the vector field Y is tangent to the integral curve exp tX (xo). Hence it is easy to construct a C°°-counter example using fiat CO°-
mappings. Indeed take f : R
R a smooth mapping such that f (x) = 0
for x < 0 and f (x) # 0 for x > 0. Consider the two vectors fields on 1R2: X = a and Y = f(x)s. At 0, DAL is of rank 1. Indeed, we have [X, Y](x) = -f'(x) = 0 at 0 and hence [X, YJ(0) = 0. The same is true for all high order Lie brackets. In this example the rank of DAL is not constant along exp tX (0), indeed for x > 0, the vector field Y is transverse to this vector field.
5.3 Nonlinear Controllability and Chow Theorem Theorem 19 (Chow). Let D be a C°°-polysystem on M. We assume that for each x E M, DAL(x) = TIM. Then we have G(D)(x) = G(DAL(x)) = M,
for each x E M.
5.3 Nonlinear Controllability and Chow Theorem
123
Proof. Since M is connected it is sufficient to prove the result locally. The proof is based on the BCH-formula. We assume M = R3 and D = {X, Y} with rank{X, Y, [X, Y]} = 3 at xo; the generalization is straightforward. Let A be a real number and consider the mapping 'Pa : (t1, t2, t3) r--' exp AX exp t3Y exp -AX exp t2Y exp t1 X (xo).
We prove that for A small be non zero, spa is an immersion. Indeed using BCH formula we have
WA(tl, t2, t3) = exp (t1X + (t2 + WY +
At3
2
[X, Y] + - )(xo),
hence
acl (0, 0, 0) = X (xo).
(0, 0, 0) = Y(xo),
09
8t3 (0) = Y(xo) + 2 [X, Y] (xo) + o(ff).
Since X, Y, [X, Y] are linearly independent at xo, the rank of (pa at 0 is 3 for
,\:A 0 small enough. 0 Definition 59. The polysystem D is called weakly controllable if the orbit O(x) of G(D) is M for every x E M. The polysystem D is called controllable if the orbit A(x) of S(D) is M for every x E M. The polysystem D is said symmetric if for every X E D, we have -X E D.
Example 4. Let D = I -L,, aL } on R2. Hence 0(0) = JR2 and A(O) = {x > 0, y > 0}. In general we have that A(x) is a strict subset of O(x). Corollary 11. Let D be a symmetric polysystem. Assume that rank DAL(x) = n (dim M) for every x. Then D is controllable. In the analytic case this rank condition is also necessary.
Proof. Since D is symmetric, we have: orbit S(D) = orbit G(D). Then we apply the Chow theorem. In the analytic case, we apply the Nagano-Sussmann theorem. 0 Remark 6. The symmetric case is the only case where we can conclude trivially that S(D) = G(D). In general the problem to decide if a semi-group is transitive is difficult. Nevertheless the following weaker result is true, see [106].
Proposition 38. Let D a polysystem. If dim DAL(X) = it (dim M) for every x E M then for each neighborhood V of x there exists a non empty open set U contained in V n A(x).
5 Controllability and Higher Order Maximum Principle
124
Proof. Let x E M. If dim M > 1, then there exists X 1 E D such that Xl (x) # 0, otherwise we would have dim DAL(X) = 0. Consider the integral curve a 1 t i--+ exp tX 1(x). If dim M > 2, then there exists in every neighborhood V of x a point y E M such that y = exp t1 X 1(x) and vector field X2 E D such that X2 and X 1 are not collinear at y, otherwise we would have dim DAL = 1. Consider the mapping a2 : (tl,t2) -. expt2X2expt1X1(x). If dim M _> 3, then there exists in every neighborhood of y a vector field X3 transverse to the image of a2 With this method we construct in every neighborhood V of x a mapping a - exp t 1 X 1(x) such that in a point z = a,,(ti, , t',) of V, a is an immersion. This construction provides a nonempty open set U contained in V fl O(x). :
5.4 Poisson Stability and Controllability Definition 60. Let X be a Coo -vector field on M. The point xo E M is said Poisson stable if for every T > 0 and every neighborhood V of x0 then there exists tl, t2 > T such that exp t1X (xo) and exp -t2X (xo) E V. The vector field X is called Poisson stable if the set of Poisson stable points is dense in M.
Theorem 20 (Poincar6 [49]). If M is a compact manifold with a volume form w, each conservative vector field X is Poisson stable.
Proposition 39. Let D be a polysystem. Assume the following: i) for every x E M, rank DAL (x) = n (dim M); ii) every vector field X E D is Poisson stable. Then the system is controllable.
Proof. Here we outline the proof, see [80] for details. Let x, y E M, we must show that there exists X1i , Xk E D and t1, tk > 0 such that -
y = expt1X1 ...exptkXk(x).
Since D satisfies the rank condition, we can apply Proposition 38 to D and
-D to find the existence of two points x', y' and two open set U and V such that x' E U, y' E V such that x' can be steered using Xi E D, -D to each point of U and each point of V can be steered to y'. To prove the proposition it is sufficient to show that there exists two points x" E U and y" E V such that x" can be steered to y". Since the polysystem satisfies the rank condition, there exists p vector fields Y1i
,
Yp and p non zero (positive
or negative) real numbers si such that y' = exps1Y1 ... expspYP(x"). In the previous sequence each element exp skYk corresponding to a negative time sk can be nearby replaced by an arc exp skYk using the Poisson-stability of Yk. The result follows.
5.5 Controllability and Enlargement Technique
125
5.4.1 Application A first application of Proposition 39 is to construct many controllable polysystems on compact manifolds using Poincard theorem.
Example 5. Take M = G a compact Lie group and D a polysystem whose each vector field is right invariant. Then the polysystem is controllable if and only if the rank condition is satisfied. In this case, observe that DAL is a Lie sub-algebra of g ^_- TG and hence is finite dimensional. There exist algorithms to compute DAL.
5.5 Controllability and Enlargement Technique See [641.
5.5.1 Problem Statement Let D be a polysystem satisfying the rank condition: rank DAL(x) = dim M
for all x E M. To study the controllability of such a polysystem, a powerful technique is the enlargement technique which was codified by Jurdjevic-Kupka, see for instance 165]. The principle is simple, we enlarge D using operations which are not modifying the controllability of D. We shall briefly explain these operations.
Lemma 17. Let D be a polysystem such that rank DAL(x) = dim M for all x. Then the polysystem D is controllable if and only if the adherence of S(D)(x) is M for every x E M. Proof. Use Proposition 38.
Definition 61. Let D, D' be two polysystems satisfying the rank condition.
We say that D and D' are equivalent if for every x E M: S(D)(x) = S(D')(x). The union of all polysystem D' equivalent to D is called the saturated of D and is denoted by sat D.
Clearly a polysystem D is controllable if and only if sat D is controllable. Now, we define the codified operations.
Proposition 40. Let D be a polysystem, then the convex cone generated by sat D is equivalent to D.
Proof. Clearly if X E D then AX E sat D for every A > 0 (reparametrisation). Let X, Y E D, using BCH formula we have
H exp!Xexp!Y =exp(t(X+Y)+o()).
n times
Taking the limit when n -. +oo, we have X + Y E sat D.
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5 Controllability and Higher Order Maximum Principle
Proposition 41. Let X E D and assume X Poisson stable, then -X E D. Proof. It is a consequence of the proof of Proposition 39.
Proposition 42. If ±X, ±Y E D, then ±[X, Y] E sat D. Proof. Apply the BCH-formula.
Definition 62. Let D be a polysystem on M. The normalizer N(D) of D is the set of diffeomorphisms cp on M such that for every x E M, O
and adkbi(Ax) = 0 for k > 2. Hence for A # 0 we have
expAbi*Ax = --1(Ax-AAb;)esat D. Taking the limit when A oo we obtain ±Ab; E sat D. Then we repeat the same operation, replacing bi by Abi. At the end we have
=7ZE sat D
i= The result is proved.
Proposition 45. Consider the following affine system on M: P
x(t) = Fo(x(t)) +Eui(t)F,(x(t)),
ui(t) E R.
i=1
Let D be the distribution x -+ Span{Fi (x),
FP(x)}.
If rank DAL(X) = dim M, for all x E M, then the system is controllable.
Proof. As before for every n E N, ui = ne, e = f1, uj = 0 if j 0- i 1(Fo + nEF;) E sat D n
and by making n - +oo we have ±F; E sat D for i = 1,
p. Applying Proposition 42 we have DAL E sat D. If DAL is of rank n, the system is controllable.
5.6 Evaluation of the Accessibility Set The Baker-Campbell-Hausdorff formula can be used to make evaluation of the accessibility set by constructing an approximating cone. In spirit it is similar to the idea of Pontryagin maximum principle where we construct the first order Pontryagin cone. This was extensively used by Hermes, see for instance [54J, to get higher order necessary optimality conditions along a singular arc.
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5 Controllability and Higher Order Maximum Principle
Definition 63. A rational polynomial is an expression of the form Fa_1 G tq, where l E N, t>_0 small enough, 4: E Q and c; E R. It is called positive if ci > 0 for all i = 1, 1. Let X, Y E V (M), D be the polysystem {X,Y}, DAL the Lie algebra generated by D. We denote by £ the set of 1
germs of vetor fields W such that there exists k E N and rational polynomials [0,e] IIt such that
exp r1(t)X exp s1(t)Y = exp (tW + 0(t°))
exp rk(t)X exp sk(t)Y
with a > 1. From the BCH formula, the set 6 is contained in DAL. We shall prove the following result.
Theorem 21. The set £ is DAL In order to prove this result we need several lemmas and the following two formulae n
k
expt1Xexpt2Wexp-t1X =exp(t2(E
(5.2)
k=0
exp t1Xexp t2 W = exp(t1X + t2W +
t 22
t12 [[X, W[, X]
[X, W]+ 924
t
2
22 [[X, WJ, WJ
[X, [W, [X, w))) + ...) (5.3)
Lemma 18. The set .6 is convex.
Proof. Let A E [0, 1] and W1, W2 E 6. Then there exists k1, k2 E N and rational polynomials such that
exp(tW1 +0(t°)) exprk2(t)X . expsi(t)Y = exp(tW2 + 0(t°)) exprk,(t)X
.
Hence we have
exp (rk1(At)X) ... exp (si (At)Y) exp (rk, (1 - A)t)X ... exp (si (1 - A)t)Y) = exp (AtW1 + (1 - A)tW2 + 0(t°)) where a > 1. Hence AW1 + (1 - A) W2 EE.
Lemma 19. The vextor fields -X and -Y belong to 6. Proof. For a, 0 E lit we have exp (atX) exp (/3tY) = exp (t(aX +,QY) + 0(t2)).
If we seta=-1,Q=0, we have -XE£,and if we set a=0, 3=-1, we have -YEE.
5.6 Evaluation of the Accessibility Set
129
Proof (Theorem 21). We must show that if ±W E E, then ±[X, W] and ±[Y, W] E E. Since ±W E E, there exists rational polynomials such that
exprk(t)X . exp si (t)Y = exp (tW + 0(t°)) exprk(t)X . . .expsi(t)Y = exp(-tW +O(t°))
witha> 1. Let33EQwith 0< 6< 1 and0 > 1. Then exp (t'-"X) exp (OW + 0(t°A)) exp (-tl-'9X) exp (-taW + 0(t°')) = exp (tpW + t[X, W] + ) exp (-tOW + 0(t°p)) = exp (t[X, W] + ). This proves that [X, W] E E. Hence by reccurence we show that ±[adk^Y, [adk^-' X, [... , [adki X, y] ...]
belongs to E. Since these Lie brackets are generating DAL the theorem is proved.
5.6.1 Legendre-Clebsch Condition We can use the same approach to get necessary higher order optimality conditions along a singular arc. We proceed as follows. Consider a single-input afiine control system
x(t) = X (x(t)) + u(t)Y(x(t))
(5.4)
where X, Y are smooth vector fields of R'. Then we have the following lemma.
Lemma 20. Let y : [0, T] - 1R be a singular arc. Assume that y(.) is reduced to a point or y(.) is a smooth one-to-one immersion and that the vector field Y is nonvanishing along y(.). Then there exists a CO -neighborhood
U of y(.) and a feedback u = a(x) + v, a(.) smooth on U, such that on U the singular are y(.) is a trajectory of the system
±(t) = X (x(t)) + v(t)Y(x(t)),
(5.5)
where f ((x) = X (x) + a(x)Y(x), corresponding to the control v =_ 0.
Proof. If y(t) is reduced to a point xo, the singular control can be taken as u(t) = uo where uo is such that X(xo) + uoY(xo) = 0 and we can take a(x) = a(xo) = uo. If y(.) is a smooth one-to-one immersion, then there exists
a smooth coordinate system along y(.) such that y(.) is reparametrized as t -- (t, 0, , 0) for t E [0, T]. Since Y is non vanishing along y(.), then the singular control u(.) is uniquely determined by y(.) using (5.4). In particular it is a smooth mapping. Since y(.) is t ' - (t, 0, , 0) it can be written u(xi ) for xi E [0,T). We set a(x) = a(x' ). In both cases -y(.) is the trajectory corresponding to v = 0. The lemma is proved.
130
5 Controllability and Higher Order Maximum Principle
Proposition 46. Consider the follouring C"-system on liY":
i(t) = X(x(t)) + u(t)Y(x(t)), I U(t) I< 1. Let y(.) be a singular trajectory defined on [0, T] corresponding to the singular control u =_ 0 on [0, T). Then for every t > 0, the first order Pontryagin cone
K(t) is the set K(t) = Span{adkX(Y)(y(t)); k E N}.
Proof. (In the C°° case we have K(t) C Span{adkX (Y)(y(t)); k E N}; the proof is straightforward) Since y(.) is singular on [0, TJ then there exists
p(t) E Rn\{0}, p(t) _ -", H(x, p, u) = (p, X (x) + uY(x)) and (p(t),Y(y(t))) = 0, Vt.
(5.6)
Differentiating recursively this relation and using u - 0 along y(.), one gets (p(t), adkX (Y)(ry(t))) = 0, Vk.
(5.7)
From Chap. 2, we know that p(t)1K(t). In the C' case both relations (5.6) and (5.7) ar equivalent, hence we have the identity K(t) = Span {adkX(Y)(y(t)); k E H}. 0
From the above proposition the first order Pontryagin cone is with empty interior in the singular case. We shall construct an extension of this cone to get better necessary conditions. For this we use the procedure of Hermes [54) which is similar to the one of the begining of this section to show the identity DAL{X,Y} = S. Definition 64. Consider the Coo-control system
.i(t) = X(x(t)) + u(t)Y(x(t)), I u(t) 1< 1 and let y(.) be the reference singular trajectory associated to u - 0 and starting from xo: y(t) = exp (tX)(xo). A vector W belongs to the higher order Pontryagin cone K'(t) at y = exp tX (xo) if there exist positive rational polynomials o2k, r2k, . ' , of , ri such that a(e) = exp o2k(E)X exp r2k(E)(X + Y) exp a2k-l (E)X exp r2k-1(E)(X - Y) 2k
exp a1(E)X exp r1(E)(X - Y) exp (- J(o;(E) + r (e)X )) expTX (xo) i=1
=exp(EW+O(e))(y) with Q > 1. By construction the left hand side belongs fore > 0 small enough to A(xo, T). Hence a(E) is a curve in the accessibility set and W is the tangent vector at E = 0.
5.7 The Multi-Inputs Case: Goh Condition
131
Proposition 47. The set K(t) is a convex cone containing the first order Pontryagin cone. Moreover the vector [Y, [Y, X]](ry(t) belongs to K'(t).
Proof. By construction K'(t) contains Span{adkX(Y)(ry(t))} = K(t). The convexity is proved in [54], see also the proof of lemma 18. To prove the last part of the proposition we have from the BCH formula exp e i (X - Y) exp 2c (X + Y) exp l (X - Y) exp - 4e X
= exp (2ead2Y(X)
- 2Ead2X(Y) + o(c)).
Hence 3ad2Y(X) - 2ad2X (Y) E K'(t) and since K'(t) is a convex cone containing ±ad2XY, we deduce that ad2Y(X) E K'(t).
0
As in the PMP, K'(t) is an approximating cone of the PMP and we deduce the so-called higher-order maximum principle.
Proposition 48. [67] If the trajectory x(t) associated to u - 0 is a time optimal trajectory then there exists p(t) E IR"\{0} such that the following equations are satisfied for every t E [0, T]:
dt (t) = X WO) d. (t) = -p(t) 8X WO)
(p(t), Y(x(t))) = 0 (p(t), X (x(t))) > 0 (Legendre Clebsch condition),
(p(t), [Y, [Y, X]](x(t))) < 0
(5.8)
Remark 7. We observe that if K(t) is of codimension one then in general K'(t) is a cone with non empty interior and is then a good approximating cone. We shall discuss later the genericity of the codimension one assumption and the sufficiency of the above optimality conditions.
5.7 The Multi-Inputs Case: Goh Condition Definition 65. Consider the multi-inputs affine control system m
x(t) = Fo(x(t)) +
>ui(t)Fi(x(t)),
x(t) E R°,ui(t) E R.
i=1
Let (x(t), p(t), u(t)) be a reference extremal defined on [0, T]. We say that it satisfies Goh condition if
132
5 Controllability and Higher Order Maximum Principle
(p(t), [Fv, Fw](x(t))) = 0
for all v, w E lRm, Vt E [0, T], where F = (Fl,
,
(5.9)
F,,,).
Remark 8. A slight modification of the enlargement technique allows to prove the following: if F denotes the polysystem {F1i - , F.} then if FAL (X) is of rank n for every x E R", for every T > 0, the accessibility set A(xo, T) from -
x0, in time T is all R" because the polysystem D = {Fo + Fu; u E Rm} is equivalent to {Fo,FAL}. In this case the time minimal control problem, where the domain of control is lR'" has no solution. Observe that [F, F] is contained in FAL. Therefore the Goh condition is somehow connected with necessary optimality conditions for the time optimal control problem and we shall precise this connection in the next section.
5.8 The Concept of Rigidity - Strong Legendre Clebsch and Goh Conditions as Necessary Conditions for Rigidity Definition 66. Consider the affine control system
i(t) = Fo(x(t)) + > ui(t)Fi(x(t)) i=1
where ui(t) E R, X(t) E M. Let x0 be a fixed point and (x(t),u(t)) be a reference trajectory defined on [0, T] and starting at t = 0 from xo. Assume that the control u(.) is left continuous at T and it is extended on IT, +oo[ by the constant u(T). The control u(.) and the corresponding trajectory are called C' -rigid on [0, T] if there exists e > 0 such that there exists no control v(.) contained in a e-neighborhood of u(.) in LOD([0,T]), v(t) 76 u(t) (a.e.) which can steer xo to x(T) in a time T' E [T - e, T + e].
Definition 67. The (timex input) end-point mapping E is the mapping R x L°°([0,T]) -, M which maps T x u(.), u(.) E LOD([0,T]) onto the end-point x(T, xo, u) of the trajectory of the system. Observe that this mapping is a slight extension of the end-point mapping defined in Chap. 3, where the time T is fixed.
As previously we can compute the Frechet derivative of E and we get the following first order necessary condition for rigidity which is a slight extension of the weak maximum principle.
Proposition 49. If (x(.), u(.)) is C'-rigid on [0,T], then there exists an absolutely continuous vector p(.) on (0, T], nonzero such that the triplet (x, p, u) satisfies the following equation a. e.:
5.8 Rigidity-Strong Legendre Clebsch and Goh Condition
i(t) = a (x(t), p(t), u(t)), At)
aH (x(t), p(t), u(t))
133
(5.10)
where H(x, p, u) = (p, Fo(x) + F(x)u) is the Hamiltonian, together with the conditions for every t: (x (t), p(t), u(t)) = 0 8u H(x(t), p(t), u(t)) = 0.
(5.11) (5.12)
Proof. The proof is similar to the proof of the weak maximum principle and required the computation of the Frechet derivative of E. Conditions (5.10) and (5.11) are the conditions of the weak maximum principle where T is fixed. The additional conditions (5.12): H = 0 uses variation of the time T, see also the conditions H = 0 in the PMP where the time is not fixed. 0
Remark 9. According to the previous proposition, a necessary condition for
C'-rigidity is that (x, p, u) is an exceptional singular extremal.
5.8.1 Intrinsic Second Variation - Morse Index Definition 68. By construction pr = p(T) is orthogonal to the image of Eu. The intrinsic second variation is Q. = pTE"(S, v) where E" is the second variation and (b, v) belongs to the kernel of the first variation (Here b is a time direction and v is a control direction for the variations of the form (T + b, u + ev)). The Morse index of the extremal (x, p, u) is the maximal dimension of the subspaces where Q. is negative definite. The Morse index of the singular trajectory (x, u) is the minimum of indices of quadratic form Qp1 for all possible pT orthogonal to Im Eu. One of the main contribution of [11] is the following.
Theorem 22. A necessary condition for C' -rigidity is that the Morse index of the reference trajectory (x, u) is finite. Both Goh-condition and Legendre Clebsch condition are necessary conditions for the finitness of the Morse index. Hence we get, see [11).
Proposition 50. Let (x, p, u) be a singular exceptional trajectory defined on [0, T]. Then the following conditions are necessary for finitness of the Morse index:
- Goh condition: (p(t), [Fv, Fwj(x(t))) = 0, Vt E [0, T[ - Legendre-Clebsch condition:
8 d2 8H au ate au
>0
134
5 Controllability and Higher Order Maximum Principle
Remark 10. The Legendre-Clebsch condition is the multi-inputs generalisation of the necessary optimality condition established in Sec. 5.6.1. Remark 11. Consider a single-input control system
x(t) = Fo(x(t)) + u(t)F1(x(t)). In this case, if (x, p, u) is a singular extremal one has (p(t), [Fo, Fi](x(t))) = 0. Hence Goh condition is a consequence of the extremality condition. Of course
it is not true in the multi-input case. We shall clarify in the next chapter the concept of C°-rigidity for single-input control systems.
Notes and Sources Higher order necessary optimality conditions for singular extremals are due to Goh, Kelley and others; see the survey article ([46]). The higher order maximum principle is due to Krener [67] and Hermes [54]. The enlargement technique exists in an heuristic form in Hirschorn [56], but was conceptualized by Jurdjevic-Kupka [65] and fully exploited to get controllability conditions for right-invariant systems on Lie groups. The problem of rigidity is everywhere present in the litterature about the abnormal problem in calculus of variations, see for instance Bliss [18]. It also appears in the article [26] in control theory and the neat analysis concerning this problem is due to [11].
Exercises 5.1 Let G be a Lie group with identity e, TeG its tangent space at e, L(G) and R(G) be the Lie algebra of left invariant and right invariant vector fields. We define as usual on TG Lie algebra structures as follows. If A, B E TeG, we define their Lie product by [A, B] = [X, Y] (e) where X and Y are right (or left) invariant vector fields such that X(e) = A, Y(e) = B. 1. Prove that the diffeomorphism on G, .0 : g -+ g-1 induces an isomorphism between L(G) and R(G).
2. Prove that the Lie product induced on TeG by the left invariant vector fields is equal to the negative of the Lie product induced by the right invariant vector fields.
5.2 Let G be the Lie group GL(n,R) of invertible n x n matrices with TeG = g1(nlR) the set of n x n matrices. 1. Prove that a right (resp. left) invariant vector field is of the form X (g) _ Ag (resp. gA) with A E gl(n, lR). 2. If X (g) = Ag, Y(g) = Bg are right invariant vector fields prove that the Lie bracket is [X, Y) (9) = (AB - BA)g and therefore gl(n, R) is a Lie algebra with the matrix commutator as Lie bracket.
5.8 Rigidity-Strong Legendre Clebsch and Goh Condition
135
5.3 Let G be a Lie group acting on a manifold M. To each X in g = TeG identify to the Lie algebra of right invariant vector fields, we can associate a vector field X+ on M by the formula:
(X+f)(m) = lim
f (exp tX m) - f (m)
t-o
(5.13)
t
for f smooth function on M. 1. Prove that [X+, Y+J = [X, Yj+. 2. Let G = GL(n,1R) and M =1R". a) If X (g) = Ag where A is a n x n matrix, compute X+. b) Let D be a polysystem of right invariant vector fields on GL(n,1R) and D+ be the induced system on 1R". Prove that D++L is a finite dimensional Lie algebra. Assume D weakly controllable (resp. controllable) on GL(n, R). Prove that D+ is weakly controllable (resp. controllable) on 1R"\{0).
5.4 Let G be a Lie subgroup of GL(n, R) and consider the subgroup E of GL(n + 1, R) of all matrices of the form: I 1 R
in 1R" andREG.
)
where v is a vector column
1. Prove that the Lie algebra of E is the set of matrices M = I 0
0), with
a being a column vector in 1Rn and A is a matrix in the Lie algebra of G, the Lie bracket being given by: [(a, A), (b, B)] = (Ab - Ba, [A, B]).
(5.14)
2. Prove that E acts on the affine space 1R" =I (1,x) where x is a column vector in R" with the law:
(v R) (x)
- (Rx+v)
(5.15)
Compute the vector field induced by (a, A) on 1R".
5.5 Consider the controlled Euler equation: c:w1(t) = W2(t)W3(t) + u(t)b1
L,2(0 = -W1(t)w3(t) + u(t)b2 W3 (t) = w1(t)W2(t) +u(t)b3
where b = (b1ib2,b3) is constant and IIu(t)II < 1.
1. Assume u = 0 (free motion). Show that the differential equation is Poisson stable. 2. Give necessary and sufficient condition on b to get a controllable system.
5 Controllability and Higher Order Maximum Principle
136
5.6 We consider the following right-invariant control system on SO(3)
dX
(t) = (A+ u(t) B)X (t), X (t) E SO(3), U(t) E IR
where
A=
-100 01010
,
0 00
B=
0 O1 0 001
.
-100
1. Show that the Lie algebra generated by {AX, BX } is the Lie algebra of SO(3). Deduce that the system is controllable on SO(3). 2. Consider the same system but with X = x E S2. Show that the system
is controllable on S2. Given any points xo and xl on S2 construct a piecewise constant control to reach xl from xo.
5.7 Consider a bilinear system on 1R2: x(t) = u(t)Ax(t) + (1 - u(t))(Bx(t) +
b) where A, B are matrices, b is a vector and u(t) E 10, 1). Classify the controllable systems.
5.8 Consider the following system in 1R3: x(t) = X (x(t)) + u(t)Y(x(t)), u(t) 1< 1. We denote by -y+,-y- the arcs corresponding respectively to u T +1, u = -1. Let A(xo, t) be the accessibility set in time t from xo and I
A' (xo) = Ut>o,small enough A(xo, t) the small time accessibility set. Assume
that X(xo),Y(xo) and [X,Y](xo) are linearly independent. Show that the boundary of A'(xo) is formed by the following two surfaces: Sl = U-Y+ -y- (X0)'
S2 = Ury_ry+(xo) and that each point in the interior can be reached by an arc y+'y-'y+ and t-'y+'Y-. 5.9 Consider a rigid spacecraft controlled by gas jets and described by the following system:
R(t) = S(w(t))R(t) P
w(t) = Q(w(t)) + E Uk(t)bk k=1
where R E SO(3) =group of rotations in 1R3 with determinant 1, S(w) is the 0
antisymmetric matrix
w3 -w2
-W3 0
wl
,
Q is given by
W2 -WI 0 a a a a3wlw2 , Q = alw2w3 aw + a2wlw3 aw + aw3 2 1
al =
a2
a3= f,I,>I2>I3>0,thebksarecon-
stant vector fields anJ uk(.) are piecewise constant mappings with values in
{-1,0,1).
5.8 Rigidity-Strong Legendre Clebsch and Goh Condition
137
1. Show that the free motion corresponding to a control identically zero is Poisson stable. 2. Prove that the system is controllable if and only if {Q, b1, , b,} is of rank 3 at each point w E 1R3.
3. Assume p = 1. Prove that the system is controllable if and only if the constant vector field b1 does not belong to one of the axis Owi, i = 1, 2,3 or one of the two planes given by the equation: a3w2 - a1w3 = 0.
5.10 Let Hk be the set of polynomial homogeneous vector fields in R' with degree k and consider a quadratic control system: P
2(t) = Q(x(t)) + E ui(t)bi
(5.16)
i=1
where Q E N2 and bi E H°. Let L be the Lie algebra generated by {Q, b1,
,
b,,} and let us define:
i) L° is the smallest subspace of 7.{° such that:
a) bi EL°, for all b) if v1, v2 E L°, then [[Q, VI 1, v2] E L°.
ii) Ll is the Lie subalgebra of W1 generated by the vector fields: {[Q, v]; v E L°}. iii) L2 is the Lie subalegbra of ek>2Hk generated by the vector fields {adL, o
..oadLm(Q); mEN;L1EL1} 1. Prove that L = L° ®L1 ® L2. 2. Let 0(0) be the orbit of 0, prove that 0(0) = L°. 3. Define the following sequence of subspaces: i) So is the linear span of {b1, , by}; ii) for k > 1, Sk is the linear span of Sk_1 U {[[Q, V11, v2]; vi, v2 E Sk_1}. Prove that L° = Uk>°Sk and that there exists k < n such that L° = Sk_1.
4. Consider the planar single-input system: i(t) = Q(a(t)) + u(t)b, u(t) E R. Prove that the system is weakly controllable if and only if we have Span{b, [[Q, b], b]} =1R2 and show that the system is not controllable.
5.11 Consider a control system in R" of the form: P
x(t) = H(x(t)) + > ui(t)bi
(5.17)
i=1
where H is an homogeneous polynomial vector field of degree m, bi are constant vector fields and the controls ui (.) are piecewise constant vector fields with values in R. Let L be the Lie algebra generated by {H, b1, , b,} and L° be the Lie subalgebra of all constant vector fields.
1. Prove that L° is equal to the smallest vector space which contains the , b,} and all element of the form adv1 o ... o advk(H) for vectors {b1, an arbitrary element v1i -, vk in the vector space.
5 Controllability and Higher Order Maximum Principle
138
2. Prove that L° is the smallest vector subspace of R" which contains the vectors bl, , by and which is invariant under the mapping H. 3. Prove that the system is weakly controllable if and only if L° =1R" 4. Assume the order of H to be an odd integer m. Use the enlargement technique to prove that the system is controllable if and only if L° = R'. 5.12 We note SL(n, IR) the Lie group of n x n matrices with determinant 1 and sl(n, R) its Lie algebra of n x n matrices with trace equal to 0. Consider the right invariant control system on SL(n, IR):
R(t) = (A + u(t)B)R(t)
(5.18)
where R(t) E SL(n, IR), A, B E sl(n, IR) and u(.) is a piecewise constant mapping with values in R.
1. Prove that the system is controllable if and only if the accessibility set from the identity contains an open neighborhood V. 2. Assume that the system is controllable on SL(n, R). Prove the controllability of the following bilinear system in IR"\{0}: ±(t) = Ax(t)+u(t)Bx(t). 3. Assume that A is diagonalizable over the set of complex numbers with imaginary spectrum:
a(A) = {±iak; ak E lR, k = 1,
, n}.
Prove that the system (5.18) is controllable on SL(n, IR) if the Lie algebra generated by {A, B} is sl(n,R).
4. Assume that the spectrum of B is real and disctinct: Al < . . . < Ai E Ill and let e 1,
, e be the corresponding eigenvectors. Let (aid)
be the matrix representing A in the basis e1. We make the following assumptions:
i) the numbers Ai - A, are all distinct; ii) a22
0forI i-j1=1;
iii) 0. Use the enlargement technique to prove that the system is controllable on SL(rt, R).
5.13 Let A, B1, , B. be n x n matrices and assume that the system ±(t) = Ax(t) + Ep, ui(t)B1(x(t)), I ui(t) I< M is controllable on )R"\101. Prove that for all b1, , br,, bi E lR" there exists R > 0 such that for each xo, xl E {x; I x 1> R} there exists a control steering x0 to x1 for the system: Wi(t) = Ax(t) + F,?1 ui(t)(B,x(t) +bi). 5.14
1. Let A1, A2 and A3 be respectively the matrices:
000 10 0 1
0-10
00-1 ,
00 0
100
0 10
-101 0 00
5.8 Rigidity-Strong Legendre Clebsch and Coh Condition
139
and denote by exp A the exponential matrix. Let R E SO(3) =group of direct rotations in 113. Prove that there exists angles Vi, 0, 95 E [0, 2ir[ such
that: R = (expi/.'A3)(exp9A1)(expOA3)
(5.19)
2. Consider a rigid spacecraft controlled by two gas jets: 3
R(t) _ (Ew;(t)A;)R(t) =1 2
w(t) = Q(w(t)) +
z-
u=(t)b=
where R E SO(3) and Q(w) =
a a1w2w3C7W1
a
a
+a2wlw38W2 +a3WIW2C7W3,
with al, a3 > 0 and a2 < 0. The controls u;(.) being piecewise constant mappings with values in [-1,+11. a) Assume b1 =' (1, 0, 0) and b2 =' (0, 0, 1). Using part 1, construct a control law to steer (Ro, 0) to (R1, 0). b) Prove that each (R, w) can be steered to some (Ro, 0) and is accessible from some (R1i 0).
c) Construct a control law to steer the spacecraft from (Ro,wo) to (RI,wl). 5.15
1. Consider the smooth differential equation on R': do (t) = X(x(t)). Let V : 1R" -+ IR be a smooth function such that:
i) V(x)>0and V(x)-* oo when Jlxii-++oo. ii) LXV < 0 for all x E lR". Let E = {x E IR ; LXV = 0} and M the largest subspace of E invariant for the solutions. Prove that all the solutions are bounded when t -+ +oo and tend towards M. 2. Consider the smooth control system on Rn:
d (t) = X(x(t)) + u(t)Y(x(t)).
(5.20)
a) Assume X (x) = Ax where A is a n x n matrix whose spectrum consists of n disctinct imaginary eigenvalues and suppose that for each x 0 0, the system (X, Y) satisfies the ad-condition:
Span{adkXY(x); k = 1, , +oo} =1R". Prove that all the solutions of th(t) = Z(x(t)), with Z = X - (x, Y)Y where (,) is the inner product are tending towards 0 when t -- +oo.
140
5 Controllability and Higher Order Maximum Principle
b) Assume that there exists a smooth function V : 1R -' R such that V >_ 0, V(x) -' +oo when DDxli - +oo, LXV(x) = 0 for all x E R". Prove that all the solutions oft(t) = Z(x(t)), with Z = X - (LXY)Y are when t oo tending towards
W = {x E R'; LXV(x) = Lad'-X(y)V(x) = 0, k = 1,
,+oo}.
3. Consider the planar differential equation:
x(t) = x2(t) - y2(t), y(t) = u2(t) with u2(t) = alx2 (t) + a2y2(t) + a3x(t)y(t), a1, a2i a3 being constant. a) Prove that this differential equation admits a positive definite first
integral V = Axe + 2Cxy + By2 if and only if: a3 > 0, a2 = f 1, a1 + a2 + a2a3 = 0b) In this case, prove that all the solutions of x(t) = x2(t) - y2 (t), y(t) _ u1(t)+u2(t), ul(t) = -LyV are tending to the origin when t +oo and draw the phase portait.
5.16 Let G be a connected Lie group, g its Lie algebra of right-invariant vector fields on G. Let A, B1, system:
,
B,, in g and consider the following control m
9(t) _ (A+>ui(t)B2)g(t)
(5.21)
awl
where u . (t) E R. We denote by go the Lie algebra: {B1,. , Bm } AL and by Go the corresponding connected subgroup of G. Suppose that go has codimension one in L. Prove the following:
1. If Go is closed in G, then the system is controllable if and only if A 0 go and G/Go is isomorphic to S'.
2. If Go is not closed in G, then the system is controllable if and only if
Ago. 5.17 Let G be a connected Lie group, with identity e and let D be a family of left invariant vector fields on G. 1. Prove that the polysystem D is controllable if and only if the accessibility set A(e) contains a neighborhood of e. 2. Assume G compact. Prove that the polysystem D is controllable if and only if D is weakly controllable.
5.18 Let G be the Heisenberg group represented by 3 x 3 upper diagonal matrices:
lxy G=
01z
001 with the usual matrix multiplication.
;
(x, y, z) E R3
(5.22)
5.8 Rigidity-Strong Legendre Clebsch and Goh Condition
141
lxy 1. Identify G to R3 via the mapping
01z
" (x, y, z) and compute the
001 induced multiplication in R3. Prove that the vector fields:
Yl=a
Y2=ay +xaz' Y3=az
,
(5.23)
form a basis of the Lie algebra of left invariant vector fields on R3. 2. Let the canonical coordinates of the first kind (x, y, z) F-+ exp(xY1 + yY2 + z[Y1iY2))
(5.24)
defined by the exponential mapping. Prove that in these coordinates, Yl and Y2 are represented by:
Fl
a ax
xa 2 az,
a F2
X49
a y + 2 az
(5.25)
Use the Baker-Campbell-Hausdorff formula to prove that the multiplication rule is:
(x,y,z).(x',y',z')=(x+i,y+y',z+z'+2(xy'-yx)
(5.26)
5.19 let Sp(1, Ill;) = SL(2, R) be the Lie group of 2 x 2 matrices with determinant 1 and its Lie algebra of right invariant vector fields identified to sp(1, R)
the set of 2 x 2 matrices with zero trace. Let P = {z E C; Im(z) > 0} be the Poincare plane; Sp(1,R) acts on P with the law: S.z = (az + b)(cz + d)-1
if S E Sp(1, R), S
=Cad
I .
Let A E sp(1, R), denotes by Ap the induced
vector field on P. 1. Draw the integral curves of Ap if: a) the eigenvalues of A are imaginary; b) the eigenvalues of A are real and nonzero. c) the eigenvalues of A are zero. 2. Let D = {A, B} be a pair of right invariant vector fields on Sp(1, IR). Prove that D is controllable on Sp(1,1R) if and only if Dp = {Ap,Bp} is controllable on P.
5.20 Let SO(n) be the group of n x n matrices of rotations with determinant 1. Construct on SO(n) a controllable pair {A, B} of right invariant vector fields.
5.21 Let K be a compact connected Lie group which acts linearly on a finite dimensional vector space V and let G = V x 8 K be the semi-direct product. If k is the Lie algebra of right invariant vector fields on K, the Lie algebra
5 Controllability and Higher Order Maximum Principle
142
g of G consists of V ® k with the Lie bracket defined by [(x, A), (y, B)] = (Bx - Ay, (A, B]) with x, y E V, A, B E k and [A, B] is the Lie bracket on k. Prove the following: suppose that V and K are such that V admits no fixed nonzero point. Then a necessary and sufficient condition of controllability of a polysystem D of right invariant vector fields on G is that the Lie algebra generated by D is g = V ® k.
5.22 Let x(s) be a (generic) curve in Rn parametrized by arc-length, v1(s), , v (s) its Frenet frame and V (s) the matrix whose columns are the vts. 1. Show that the Frenet formula: dx V1
TS dv1
ds = dv;
s=
k1v2
-kt_1vi_1 + k;vt+1, 1 < i < n dvn
ds
-
-kn-lvn-i
can be written ds
where e1 =c (1, 0,
(s) ,
V(s)A(s),
ds
(s)
V(s)e1
(5.27)
0) and A(s) is then x n matrix
0
... kn_1
0
and interpreted as a differential equation on the semi-direct product SO(n) x, Rn. 2. Let n = 3, k1(s) = k(s) be the curvature and k2(s) = r(s) be the torsion.
0-k 0
a) Assume that k, r are constants and that A =
k 0 -r Ur
. Prove
0
that exp sA is a rotation with axis tw = (r, 0, k) and angle sr + k2 and x(s) is an helix along w. b) Assume k(s) =(nonzero) constant and r(s) = r1 or 7-2, r1 0 r2. Prove that each helix is a concatenation of helices along: wl =t (r1, 0, k), W2 =1' (r2, 0, k).
6 The Concept of Conjugate Points in the Time Minimal Control Problem for Singular Trajectories, C°-Optimality
6.1 Single-Input Case 6.1.1 Preliminaries In this section we consider a real analytic (or C°°) single-input affine control system:
x(t) = X(x(t)) + u(t)Y(x(t)), u(t) E R. Let ry
:
(6.1)
[0, T] -+ lR" be a reference singular trajectory corresponding to a
control u(.) E L°°([0,T]) and starting at t = 0 from y(0) = x°. We shall give necessary conditions for time-optimality under some assumptions. The topologies we consider are the following: C°-topology and strong C°-topology.
Definition 69. The C°-topology is defined as follows. Let y(.), 7(.) be two C°-curves defined respectively on [0,T], [0,T] and with extremities Po =
(0,7(0)), Pi = (T,y(T)), Po = (0.-;Y(0)). P1 = (T,=y(T)) in the time extended state space. The distance between y and
is given by:
P(y,y) = max 11y(t) -7(t)II + d(Po, Po) + d(P1, Pt)
is any norm in IRn. The curve y(.) is said to be C°-time minimal (resp. maximal) if y(.) is a time minimum (resp. maximum) trajectory with respect to all solutions of (6.1) with extremities (0, y(0) = x°), (T, %T) = y(T) = xl) and contained in a C°- neigborhood of -y(.). A stronger optimality concept called strong C°-optimality is defined by considering all piecewise C°-trajectories ry(.) with extremities ro and xl and contained in a tubular neigborhood of the image of y(.). where
Jump direction and generalized control. Since the system is affine, the set of singular trajectories solutions of the maximum principle with u(t) E R is not sufficient to provide optimal solutions for every boundary points. This leads to the following definition. Definition 70. The control direction JRY is called the jump direction. A control jump of height a at time t is the control impulse defined as the limit when
144
6 The Concept of Conjugate Points
n - +oo of the sequence un(s) = na for s E [t, t + ,,] and 0 otherwise. We extend the class of admissible control U = L°° to the set of generalized controls U. defined as follows. The control u(.) defined on [0, T] belongs to U9 if there exists 0 < t1 < < tk < T such that the restriction of u(.) to ]ti, ti+1 [ belongs to L°° and at every ti, u(.) has a control jump of height ai. This leads to well defined generalized trajectory solution of (6.1).
Basic assumptions. Let (z(), u(.)), z = (x, p) be a singular extremal defined on [0, T]. We shall assume the following.
Assumption (Ho). z(.) is contained in the set .R = {z; (p,ad2Y(X)(x)) 34 0}
and x(.) is contained in the set if = {x; X (x) and Y(x) are linearly independant} and is one-to-one. Then according to Proposition 22, t i-, x(t) is a C°'-curve not reduced to a single point and the mapping t i-, x(t) is an one-to-one immersion. According to (Ho), z(.) is a singular extremal of minimal order and is a solution of the
Hamiltonian vector field H where H(x, p) = (p, X (x) + u(x, p)Y(x)) and (p. [ Y,X ,X x u(x , P) Y (x p. We shall consider the following assumptions:
Assumption (HI). For each t E [0, T] the linear space 71
-
K(t) = d7r (Span{adk H (HY)(z(t)); k = 0,
,
+oo})
is of codimension one and is generated by the vectors corresponding to k = 0,. , n-2 where Hy (x, p) = (p, Y(x)) and ir is the projection (x, p) '- x. Assumption (H2). If n > 3, for every t E [0, T], we have
X(z(t)) 0 dlr(Span{adk H (HY)(z(t)); k = 0,
n - 3}).
Geometric interpretation of the condition (HI). The condition (H1) means that for t > 0 the first order Pontryagin cone denoted K(t) is of codimension one and is generated by the first (n - 1) vectors.
6.1 Single-Input Case
145
Non-intrinsic formulation. If the singular control is identified to u = 0 the previous assumptions take the following simple form:
(H1) Vt E [0, T], Span{adkX (Y)(x(t)); k = 0, , +oo} is of codimension one and generated by Span{adkX (Y)(x(t)); k = 0, , n - 2};
(H2) If n > 3, Vt E [0, T], X(x(t)) V adkX (Y)(x(t)) for k = 0, - n - 3.
Definition 71. Let (z(.), u(.)), z = (y, p) be the reference extremal defined on [0, T] and assume that the previous assumptions (H°), (H1) and (H2) are satisfied. According to (H1) the vector p is unique up to a scalar and it can be oriented such that H(x, p) = (p, X 0. We say that y() is exceptional if H = 0 along y. Let D be the quantityBua -7 7V = (p(t), [[Y, X], Y](x(t))).
The trajectory y(.) is said to be hyperbolic if D is strictly positive along y and said to be elliptic if D is strictly negative. Remark 12. Recall that from Proposition 48, the condition
e d2OH>0 Ou dt2 8u called the Legendre-Clebsch condition is a necessary time-optimality condition. We can now formulate our main result.
Theorem 23. Let y(.) be a reference singular trajectory defined on [0, T] and satisfying assumptions (Ho),(H1),(H2). Then 1. For T small enough, y(.) is a strong C°-time minimum (resp. maximum) for the generalized trajectories of (6.1) if -y(.) is exceptional or hyperbolic (resp. elliptic). 2. If -y(.) is an exceptional or a hyperbolic (resp. elliptic) trajectory, it is a strong C°-time minimum (resp. maximum) if and only if T < t1c where t1c is called the first conjugate time along -y.
Remark 13. The previous theorem gives a necessary and sufficient strong C°optimality condition for a singular trajectory satisfying (Ho), (Hl) and (H2)
and we shall give an algorithm to compute the first conjugate time. An important point which must be emphasized is that we have two different algorithms: one in the hyperbolic-elliptic cases and one in the exceptional case.
Sketch of the proof: The complete proof is given in [26] and we shall only outline the different steps and indicate the geometric intepretation. The main idea of the proof is to make a direct computation of the accessibility set A(x°i T) along the reference singular trajectory. This requires the computation of a feedback semi-normal form and the theory of cheap LQ-control using differential operators.
146
6 The Concept of Conjugate Points
6.1.2 Feedback Semi-Normal Forms in the Hyperbolic and Elliptic Cases Proposition 51. Assume that -y(.) is a hyperbolic or an elliptic singular trajectory. Then the system is feedback equivalent in a C°-neighborhood of -y(.) to a system (X1,Y1) of the form: X1 = a
8x1
+n-I E x'+'
a
:=2
n
E a;j(x')x2)
1
+R
:,3==2
a Y1
axn
where ann is strictly positive (resp. negative) on [0, T] if 'y(.) is elliptic (reap. i R, i is a vector field such that the weight of Ri hyperbolic) and R =
is of order greater or equal to 2 (resp. 3) for i = 2, , n - 1 (reap. i = 1), the weights of the variables x' being 0 for i = 1 and 1 for i = 2,- , n.
Geometric interpretation. - The reference singular trajectory y(.) is identified to t '- (t, 0, corresponds to u = 0. - The first order Pontryagin cone along y is given by KI, = Span{
... , a--
xe
IY
,
0) and
}
17,
and the linearized system is autonomous and in the Brunovsky canonical form l2(t) = t;3(t),. .. , £n(t) = u(t).
- If py(.) is the adjoint vector associated to y(.), then we can set p.1 = (e, 0, , 0) where e = +1 in the elliptic case and c = -1 in the hyperbolic case, the Hamiltonian being e. The second order variation at T is then e
with t2(t) =
f
aij(t)e'(t)(t)dt
(6.3)
i,j=2
en(t),tn(t) = v(t) and the intrinsic
derivative is obtained by taking the boundary conditions 2(s) = . . . = l;n(s) = 0 for s = 0 and s = T. Moreover (py,ad2Y1(Xl),,1) = wan(t)-
Exceptional case. Proposition 52. Let -y(.) be an exceptional trajectory. Then n > 3 and there exists a C°-neigborhood of y(.) in which the system (6.1) is feedback equivalent to a system (X2, Y2) of the form:
6.1 Single-Input Case
X2 = s1 +
n-2
E
147
n-1
xt+1
ax' +
i,j=2
i=1
an + R
aij (x
Y4 2 =
where an-1,n-1 > 0 on [0,T], R = E 1 RI
a axn-1
;, Rn-1 = 0 is a vector field
such that the weight of Ri has order greater or equal to 2 (resp. 3) for i
, n - 2 (resp. i = n), the weight of the variables xi being zero for i = 1, one for i = 2, , n - 1 and two for xn. 1,
Geometric interpretation. - The reference trajectory y(.) is identified to t H (t, 0, sponds to u 0.
,
0) and corre-
- The first Pontryagin cone along y is given by: KI.y = Span{
a ax117'
... ,
a I,y}.
axn-1
- The adjoint vector p.y associated to -y can be normalized to (0, moreover (p.y, ad2Y2(X2)17) = -an-1,n-1(t) - The intrinsic second-order derivative at time T is identified to
-
,
0, -1)
rTn-1
J° Ei,j=2aii (t)t`
(t)dt
(6.4)
/ t Sn-1(t) = u(t), n(t) = v(t) and with 41(t) = t 2(t), ... , n-2 (t) = en-1(t), the boundary conditions at s = 0: c1(s) = . . . = to (s) = 0.
6.1.3 LQ-Model Having computed the semi-normal form we can identified in both cases the LQ-model which will be used to evaluate the end-point mapping. It is simply obtained by taking Ri = 0 in the semi-normal forms and replacing x1 by t. We get the following. Hyperbolic-Elliptic case:
21(t) = 1 +
aij (t)xt (t)xi (t) i,j=2
x2(t) = x3(t)
xn(t) = u(t) Exceptional case:
148
6 The Concept of Conjugate Points
xl (t) = 1 + x2(t) x2(t) = x3(t)
in-1(t) = u(t) n-1
in (t) = > a:; (t)xi (t)xi (t) i,j=2
This leads to a concept called accessory problem similar to the one in the calculus of variations introduced in Sect. 2.1.6. Definition 72. Let 0 < t f < T be fixed. The accessory problem is - Hyperbolic-elliptic case:
e = f 1, subject to . x2(t) = x3(t), . . , xn(t) = u(t)
min ex' (t f ),
and the boundary conditions
x(0) = A, 1(tj) = B where i = (x2, ... , xn-l - Exceptional case: minxn(t1), subject to
xl(t) = 1 + x2(t), x2(t) =
x3(t)...... n-1(t) = u(t)
and the boundary conditions
x(0) = A, 1(tt) = B where x =
(x1,...,xn-2)_
We observe that the previous equations are integrable and they will be used to approximate the end-point mapping of the original problem in a C°neighborhood of the reference singular trajectory y(.). We proceed as follows.
6.1.4 Approximation of the End-Point Mapping Using the LQ-Model in the Elliptic Case The model is written
i1(t) = 1 + q(t, x2, ... , xn) ±2(t) = x3(t)
in-1(t) = xn(t)
6.1 Single-Input Case
149
where q is the quadratic form n
q(t,x2,...,x^) = r aij(t)x.(t)x'(t) i,j=2
We set x1(t) = t + 1(t) and we get n
bl(t) = E
=Sn(t)
i,j=2
and the accessory problem is, using the notation l; = f 2, equivalent to min f (, -2)
ref
J0
q(t, c
,
&-2) )dt
where q() = rii 20 bij(t)F,(i) (t)e(j) (t) and the bij are symmetric and defined by
bi-2,j-2 =
aij + aji 2
if i 0 j,
bi-2,i-2 = aii.
Observe that the control is taken as &-2) and this corresponds to make the Goh transformation for the system where xn taken as control. We have the following standard concept and results concerning the above problem. Definition 73. Let 0 < t < T and let us denote by Ct the set of C2(n-2)curves defined on [0, t] and satisfying the following boundary conditions: C(0) =
= &-3) (t) = 0.
_ C(n-3)(0) = C(t) =
Let D be the differential operator of order 2(n - 2) defined by n-E=1 2(-1)iai8
D(C)-2 S
()(S) M(C)
This operator is called the Euler-Lagrange operator associated to the accessory
problem. We denote by b the restriction of D to V. The time t, is said to be conjugate to 0 along the elliptic trajectory ry(.) for D if there exists a non trivial solution (.) E V such that D(1:) = 0. The differential operator b is self-adjoint and we have the following results.
Proposition 53 (Hilbert-Schmidt theorem [89]). For every t E]0,T] there exists a sequence (ea, Aa), a = 1,
, +oo such that:
1. ea(.) E Ct, (ea, ep)La((o,tJ) =6! (Kronecker symbol) and Dea = Aaea.
2. AI t1c where t1, is the first conjugate time, then the minimum is -oo.
We represent on Fig. 6.1 the end-point x1(t f) for t f < tic and t f > t1,.
Fig. 6.1. Elliptic case
Remark 14. In the hyperbolic situation, the analysis is similar replacing the minimization of x1 (tf) by the maximization of x1(t f ), see Fig. 6.2.
6.1 Single-Input Case
153
Fig. 6.2. Hyperbolic case
6.1.5 Conclusion By Inspecting Fig. 6.1,6.2. we can deduce the following. For the LQ-model, we have:
- in the hyperbolic case, the reference singular trajectory y(.) is C°-time minimal up to the first conjugate time tl, of the operator D. - in the elliptic case, the reference singular trajectory is C°-time maximal up to the first conjugate time ti of the operator D. Clearly, those results are still true for the strong C°-topology. Also they are valid for the original system when we consider the generalized trajectories, see [26] for the details. Moreover, our analysis proves the following.
Proposition 56 (C°-one side rigidity). Assume T < tl,,. Then the reference singular trajectory y(.) defined on [0, T] is in the hyperbolic (resp. elliptic) case the only trajectory ry(.) contained in a C°-neighborhood of y(.) and satisfying the boundary conditions ^y(0) = y(0), y(T) = y(T) in a time
T < T (resp. T > T).
6.1.6 The Exceptional Case We shall use a similar technique to evaluate the end-point mapping in the exceptional case. The model is then i1(t) = 1 + x2 (t)
x2(t) = x3(t) thn-2(t) = xn-1(t)
in (t)
= 9(t,
x2 (t), .
. .
,
xn-1(t))
154
6 The Concept of Conjugate Points
where q is the quadratic form n-1
q(t, x2 ... , xn) _
aij (t)x`xj i, j =2
which can be written as n-1 q(t,x2...... n) =
bij(t)xiy
i,j`=2
where bij(t) = bji(t). We write x1(t) = t + fi(t)
and the quadratic form can be written with x2(t) = e(t) as n-1 i,j=2
t(n-2)) not depending on S
It is a quadratic form on the space n = t;. If 0 < t < T we introduce
Q(t) =
(n-2)(T))dT
J
and B(C, r7) the associated scalar product. As previously if C(.),77(.) are C2(n-2)_curves and n(.) E V, i.e. 77(0) = ... = 77 (n-3) (O)
= n(t) _ ... =
7(n-3) (t) = 0
we can write
B(t, n) = 77)Lwhere D is as in Definition 73. Its restriction to G` is denoted f) and is self-adjoint. We denote by t1c the first conjugate time for the operator D. Assume that t < t1c. Using the Proposition 55, we can expand each curve l;(.) E
C2(n-2) such that
e(O) = ... = as
+00
&-3)(O) = O
n-2
_ E C. e. + ?.wiJi 1
where (O) = 'w1, ... ,
&-3)(0) _ Wn-2
6.1 Single-Input Case such that
= +0 AaSa CC
Q(tt)
155
n-2
+ E Bijwiwj. i,j=1
C.=1
Hence the end-point mapping for the model, where xn-1(t) = (n-2) (t) is the control, is then given a t time t f < t1c by:
n-2 (tf + w1, w2, ... , wn_2, E Aac2a + > bijwiwj) too
C
O(S(n-2))
a=1
(6.10)
i,j=1
where Aa > 0, Va = +1, , +oo. Let x(.) be a trajectory of (6.8) defined on [0, t1] and satisfying the boundary conditions: i(0) = 0, x1(t f) = T, x2(tf) = xi-2(t f) = 0. Then we have using (6.10): x(tf) = (T, w2 = 0, ... , wn-2 = 0, xn(tf )) where tf+w1 = T, xn(tf)
_ EQ° Ab ara2 +wiB11 and B11 is given by Q(J1).
In particular, we have
min in(t f) = wiB11 = (T - tf)2B11
=n-1
and this is consistent with Hamilton-Jacobi-Bellman equation
8minin(tf)
f
Itf=T
Definition 74. The time tcc is said to be conjugate to 0 along the exceptional Jin-2) )dt = 0. trajectory -Y if Q(J1) = focc q(t, J1: Ji,... On can prove the following, see [26].
Proposition 57. We have: 1. The first conjugate time satisfies the inequality: 0 < tlcc < tic. 2. Assume tlcc < t1ci then Q(J1) = fog q(t, J1(t), .. , Jin-2)(t))dt > 0. 3. If n = 3, Q(J1) > 0 for every t f and if n > 4, tcc are the conjugate points of the differential operator D of order 2(n - 3) defined by D = - A(D A )
Geometric representation of the accessibility set. On Fig. 6.3 we represent xn(t f) for the model with the boundary conditions w2 = = wn-2 = 0 for t f < tlcc and t1cc < t f < tic, the control being xn_1.
6.1.7 Conclusion Inspection of the previous figures shows the following for the model:
156
6 The Concept of Conjugate Points
I1\1
tf=T
tf 0 containing the hyperbolic trajectory satisfying the assumptions (Hl) - (H2) - (H3).
Definition 75. The function K(.) is called the curvature in the hyperbolic case.
6.3.5 Conjugate Points and Time Minimal Synthesis Let 0 < M < +oo and consider the CO-control system v(t) = X(v(t)) + u(t)Y(v(t)), where the set of admissible controls is the set UM of bounded
measurable mappings taking their values in [-M, +M). Let S be given by S(X, Y) = X - aY where D = det(Y, [Y, X], [[Y, X], Y]) and D' _ det(Y, [Y, X], [Y, X], X]). Let -y(.) be the reference hyperbolic singular trajectory defined on [0, T] and satisfying (Hl) - (H2) - (H3); we assume the following:
164
6 The Concept of Conjugate Points
Assumption (H4) t1, E) - M, +M[. Let V(t), 0 < t < T be the solution of the variational equation
6v(t) = 7(.y(t))ov(t)
(6.15)
with initial condition v(0) = Y(y(0)). Let E, E' = ±1 and let f be the mapping f : (ti, t2) t3, E, E') 1- exp t3(X + E'MY) exp t2S exp t1(X + EMY)(y(0)).
from the analysis of Chap. 3, such a trajectory is an extremal for t1, t3 _> 0, small and M < +oo. Let F be the image of f for t2 E [0, T] and t1, t3 sufficiently small. According to [88] if det(V(y(t)), Y(y(t)), S(y(t))) never vanishes on ]0, T], then F is an extremal field about the arc y(.) in the following sense. There exists a C°-tubular neighborhood U of y(.) such that each point of U is the image of an unique (t1, t2, t3, E, E'). Moreover it is the time minimal synthesis in a neighborhood of the reference singular trajectory for the fixed end-point problem. Our previous analysis shows that this result is still valid for M = +oo if we consider the generalized trajectories.
Definition 76. Let us denote by t1 the first time in 10, TJ such that det(V(t), Y(y(t)), S(y(t))) vanishes.
The following lemma is proved in [24].
Lemma 21. We have the following:
1. V(t) E Span{Y(y(t)),[X,Y)(y(t))} 2. det(V(t), Y(y(t)), S(y((t))) = 0 for 0 < t < T if and only if V (t) and Y(y(t)) are colinear. 3. tic = tic Hence we get an equivalent definition of the concept of conjugate point. Also it shows the relation with the time minimal synthesis when I u(t) I< M.
6.4 Connection with the Liu-Sussmann Example 6.4.1 Preliminaries In 179J Liu and Sussmann consider the following system:
x(t) = ui(t) y(t) _ (1 - x(t))u2(t) z(t) = x2(t)ul(t)
(6.16)
6.4 Connection with the Liu-Sussmann Example
165
and the time minimal control problem. The system is written as
v(t) = ul (t)F1(v(t)) + u2(t)F2(v(t))
(6.17)
where v = (x, y, z), F1 = YX- , F2 = (1- x) + x2 . The system is symmetric and the singular trajectories satisfy the following equations: (P, Fi (v)) = (p, F2 M) = 0
=0 u1(, [[F1, F21, F1J(v)) + u2 VI, [[Fl, F21, F21) = 0
Hence they are exceptional and are contained in det(Fi, F2, [F1, F21) = x(x - 2). Moreover
D1 = det(Fi, F2, [[Fl, F2J, F1]) = 2(x - 1), D2 = det(Fi, F2, [[F1, F2J, F1J) = 0.
Hence they are contained in x = 0 or x = 2 and the singular control satisfies ul = 0 almost everywhere. If they are parametrized by the length ui (t) +
u2(t) = 1, we get u2(t) = ±1. We observe that the system is left invariant by:
S : (x, y) z, ul, u2) '-' (x, -y, -x, u1, -u2) and we can assume u2(t) = +1. We consider the reference singular trajectory ry : t F-4 (0, t, 0) starting from 0 and corresponding to u2 = +1. The connection with the Sect. 6.1 is the following.
Remark 15. The system (6.17) with v = 1 is the model corresponding to z = a(y)x2, a(y) = a(0) (small time model) used in Sect. 6.1 to study the C°-time optimality of the reference singular trajectory. In dimension 3 the following result is obvious but it can be generalized to R".
Corollary 14. The reference singular trajectory ry t 14 (0, t, 0) is strongly C°-time minimal for the system (6.16) when ul =_ 1 and u2(t) E R. :
We deduce the following.
Corollary 15. The reference singular trajectory y : t ' - (0, t, 0) is strongly C°-time minimal for system (6.16) when 0 < u1(t) < 1 and u2(t) E R. Proof. The system (6.16) is symmetric, hence the set of trajectories is invariant for the following control homothety: (ul, u2) '-+ (Aul, Au2). In particular each trajectory such that 0 < ul (t) < 1 at a Lebesgue point can be reparametrized on a smaller interval as a trajectory with ul (t) = 1. Remark 16. The above proof shows that we cannot extend the control domain beyond the point u1(t) = 1 and keeping the time optimality. But we leave as an exercise to the reader the possible extension to ul (t) < 0, using integrability of the equations.
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6 The Concept of Conjugate Points
6.4.2 Conclusion The Liu-Sussmann example fits in our framework and is really equivalent to
the C°-rigidity of the exceptional trajectories satisfying (Hl) - (H2) (H3). In the remaining of this section we shall present the original proof of both authors which is different of our approach in which we parametrize the boundary of the accessibility set using extremals. Also our approach proved that the example is a generic model, compare with the extension of Sussmann in [79].
6.4.3 Statement and Proof of Liu-Sussmann Result Lemma 22. Let T = 3 and 0 < t < T. Consider two measurable mappings ul, u2 : R -+ [-1, +1] such that:
i) x(t) = fo ul (s)ds = 0; ii) z(t) = fo x2(s)u2(s)ds = 0.
Then fo(1 - x(s))u2(s)ds < t and the equality is true only if ul - 0 and u2 =I on [0, t]. Proof. We set A = fo(1 - x(t))u2(t)dt and h(t) = fo u2(s)ds. Then -t < h(t) < +t. Let a = t - h(t) and /3 = sup,Elo,tl I x(s) [. Since I u2(t) 1< 1, we have It
J0
x(s)u2(s)ds I0,wehave A 1. It is the case physically interesting where the optimal problem is nontrivial.
7.3.1 Projected System We use the invariance of the system by the transformation (x, y, v) p--' (Ax, ay, v), A E IR\{0}. Hence it can be projected onto 1P' x R where IP' = projective space. More precisely, since x never vanishes in the physical space, it becomes in the coordinates (x, z = s, v) dx
dv
dz
= -vx' dt = v - /3v°z + vz dt = h(v)u at and the system
dt = v - /3v°z + vz, dt = h(v)u
(7.9)
is the projected system. Now we can project the differential equation (7.7) onto dz
dt
_
- v - /3vax + vz,
dv
dt
_
v2
az
(7.10)
whose solutions are the projections of the singular trajectories. As observed in Sect. 7.2.4, not every solution of this equation corresponds to a singular
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7 Chemical Batch Reactors
trajectory of the projected system. Indeed if (X, Y) is a planar system the singular trajectories are contained in the set S : det(Y, [X, Y]) = 0
the singular control being given by a feedback on S. By computing for the projected system we get the following lemma.
Lemma 26. The singular trajectories for the projected system (7.10) are contained in S = {(z,v); z(a,3v°-1 - 1) = 1} and the singular control is given by u(z, v) = - h v2QZ . The differential equation describing the evolution on the singular arc is di (t) _ ! ! (1 - a,Ov°-1(t)).
Conjugate points and curvature. Since the singular trajectories corresponding to the system (7.4) of J3 are satisfying assumptions (H1) - (H2) (H3) of Chap. 6, we can apply our algorithm to compute the first conjugate time t1c where a reference singular trajectory ry(.) ceases to be optimal. The variational equation associated to (7.7) is the equation ax = -vbx - xbv az = v(1 - ,Ov°-1)6z + T/i(z, v)6v
by =
v2
az2
(7.11)
6z - 2v 6v az
where t'(z, v) = 1 + z(1 - a,6v°`1) and 0 = 0 is the set S. Let Q = S x 1R and G = {(x, z, v); z = 0}. To compute the conjugate points we must compute the Jacobi field V defined by integrating the previous equation with 6x(0) = bz(O) = 0 and bv(0) = 1. The Jacobi field V is contained in Span[Y, [X, Y]] along y(.) and ti, is the first conjugate time t such that V(t) is collinear to Y = ]R& To simplify our computations we can use the projected system. We have to distinguish two cases.
Lemma 27. A singular trajectory in P n Q, where P is the physical space, is without conjugate point.
Proof. Integrating (7.11) with the initial condition v(0) E IRS and with = 0 we get:
6z=0
6v=exp
2v(s))ds (- az(s) /
I0 \
and clearly 6x(t) < 0 for every t > 0. Therefore we cannot have 6x(tic) = 0. Now we consider the singular trajectories not contained in PnQ. The system (7.7) is written as th(t) = X(w(t)) +u(t)Y(w(t)) where w = (x, z, v). Let us denote by it the projection (x, z, v) H (z, v) and let X'", Y' being respectively the ir-projection of X, Y. Let y(.) be a singular trajectory contained
7.3 Singular Extremals - Curvature - Conjugate Points
179
in P\Q. The vector V(t) can be written A1(t)Y(7Y(t))+.2(t)[X,Y](ry(t)), its projection on the space (z, v) is d7r(V(t)) = )1(t)Y"(ir(7(t))) + A2(t)(Xx,Y"](ir('Y(t)))
and t1, is the first t such that .1z(t) = 0. Since on P\Q, V(t) is collinear to Y(ry(t)) if and only if d7r(V (t)) is collinear to Y'"(ir(ry(t))), we have proved the following lemma.
Lemma 28. The time t1c is the first time such that the solution of the projected variational equation bz = v(1
- /3v°-1)oz + O(z, v)8v (7.12)
by =
vz
azz
oz -
2v
az
by
passing through (0,1) at t = 0 is such that bz(tlc) = 0. 6 Computations. By setting J = 70T v-
previous equation can be written in the canonical form: J + (K o -y) J = 0 where K is the curvature for z 0, s > 0. The switching points for BC-extremals -y+-y- are then (x(t), y(i)) _ The switching times
s(- 2-k, 1 + -k) + o(s). Hence we have proved the following result.
Lemma 29. Assume k # 0. Hence the switching points of BC-extremals y+y_ are located on a smooth curve K whose tangent space at 0 is the line R(- ak, I+ ak ).
0 and k Lemma 30. Assume k -S.. Then a BC-extremal -y+-y- is crossing K if k > 0 or - a < k < 0 and it is reflecting on K if k < - a Proof. At 0, the slope of the tangent to K is -1 - sk and the slope of 'y+ is +1. Hence if k > 0, the slope to K is less than -1 and if k < 0, -1 - -' > 1 if and only if - a < k. Hence the geometric situations are given in Fig. 7.5.
k>O
-a 0 and
d(s) = exps(X f Y)exp(T - s)X(y,(0)) where X = X + uY. Since y, (T) = exp TX (y, (0)) we get
d(s) = exps(X ± Y) exp -sX (y,(T )) and by using the Baker-Campbell-Hausdorff formula we have
d(s) = exp [s(±1 - u)Y + 2s2(X, X ± Y] + o(s2)J I (y, (T)). The curve d(s) can be evaluated using Chen formula
7.4 Time Minimal Synthesis for Planar Systems
187
k 8nZn exp sZ(v) =
n. n=o
(id) (v) + o(sk)
for s sufficiently small where Z is any vector field acting by Lie derivative on
the mappings and id is the identity. If Y = a we have Yn(id) = 0 if n > 1. Since along a singular trajectory Y and [X, Y) are collinear we get d(s) = -f. (T) + (s(±1 - u) + f (s))Y(ye(T)) + o(s2)
where f (s) = o(s). Using the parameter s' = (±1-u)(s+ f (s)), the boundary is given by d : s' +- (-s(T) + o(si2), s' + o(s12)). Hence it is C2, d'(0) is collinear to Y and d"(0) = 0. Higher-order expansions would tell us the nature of its singularity. For
instance if the system is given by x = 1 - y2, y = it and y8 (0) = 0, we get d(s) = (T - 3 , es) with e = f 1. Hence the boundary is the graph of
x=T-1.
Proposition 64. If k 0 0, the optimal syntheses are given by Fig. 7.7. Moreover in the flat case, the optimal synthesis is given as the case k > 0.
k>O
kk o bk adkP)(V0) - Z(vo), where 6 E R and P is any vector field. If y(.) is optimal, then for every b > 0 small enough and any P in the polysystem {X + uY; I u 1< 1} we must have (n,.1(b, P)) < 0 where n is the normal to N at vo outwardly oriented with respect to y(.) Proof. Our demonstration is based on the proof of the maximum principle. We construct along the reference trajectory -y(.) an approximation of the accessibility set. Since the terminal manifold is of codimension one, this approximation has not to be convex to decide about optimality. We proceed as follows. The arc y(.) associated to uo is such that -y(O) = v1, y(T) = vo and we have exp TZ(vi) = vo. Let b, e > 0 small and P be a vector field in { X + uY; I u I< 11. We fix b and we consider the curve
a(e) = exp5ZexprPexp(T- a - E)Z(vl). We have a(0) = vo and a(e) belongs to the accessibility set A(v1,T). Using the Baker-Campbell-Hausdorff formula we can write K ak
a(e) = exp (E(E k>O
kj
adkZ(P) + o(5K) - Z) + o(e)) (vo).
Hence a'(0) = Ek>0 4k ad1Z(P) - Z+o(SK). If (n, a(8, P)) > 0 the reference trajectory is not optimal because we can construct a trajectory reaching N in less time.
7.4.7 Generic Exceptional Case Assumptions. We analyze the case where the assumption Cl is not satisfied and one of the arcs y+ or y_ arriving at 0 (and denoted y+, y°) is tangent to N at 0. We may assume it is y°.. We make the following assumptions: the two vector fields Y and X - Y are not tangent to N and the contact of y+ with N is minimum.
7.4 Time Minimal Synthesis for Planar Systems
Model. We choose near 0 a coordinate system such that Y = the curve s i- (0, s). The system can be written
31-
189
and N is
x(t) = X1(x(t), y(t)) + u(t), y(t) = X2(x(t), y(t))
with X1(0) = -1,
-a 0 0 and X2(0) 36 0. Moreover we can assume
X2(0) > 0.
Proposition 65. Under the previous assumptions and normalizations, the optimal synthesis is represented on Fig. 7.8.
a>O
a 0. The arc ry+ is a BC-extremal with -n = (1, 0) as adjoint variable at 0. We prove it is not optimal using Lemma 32. The normal to N at 0 oriented as in the lemma is n. We have Z = X + Y
ands we take P = X - Y. We get a(8, P) = -2Y(0) + o(1). Hence for 6 small, (n, -2Y(0) + o(1)) > 0. This proves the assertion. Moreover a simple computation shows the following. Assume we are at a distance a from N in
the domain x > 0. The time to reach the target N is of order f along ry+ and of order c along 'y (the contacts are different). In the domain x < 0, the optimal control is +1, the value function v F- T'(v) being not C°. When a < 0, the analysis is similar but the target N is not accessible from the points in the sector x < 0, above ry+.
7.4.8 Generic Flat Exceptional Case Assumptions and normalizations. The terminal manifold N is identified to the curve s I- (0, s) and Y is assumed everywhere tangent to N. We
190
7 Chemical Batch Reactors
assume X tangent to N at 0. We suppose that Y, X ± Y are nonzero at 0 and that [X, Y] is not tangent to N at 0. Then locally the system can be written ax + by + o(x,y)
y=X2+u with b = (n, [X, Y](0)) 0 0. We may take the following normalizations: a =
O,b=land1+X2(0)>0. Proposition 66. Under the previous norm.alizations, the optimal synthesis is given by the Fig. 7.9.
X2(0)>1
X2(0) 1 the target is not accessible from the domain located above y° in the domain x > 0. In the case X2(0) < 1 the origin is locally controllable.
7.5 Global Time Minimal Synthesis
191
7.5 Global Time Minimal Synthesis 7.5.1 Preliminaries Our objective is to reach the target N = {(z, v); z = d, d > 0 fixed} in minimum time. The system is t
dv
(t) = MO - Qv'(t)z(t) + v(t)z(t))'
(t) = h(v(t))u(t)
where u E [u_,u+], u_ < 0 < u+. We study the nontrivial case a > 1. Moreover we make the following assumption on the parameters set:
Al>p
,
0. We represent on Fig. 7.11 the phase portraits.
u=u+
Fig. 7.11.
7.5 Global Time Minimal Synthesis
193
7.5.4 Optimal Synthesis in the Neighborhood of N The analysis of Sect. 7.4 allows us to analyze the optimal synthesis in the neighborhood of N. It is represented on Fig. 7.12. We have two situations of codimension one:
- At P1, the hyperbolic singular arc meets the target and the problem is flat. The optimal synthesis is given by Proposition 63;
- Let P3 = E f P. Then in these points we are in the flat exceptional case and the optimal synthesis is deduced from Proposition 65.
Fig. 7.12.
Remark 20. In this problem z = d has to be considered as a varying parameter. Hence in our analysis we must also consider the situation where the singular control is saturating for z = d. This situation can be easily analyzed as in Sect. 7.4, see for instance [28]. In our analysis it will be a consequence of the global synthesis near P2, see also [98].
7.5.5 Switching Rules Switching function. One of the main problem to be solved in order to compute the global synthesis is to find global bounds on the number of switchings. In our case it is related to pseudo convexity properties of the switching functions. Let p = (PI, P2) be the adjoint vector for the system. We have 171 = plv(,3va-1 - 1)
02 =
pi(a/va-iz
- 1 - z) - p2h'(v)u
7 Chemical Batch Reactors
194
and the switching function P(t) = p2(t)h(v(t)) satisfies the following equation
fi(t) = plh(v)[z(af3va-1 - 1) - 1] fi(t) = p1h(v)[(3v°(a - 1) +ua(a - 1)pv°-2zh(v)] + uh`(v)-h(t). The set fi(t) = 0 plays an important role when the switching points are computed. If pl is nonzero, its projection on the state space is the curve S. The adjoint variable corresponding to a BC-extremal defined on [0, T] must satisfy p(T) E ]R(1, 0). Moreover from the analysis of Sect. 7.4, it can be oriented according to the principle p(T) = (1, 0). The switching rules are the consequence of the following lemmas. Lemma 33. If O(0) = P(T) = 0, then a smooth BC-extremal y(.) meets the set S in a time 0 < t < T. Proof. We have pi (T) = 1 and pl has a constant sign on [0, T], hence pi > 0. Since iP(0) = 45(T) = 0, using Rolle's theorem there exists t E]0, T[ such that z(a,l3va-1 -1) = 0. fi(t) = 0. Since pi is not vanishing on [0, T], we have at t: Hence y(t) E S.
Lemma 34. If (-y, p, u) is a BC-extremal with u = u+ defined on [0, T] and 4i(T) = 0 then P(0) is nonzero.
Proof. Let cp(t) = h a evaluated along the given extremal. We have cp = pi (a - 1)(I3v" + u+a13v°-2zh(v)). Hence signcp = sign pl > 0 and gyp(.) is a strictly increasing function. Since
(y, p, u+) is an extremal and O(T) = 0, we must have 4t(T) < 0. Hence V(0) < V(T) < 0. Let us assume 45(0) = 0, then again we must have 4t(t) > 0.
This contradicts (p(0) = h o < 0. Proposition 67. Every optimal trajectory is of the form y+y_y, where each arc of the sequence can be empty.
Proof. The proof follows from the two previous lemmas and from the following remark. An optimal arc of the form y,'y_ where -y, and y_ are not empty
is not possible because the subarc y_ must meet twice the set S, which is absurd.
7.5.6 Optimal Synthesis Using our analysis it is straightforward to construct the synthesis represented
on Fig. 7.13. The switching curve W is the union of y, with a curve WThe curve W_ is the set of switching points of the arcs y+y_. This curve is containing the point P2 as a consequence of the saturation of the singular control at P2.
7.6 State Constraints due to the Temperature
195
Fig. 7.13.
7.6 State Constraints due to the Temperature In the previous synthesis, we did not take into account constraints of the
temperature: 0 < Tm < T < TM which imply constraints on the state v = Al exp -b) of the form v E vM]. It can be done in general using the maximum principle for systems with state constraints, see (91). Here we present an analysis valid for planar systems and using the clock-form.
7.6.1 Preliminaries The system is written as
z(t) = f(z(t),v(t)), v(t) = h(v(t))u(t) with u(t) E [u_, u+], u_ < 0 < u+. We denote respectively Lm and LM the vertical lines v = vm and v = vM in the plane (v, z). The control which allows to stay in these lines is the control u =_ 0. We note E the set z = 0. Let 7b(.)
be a trajectory located in the boundary. It is oriented towards the target if z > 0 and with the inverse orientation if z < 0, see Fig. 7.14. A crucial step is to understand the time optimality status of a boundary trajectory -yb(.) (it is one of the main aspect of the maximum principle with state constraints). For this we can use the clock form w defined for a planar system (X, Y) by the relations: (w, X) = 1, (w, Y) = 0. Computing we get
f (zzv)
and dw = - fa(y}
v)
of dv n dz,
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7 Chemical Batch Reactors
Fig. 7.14.
Fig. 7.15. the two form dw vanishes on the singular arc S and the sign of dw in the state domain is represented on Fig. 7.15. Using this clock form we can analyze the optimality status of a boundary trajectory for the fixed extremities problems
and state constraints and deduce the optimal synthesis with N as terminal manifold. We present the optimal synthesis when the constraints are given by Fig. 7.16, the others situations could be deduced from this analysis. We note Bl = (z, vM) the intersection of the line: v = vM with the switching
7.6 State Constraints due to the Temperature
197
V
Fig. 7.16.
curve W_ computed in Sect. 7.5. We design by B2 and B3 the respective intersections of S and E with the line v = vM. Lemma 35. The optimality of a boundary trajectory for the fixed extremities problem is represented on Fig. 7.17.
Z
N
P1
I-- non optimal
,,l S non optimal
B3
dw >O
non optimal
-- ---------dw d. In the concentrations space n = (0, 1, 0) and in the canonical coordinates we have n = (d, exp x, 0). The main steps of the analysis are the following. First we stratify the target N by computing the optimal control feedback at the terminal points. Then we must compute the optimal synthesis near the terminal manifold. This results are the extension of the analysis of Sect. 7.4 for planar system. This extension is not straightforward and the results complex. It has justified several articles, see for instance [28] and [27]. Finally we generalize the results of Sect. 7.5 concerning a glgbal bounds on the number of switchings. These three steps allows to compute the optimal synthesis using numerical simulations, where the local analysis near N plays the role of the analysis of the singularities.
7.7.2 Stratification of N by the Optimal Feedback Synthesis We write the system as (X, Y) and we have two sets where the optimal synthesis is not straightforward.
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7 Chemical Batch Reactors
Exceptional set. It is the set of points where X is tangent to N. It is given by the relation E : { (n, X) = 0} fl N. Computing in the canonical coordinates we get E : 1 - ,Ov'-1 z = 0, z exp x = d.
The optimal synthesis near E is complex and we can give a geometric explanation. Consider the system written in the space of concentrations. The terminal manifolds is y = d, where d < 1. If we consider the system we have th + y < 0. Hence if x(0) < d, the point (x(0), 0) cannot be sent to N. Hence the set of points which can be steered on the target has a boundary.
Singular set. The singular trajectories corresponding to BC-extremals are located on the set S : {(n, [X, Y]) = 0} n N. Computing in the canonical coordinates we get S : a)3v°-1 z - 1 = 0, z exp x = d.
If a > 1, the set S in the physical space is a simple smooth curve transverse to Y. The singular feedback is given by u = - ti v and its restriction has the following properties: it is strictly decreasing from 0- to -oo when v goes from 0+ to Al . There exists an unique point denoted P0 where the singular control is saturating: u(Po) = u_. Moreover we know from Sect. 7.3 that all the singular trajectories are hyperbolic. Hence using the terminology of Chap. 3 if P E S, then P is lifted into the point (n, P) of N' contained in E : (p, Y) = 0 and we have
a
- if u(P) E]u_, u+[, the point (n, P) is an hyperbolic switching point; - if u(P) V [u_, u+] the point (n, P) is parabolic. At Po, the singular control is saturating. We represent on Fig. 7.19 all these results, the target N being identified to the plane (v, z).
7.7.3 Orientation Principle Outside E, the adjoint vector at the terminal point can be taken as n, according to the maximum principle. At a point of E, the Hamiltonian H is zero and we need the following lemma to take p as n. Lemma 36. Let y+ or y_ be a BC-extremal defined on [0, T] and tangent to N at xo but not included in N. Let n be the normal to N oriented towards
the domain not containing y+ or -y-. Assume that (xo, n) is an ordinary ching point, i.e. (n, Y(xo)) = 0 and (n, [X, Y](xo)) # 0. Then if y+ or y_ is optimal, the adjoint vector at x0 is necessary n.
7.7 The Problem in Dimension 3
201
S
E:=O
hyperbolic
1. Then each optimal control has at most two switchings and each optimal trajectory is of the form ry+y_rya where each are of the sequence can be empty.
The proof follows from a sequence of lemmas. If p = (Pi, p2, p3) is the adjoint vector in the canonical coordinates we have the equations
pl =0 P2 = P2(3va - v)
03 = P1 +
p2(01/3v°-' x - 1 - z) - P3h'(v)u
Hence we get.
Lemma 37. The ajoint vector p has the following properties. The component p, is a first integral: pi (t) = Pl (0); p2(t) = (expf(/3v0 v)ds) p2(0) and either p2(t) __ 0 or its sign is constant. In particular along a singular arc we have p2(t) > 0 and p, (t) > 0.
-
202
7 Chemical Batch Reactors
Proof. The first assertion is a direct consequence of the equations. For the second part we use the previous orientation principle. Along an optimal arc we can choose at the terminal point p(t) = n = (d, exp x, 0).
Lemma 38. The switching function P(t) = (p, Y) defined on IT, 0], T < 0 along a BC-extremal satisfies the equations:
1. 4 = h(v)p3, 45(0) = 0 since the problem is flat. 2.
= h(v)[pi +p2(af3v°-1 z - 1 - z)], = h'fiu + hp2[(a - 1)/3v° + ca(a - 1)/3v°-2huz], a.e.
3. If tp = n we have cp = p2(a - 1)/3v°-2[v2 + uazh(v)], a.e.
Lemma 39. Let y, be a singular subarc of an optimal trajectory. Then y, is contained in the sector aj3v°-1 z -1- z < 0 and satisfies the condition z > 0.
Proof. By definition fi(t) = 0 along a singular arc and P1, P2 > 0. Hence z - 1 - z < 0 along the arc y,. Since a > 1, in the we must have physical space we have 0 > a,33v°- 1 z - I - z > ,3va-1 z - 1 - z and with a,3v°-1
z=v(1+z-/3v°-1z)wegetz>0. Lemma 40. Let y_ be an optimal arc defined on IT, 0]. Assume that 0 and z - 1 - z > 0. Hence. -y- must T are switching times and -y_ (0) E meet the locus: a0va-1z - 1 - z = 0. a/3v°-1
Proof. Since 0(0) = O(T) = 0 there exists t E IT, 01 such that fis(t) = 0. z - 1 - z < 0. By assumption Since p1 i p2 > 0, at time t we have af3v°-1
y_ (0) E a/3va-1 z - 1 - z > 0. The assertion follows. Lemma 41. Let 'y_ be an optimal arc defined on IT, 0]. Assume that 0, T are switching times and moreover i'(T) = 0. Then y_ must meet the saturation
setC: u=u_ defined byu_2V Proof. Since P(0) = O(T) = 0 there exists t E]T, 0[ such that rh(t) = 0. satisfies tp(t) = V(T) _ Moreover by assumption (T) = 0. Hence c' = 0 and there exists t' E]T, t[ such that V(t') = 0 i.e. p2(a - 1),Qv°-2(v2 + u_azh(v)) = 0. Since p2 0 0 we have v2 +u_azh(v) = 0, i.e. u = u_. Lemma 42. Let -Y+ be an optimal arc defined on IT, 0]. Assume that 0 is a switching time. Then T is not a switching time. Proof. See lemma 34.
7.7 The Problem in Dimension 3
203
Proof (Proof of the theorem 24). We can project a trajectory y(.) in the space (z, v) and the projection is denoted We denote by G' the set i = 0 i.e.
1 - $v"-lz + z = 0 and by C the projection of the saturating set u = u_ which is the graph z = - .u h v and is a strictly increasing function of v. First we consider the case AI < J3T where i > 0 in the physical space. To prove our result it is sufficient to prove that no subarc 'Y,-y+ where y y_ are non empty is optimal. If such an arc is optimal, using Lemma 41 the arc y_ must meet the saturation set C : u = u_. If it is the case the piece of y_ before the intersection is contained in the sector u < u_ and the connection with y, is in this sector. This is not possible because in this sector the singular control is not admissible. The case Al > ATh can be analyzed in a similar way.
7.7.5 Local Classification near the Target Hyperbolic case. Under generic assumptions the system can be written as th = 1 + a(x)z2 + 2b(x)yz + c(x)y2 + R1 y = d(x)y + e(0)z + R3 z = (u - fi(x)) + f(x)y + g(0)z + R3
where R1 (resp. R2, R3) are terms of order > 3 (resp. > 2) with respect to (y, z). In this semi-normal form Y is identified to , the set L where [X, Y]
is tangent to N to the axis Oy and the target N to the plane x = 0. The reference singular arc y, is identified to the curve t -4 (t, 0, 0). Moreover [X,Y]I,. = [X,Y](0) In the hyperbolic case we have a(0) < 0 and u(0) E] - 1,+1[. Using the previous normalization we can construct the optimal synthesis near an hyperbolic point and we get the following result.
Proposition 69. The BC-arcs y+ and y_ are not cutting themselves near 0. The optimal synthesis is C°-equivalent to the synthesis using the model x = 1 + a(0)z2, y = 0, i = (u - v.) where each plane y = constant is an invariant leaf and in each leaf the synthesis is represented on Fig. 7.20.
Parabolic case. At a parabolic point P the singular feedback control is such that u(P) V [-1,+1]. We can construct the optimal synthesis as previously.
There exists a CO invariant foliation and in each leaf y = constant, the synthesis is given by Fig. 7.21.
Saturated case. Near the point P° where 11(P°) = u_, the optimal synthesis is more complicated and is described in [28]. The geometric situation is the following. As in the previous situations, there exists a CO invariant foliation
204
7 Chemical Batch Reactors
Nn y=constant
Fig. 7.20.
N l1 y=constant
Fig. 7.21.
whose leaves are given by y = constant. The point Po represents a bifurcation between the hyperbolic and the parabolic situation and the synthesis is represented on Fig. 7.22. In the leaves associated to a saturated point or a parabolic point the synthesis are equivalent, every optimal trajectory is of the form -f+-y- and the switching points are located on W+. In the leaves associated to an hyperbolic point the situation is more complex. Every optimal trajectory has at most two switchings and is of the form 'y+ry_-y, where each arc of the sequence can be empty. The switching locus is the union of these curves denoted W+, W, and ry,. The set W+ is the switching locus of optimal policy of the form ry+'Y_ and W. is the set of first switching points of optimal trajectories of the form -y+-y--y,.
7.7 The Problem in Dimension 3
205
N A y=constant
;p
hyperbolic
saturated
parabolic
Fig. 7.22. Remark 21. In the theorem 24 we prove that every global optimal trajectory has at most two switchings and is of the form -y+-y--y.. In the previous analysis we show that near the saturating point Po we have such optimal trajectories.
Generic exceptional case. We can easily compute the optimal synthesis in a generic point of E where x is tangent to N. A semi-normal form is constructed as follows. The point is taken at 0, the target N is identified to the plane x = 0. Moreover we can assume that the locus of tangent points is a simple curve transverse to Y and it is identified to the axis Oy. We choose Y = YZ- and moreover we can assume [X,Y](0) = M-10. The system can be written
x=z+R b+o(1) c + u + 0(1)
where R is of order > 2, b and c are constants c + 1, c - 1 nonzero and we can assume c + 1 > 0. There exists a C°-invariant foliation identified to y = constant and the optimal synthesis in each leaf is as the planar case and is described on Fig. 7.23. In our problem there is a point of codimension 2 where the condition c # 1 is not satisfied. The analysis is complicated and is presented in [27]. At such a point the curve 7_ has a contact of order 2 with the target.
206
7 Chemical Batch Reactors
c1
Fig. 7.23.
7.7.6 Focal Points To deal with the time minimal control problem where the terminal manifold N is not necessary a point we must generalize the concept of conjugate point along an hyperbolic singular arc into the concept of focal point. The extension is straightforward and is the following in our problem. Let 'y, defined on [T, 0] be an hyperbolic singular arc and assume that the singular control it satisfied the bounds u_ < u < u+ (i.e. is not saturating). Then the local synthesis described in proposition 69 can be extended in a tubular neighborhood of y, up to a point t1 f called the first focal point. It is defined as follows. Let W (O) be a tangent vector at y, (0) to the curve a where [X, Y] is tangent to the manifold N and let W (t) be a solution on [T, 01 of the variational equation (7.11) obtained by linearizing the singular flow x = S(x) along the singular trajectories. The first local point t1 f is defined as follows.
Definition 79. Let T < t1 f < 0 be the first time t < 0 such that det(W(t), Y(ys(t)), S(y9(t))) = 0. Then y,(t1f) is called the first focal point along y,.
In the C" case, W(t) as the Jacobi field V(t) associated to the concept of conjugate point is contained in Span{Y, [X, Y1}1,y,.
Its numerical computation is straightforward.
7.7 The Problem in Dimension 3
207
Notes and Sources All the results presented in this chapter are coming from a series of articles. In [24], the algorithm for computing the conjugate and focal points are given and the planar case is solved. The generic local classification near the terminal manifold is intrincated and is presented in two articles [28] and [75] which deals with the exceptional case. This case has a connection with subRiemannian-geometry which shall be explained in Chap. 9. The problem in dimension 3 for batch reactors, where the reactions are not necessary of firstorder is analyzed in full details in [27]. Also this article contains numerical simulations which solve the problem.
8 Generic Properties of Singular Trajectories
8.1 Introduction and Notations Let M be a o-compact COO manifold of dimension d > 3. Consider on M a single-input control system i(t) = Fo(x(t)) + u(t)F1(x(t)) where Fo, F1 are C°O-vector fields on M and the set of admissible controls U is the set of locally integrable mappings u : (0, Tu] - IR, Tu > 0. The aim of this chapter is to show that for an open dense subset of the set of pairs of vector fields endowed with the C°°-Whitney topology, all the singular trajectories are with minimal order and normal. It is based on the article [25]. Also we analyze in dimension 3 the singularities of the vector field whose solutions are the singular trajectories of minimal order. This study was initialized in (22] for quadratic systems connected to the attitude control problem of a rigid spacecraft. This analysis is related to two important control problems: first of all the feedback classification, see Chap. 4; secondly the existence of broken singular minimizers in optimal control e.g. in sub-Riemannian geometry.
Before to state our results we must introduce very precise notations and slighty extend the concept of singular trajectories to deal with the singularities.
We shall denote by M a c-compact C°° manifold of dimension d > 3. We shall use the following notations:
- TM: tangent space of M, TmM : tangent space at m E M. - T'M: cotangent space of M, T,, M: cotangent space at m E M. - The null section of T'M is denoted by 0 and (T*M)o = T'M\0, JPT'M: projectivized cotangent space, i.e. JPT'M = T*M\1R`, [z]: class of z in for any integer N > 1, JNTM: space of all N-jets of COO-vector fields: JIM : JNTM - M: canonical projection. - VF(M): vector space of all C°O-vector fields endowed with the Whitney topology.
- E xM F: fiber product of two fiber spaces (E, JIE, M) and (F, 17,F, M) on M. - H: given any COO-function defined on an open subset (1 C T' M, H will denote the Hamiltonian vector field defined by H on .R.
210
8 Generic Properties of Singular Trajectories
- (HI, H2 }: given any two C°°-functions on 17, {H1, H2 } will denote their Poisson bracket: { H1, H2 } = dH1(H2) -
- SpanA: if A is a subset of the vector space V, it is the vector subspace generated by A. To each couple (FO, F1) of C°°-vector fields on M, we associate the control system:
dt (t) = Fo(x(t)) + u(t)FI (x(t)),
u(t) E R.
(8.1)
The study of time minimal trajectories leads to the consideration of the following solutions of Pontryagin's maximum principle (z, u) : J -+ T'M x lR, J interval [TI, T2], T, < T2 and 1. z is an absolutely continuous curve and u(.) is locally Lebesgue integrable;
2. z(t) # 0 (0 = null section) for all t E J;
3. T, (t) =Ho (z(t)) + u(t) Hi (z(t)), for a.e. t E J where Hi : T`M R,i = 1, 2, Hi(z) = (Fi (17T. M (z)), z) (canonical Hamiltonian lift of Fi); vH1(z(t))}.
4. for a.e. t E J, Ho(z(t)) + u(t)H1(z(t)) =
We observe that 4 is equivalent to Ho(z(t)) = 0 for a.e. t E J, but since z(t) is C°, this is equivalent to 4'. Ho(z(t)) = 0, Vt E J. t
Definition 80. A curve (z, u)
:J
T*M x R satisfying the conditions
1.,2.,3.,4'. will be called a singular extremal and (IIT M (z), u) a singular trajectory. This definition is only a slight extension of the concept of singular trajectories in the preceedings chapters, valid because system (8.1) is affine. It will allow to include the singularities concerning singular trajectories. Next we give a general algorithm to compute the singular algorithm in
the single-input case.
8.2 Determination of the Singular Extremals with Minimal Order Let (z, u) : J -- T'M x R be a singular extremal. Using the chain rule and condition 4'. we get
. y (Ho(z(t))) = dH1(z(t))(Ho (z(t)) + u(t) Hi (z(t))) dt d
0
for almost every t E J. Since t i- z(t) is C°, this implies the relation 0 = {Hl, Ho}(z(t)). Using the chain rule again we get
8.4 Geometric Interpretation and the General Concept of Order
211
0 = dt {H1, Ho}(z(t)) = {{Hl, Ho}, Ho}(z(t)) + u(t){{H1, Ho}, H1 }(z(t))
for almost every t E J.
Definition 81. For any singular extremal (z, u) : J -. T*M x R, R(z, u) will denote the set {t E J; {IH1, Ho, }, Hl}(z(t)) 0}. A singular extremal (z, u) : J - T'M x R is called of minimal order if R(z, u) is dense in J. The following propositions are an immediate consequence of the above considerations.
Proposition 70. If (z, u) : J
T*M x IR is a singular extremal and 1Z(z, u)
is not empty, then 1. z is restricted to R(z, u) is C°°; z t for all t E R(x, u); 2. a(t) =Ho (z(t)) + (JH1,HoJ,H1)(z(t)) 3. u(t) = Ho,H1 ,Ho zztt for almost every t E R(x, u). JJff1,Ho),H,
Proposition 71. 1. Let (Fo, F1) E VF(M) x VF(M) be a pair such that the open subset 17 of all z E (T'M)o such that {{Ho,Hl},H1}(z) = 0 is not empty. If H: 17 -+ R is the vector field Ho + {Hj,Ho,HAH1, then any trajectory of H starting at t = 0 is a minimal order singular extremal
of (Fo,Fi) 2. There is an open dense subset of VF(M) x VF(M) such that for any couple (F0, F1) in that subset the set 17 is open dense in T* M.
8.3 Statement of the First Generic Property Theorem 25. There exists an open dense subset G of VF(M) x VF(M) such that for any couple (Fo, F1) E G the associated control system (8.1) has only minimal order singular extremals.
8.4 Geometric Interpretation and the General Concept of Order We shall give a geometric justification of the above theorem; moreover it will
lead to define the concept of order to compute in general the singular extremals in the single-input case. If (z, u) is not of minimal order then there exists a non empty open subinterval J' on which {{H1, Ho}, Hl}(z(t)) = 0.
Combining with the relation
8 Generic Properties of Singular Trajectories
212
{{Hl, Ho}, Ho}(z(t)) + u(t){{H1i Ho}, Hi}(z(t)) = 0 it shows that there exists a non empty open subinterval J" on which we have {{H1, Ho}, Hl }(z(t)) = {{Hi, Ho}, Ho}(z(t)) = 0.
Differentiating both relations with respect to t we get a.e. on J" {{{H1i Ho}, HI), Ho}(z(t)) + u(t){{{Hi, Ho}, H1}, Hl}(z(t)) = 0, {{{Hi, Ho), Ho}, Ho}(z(t)) + u(t){{{Hi, Ho}, Ho}, Hi}(z(t)) = 0. This is a compatibility relation which means geometrically the following: there exists a singular extremal arc contained in both surfaces: {{H1,Ho}, Ho} = 0 and {{H1iHo},H1} = 0.
An instant of reflexion shows that this property cannot be generic but it needs a complete proof.
Also the previous computation gives us the general stratification to computed the singular arcs at any order. Notation. If X, Y E VF(M), the Lie bracket is computed with the convention [X, YJ (m) = - ay(m)X(m) and the adjoint mapping is defined by adX(Y) = [X, Y]. For any multi-index a E {0,1 }", a = (a1, , a"), length of a: I a 1= n, we note I a 10= card{i; ai = 01, a I I= card{i; a; = I}. The Lie bracket denoted by Fe0 is defined inductively Fan]. Similarly the function HQ : T'M -+ R is defined by: F. = [F(01,...
inductively by H0 = {H(01,...,0 _,l, H,,,.). Observe that using the relation {Ho,Hi}(z) = ([Fo,F1](UT.M(z)),z) we have H. = Definition 82. The singulat arc (z, u) defined on the open non empty interval J is said of order q > 2 if 1. H0(z(t)) = 0 for all a = (1, i2, , ik) E {0,1}k+1, I a 1< q. 2. Then exists 3 = (1, ii, -, iq_1 i 1) E {0,1}9+1 such that
{t E J; Ha(z(t)) # 0) is dense in J. Algorithm This gives us the stratification to compute the singular arcs of any order
- Order 2 H1 = H1o = 0, H1o1
9& 0,
u = -H.
- Order 3: Hi = Hio = H1o1 = H1oo = 0, 1on or H1o01 # 0 and u is solution a.e. of both equations Hio1o + uH1o11 = 0 H1ooo + uH1oo1 = 0 etc...
8.5 Proof of Theorem 25
213
8.5 Proof of Theorem 25 To prove Theorem 25 we are going to define for each N sufficiently large,
a bad set B(N) in JNTM X JNTM having the following property: any couple (Fo,FI) E VF(M) x VF(M) such that (jNFo, js Fl) 0 B(N) for all x E M has only singular extremals of minimal order. Then we shall show using transversality theory that the set G of all couples (Fo, FI) E VF(M) x VF(M) such that (jz Fo, jz FI) 0 B(N) for all x E M is open dense in VF(M) x VF(M). To construct the stratification of the bad set we have to analyze two cases: the points x where FO, F1 are linearly dependent and the points x where FO, F1 are linealy independent.
Construction of the bad set. Definition 83. For N > 2d - 1, let Ba(N) be the subset of JNTM x M JNTM of all couples (jx Fo, j2 F1) such that
dim Span{adFo(FI)(x); 0 < i < 2d - 1} < d.
Definition 84. 1. For N > 1, BB (N) is the subset of JNTM x M JNTM of all couples (jx Fo, j. F1) such that dim Span {Fo(x), F1 (x), [Fo, Fi](x)} < 1.
2. For N > 2, let Bi'(N) be the subset of JNTM xM JNTM x R of all triples (jNF0, jx F1, a) such that a) Fi(x) 34 0;
b) Fo(x) = aFi(x),
c) dim Span{adGa(Fi)(x),0 < i < d- 1,[[FoF1],Fi](x)} < d where
Ga=Fo-aF1 S. Denote by Bj'(N)the canonical projection of BI"(N) onto JNTM xM JNTM. 4. B1(N) = Bi(N) U BI" (N).
Definition 85. Using the notations 8.4 define the following. (i) For any integers c >0, any
a
=1
a 0 101"-2) any integer N > n + c - 1, let
B(N, a, c, 0) (resp. B(N, a, c, 1)) be the subset of JNTM X M JNTM x m PT*M of all triples (jz Fo, jx F1i [z]) such that 1. Fo(x), F1 (x) are linearly independent;
2. HQ0(z) 0 0, Ho, (z) 0 0;
8 Generic Properties of Singular Trajectories
214
3. 9(ZQ)kHlo.,_1(z) = 0 (resp. 9(ZQ)kH101.,_2(z) = 0 for 0 < k< c, where Za is the vector field H01 Ho -H00 Hi on T'M, and 9 denotes the Lie derivative.
(ii) B(N, a, c, a) (a = 0, 1) will denote the canonical projection of B(N, a, c,
a) onto JNTM xM J^'TM. Definition 86. We set B(N) = Ba(N) u B1(N) u (uB(N, a, 2d, a));
3I 2 and since Fo(Y(t)) = a(t) F1 (Y(t)), F1 (2(t)) and [Fo, F1](T(t)) are linearly
independent for all t E J\S. Suppose that S 54 0. Then the open set J\S contains an interval ]a, p[=
{t; a < t < Q}, where either a E S or 3 E S. Assume that a E S (p E S is similar). Since a(t) = a(t) + u(t) = 0 for a.e. t E]a, 8[, a is constant on ]a,,3[ and -u(t) = -a for a.e. t E]a,1[. Hence x(t) = 0 for a.e. So Y(t) = xo for all t E]a, Q[. This leads to the contradiction t 0 = F1(x(t)) = limt.a,gEi«,13iFi(Y(t)) = F1(xo) 54 0. Hence S = 0 and Y(t) = xo for all t E J. This proves 1.
8.5 Proof of Theorem 25
215
2. Let ('z, u) : J -' TOM x R be a singular extremal such that z(t) E T0M for all t E J. The assumption of Lemma 43 implies that dim Span {Fo(xo), Fo(xo), [Fo, Fi](xo)} > 2.
If Fo(xo) = 0, then Fo(xo) # 0 but we have for a.e. t E J : 0 = _X(t) _ Fo(xo). We have a contradiction. Since 0 = i(t) = Fo(xo) + u(t)F1(xo) for a.e. t E J, there exists an a E IR such that Fo(xo) = aFi(xo) and
u(t) = -a for a.e. t E J.
Let C = Pb - aF1
:
G(xo) = 0 and set H(z) =
By definition z(t) =H (I(t)) for all t E J and Hi (z(t)) = 0. Hence 0 for all t E J adkH(Hi)(z(t)) = and all k E N. Since G(xo) = 0, Span{adcG(F1)(xo); k E N}= Span{adkG(Fi)(xo),0 < k < d - 1}. By assumption xo is a singular trajectory, hence this space is at least of codimension one. Since (j o Fo, j o Fl) V BI (N), it is exactly of codimension one, and moreover ([Fi, [Fo, F1]](xo), z(t)) 0 0. Therefore z(t) belongs to a line l E TT0M and (7(t),11(t)) is of minimal order. By definition z(t) is solution of a linear system and since G(xo) = 0, it is autonomous. Hence 2 is proved.
Proposition 72. Let a couple (Fo, F1) E VF(M) x VF(M) which satisfies the condition: there exists an integer N such that for all x E M (is Fo,j2 Fi) ¢ B(N). Then the control system associated to (Fo,FI) has only minimal order singular extremals.
TOM x ]R is a singular extremal not J of minimal order. This shows that there exists an open subinterval J0 of J,Jo not empty, such that {{Ho,H1},H1}(z(t)) = 0. Then the closed set has an empty inIt E Jo; dim{Fo(1(t)),Fi(Y(t))} < 1}, Y = terior: otherwise it would contain an open non empty interval Jot C Jo. Then Lemma 43 applies to the restriction (z', u') of ('z,-u) to Jol. But since Proof. Assume that (11,'11)
:
{I Ho, Hi }, H1 } (z') = 0 we get a contradiction. Replacing J by an open non empty subinterval we can assume that for alll t E J:
1. {{Ho, Hi}, Hi}(z(t)) = 0; 2. dim Span {Fo(r(t)), Fi (Y(t))} = 2.
Since If Ho,Hi},H1}(z) =0, then {{Ho, Hi }, Ho}(z) =
d
({Ho, Hi }(-z)) = 0.
We claim that there exists a multi-index a E {0,1}n, a = (a1, al = 1 such that
(i) 3 0.
It is easily seen that this is equivalent to (B(Za)k(Hy))(z) = 0 for all k > 0, and ?a and Za are collinear. This shows since adk9-la(Hy) = that for all t E J, (j=ctiFo, j(t)F1.[z(t)]) E B(N, a, c, a), a = 0 if a 76 and a = 1 if a = 10"-1, where x(t) = and [z(t)] denotes the class of z(t) in IPT*M. 101-1
Now we shall compute the codimension of the bad set. It can be done by using semi-algebraic sets, see [17].
8.5 Proof of Theorem 25
217
8.5.1 Partially Algebraic and Semi-Algebraic Fiber Bundles Definition 87. A VP bundle on M is a locally trivial fiber bundle on M
,W 1P(Wn), where V, WI, whose typival fiber is a product V x IP(W1) x , IP(Wn) the associated proare finite dimensional vector spaces, IP(W1), x jective space and whose structural group is Aut(V) x Aut(IP(WI)) x -
Aut(IP(Wn)) (Aut(V) = GL(V), Aut(IP(W;)) = GL(Wi)\1R').
Definition 88. A partially algebraic (resp. semi-algebraic) subbundle of a VP bundle on M is a locally trivial subbundle whose typical fiber A is an algebraic (resp. semi-algebraic) subset of the typical fiber V x 1P(W1) x . . . 1P(Wn )
of the VP bundle.
Lemma 44.
(i) JNTM xM JNTM, JNTM xM JNTM x R, JNTM XM JNTM xM 1PT*M are VP bundles on M whose typical fibers are respectively P(d, N)
x P(d, N) x, P(d, N) x P(d, N) x R, P(d, N) x P(d, N) x IP(Rd) where P(d, N) denotes the set o f all polynomial mappings P = (P', , Pd) Rd such that degPt < N for 1 < i < d. Rd (ii) Ba(N), Bi(N), b" (N), B(N, a, c, a) are partially algebraic (for the first two) and semi-algebraic (for the last two), subbundles of the VP bundles JNTM xM JNTM, JNTM XM JNTM, JNTM XM JNTM x R,
JNTM XM JNTM xM IPT'M. Their typical fibers .F.(N), .F,(N), f' (N), P(N, a, c, a) can be described as follows: X. (N) = {(Po, PI); dim Span {{adk Po (PI) (0), 0 < k < 2d -1} < d}; P, (N) = {(Po, Pi); dim Span {Po(0), P1(0), [Pi, Po](0)} < 1}.
XI" (N) is the set of all triples (Po, P1, a) E P(d, N)2 x R such that 1. P1(0) 0 0, Po(0) = aP1(0);
2. dim Span{adkRa(P1)(0),0 < k < d- 1} and [[Po,P1J,Pij(0)} < d where Ra = Po - aP1. For the definition of the last fiber we shall use the following notations:
for a E {0,1}n, a = (al,
, an) the function Ha : Rd x Rd* --I' R is defined inductively as follows: if i = 0 or 1, Hi (x, C) (Pi (x), t). If a = (al, , an), Ha = {H.Is...san-I, Han } where {, } denotes the
Poisson bracket.
For any integer c > 0, any a E {0,1 }n, n > 3, a = (al, , an), al = 1, N > n + c - 1, .F(N, a, c, a) is the set of all triples (Po, PI, [l;]) in P(d, N)2 x IP(Rd) such that (i) Po(0),P1(0) are linearly independent; ( i i ) Hao(0, e) 0 0, Hal (0, E) -A 0;
218
8 Generic Properties of Singular Trajectories
(iii) (O(Z.)k(H.y))(0,1;) = 0, 0 < k < c; where ry = 10s-1, a = 0 if a 0 10n-1 a n d
8.5.2 Coordinate Systems on P(d, N) First let us explain a few facts about coordinate systems on homogeneous polynomials. For m > 1, the space Pm(d) of all homogeneous polynomials of degree m in d variables can be identified with the space of m-multilinear symmetric mappings on Rd as follows. Let f E Pm(d), (1), , b(m) E Rd, define the total polarization of f as (P f) (ill i, . . ,(r,,)) = D{(,) . . DC(-) f where ed). Clearly f and Pf can be identified since DC f = , _ , xm ). Given a basis e1, f (x) = -1=, P f (xi, , ed of l[td we define a system
'
of coordinates {X,,; v E Im} as follows. The set Im is the set of sequences v= ik E [1,d] where (i1,.. , im) and (iolll) , i,lml are identified for any permutation a. Hence we can order with it < < im. Let I v 1= m denote the length of v and I v J;= the number of occurence of i. Define X as follows. For m = 0, set Im = {0} and define X (f) = f (0). If m > 1, (Pf)(e1...... eim) Now let the couple (A, B) E P(d, N). Let U be a neighborhood of (A, B)
in P(d, N) and let e : U - (Rd)d be a smooth mapping such that for any (Q, R) E U, e(Q, R) _ (el (Q, R)), , ed(Q, R)) form a basis of Rd. Then to e we can associate a coordinate system
as follows
X '(Q, R) = (PQL)(ei, (Q, R), ... ea.,, (Q, R)) (Q, R), ... , e;,.. (Q, R)) R) = where Q;n (resp. R;,i) is the ith component of the homogeneous part of degree m of Q (reap. R). This system of coordinate is a curvilinear system of course.
We set X _ (X,1
,
Xd) and Y _ (Y1,
Yi).
8.5.3 Evaluation of Codimension of the F(N) Each F(N) being semi-algebraic in their corresponding spaces, the concept of dimension is well defined. We shall estimate their codimensions.
Lemma 45. cod(Fa (N); P(d, N)2) = d + 1
cod(F1(N); P(d, N)2) = 2d - 2
8.5 Proof of Theorem 25
219
Proof.
.F. (N) _ .T'a(N) U.F (N) UY.'(N) .F.' (N) _ -PT(N) n {(P0, Pi)/Po(0) 0 0) .Pa (N) = FP.(N) n {(Po, P1 )/P1 (0)
0, Po(0) = 0}
.F."' (N) _ )1. (N) n {(Po, Pi)/Po(0) = P1(0) = 0).
It is easy to see that cod(.P4'(N); P(d, N)2) = 2d. To compute the codimensions of .Pa (N), .P.(N) let us introduce the following semi-algebraic sets
C C Rd x End(R'), C = J (v, A); v 0 0, dim { A"v/0 < n < d - 1) < d}
and D C (Rd)2d = {(v,.. . , v2d_ 1); vi E Rd}, dim D < d. Then clearly cod(C, Rd x End(lRd)) = 1 and it is well known that cod(D, (Rd)2d) = d + Rd x End(Rd) x Rd and p 1. Consider the mappings A : P(d, N)2 P(d, N)2 -+ (Rd)2d defined as follows MPo, PI) = (PI (0), Poi, Po(0))
where P01 E End(Rd) is the linear part of Po at 0,
µ(Po,PI) =
(Pi(0),adPo(Pi)(0),...,ad2d-'Po(Pi(0))
Then.Fa (N) =,\-'(C x {0}) and.PQ(N) = µ-' (D). Since \ is a projection, it is a submersion and hence
cod(F. (N); P(d, N)2) = cod(C x {0}; Rd x End(Rd) x Rd) = d + 1.
We prove that v restricted to the open semi-algebraic subset fl of P(d,N)2, fl = {(Po, P1)\Po(0) 34 0} is a submersion. Since .F.(N) _ µ-' (D n fl) it follows that cod(.P.(N); P(d, N)2) = d + 1. To study µ, take a couple (Qo, Qi) E fl. There exist vectors e2,
, ed E
Rd such that (Qo(0) = e1, e2, , ed) is a basis of Rd. Then on a neighborhood V of (Qo, Q 1) contained in fl the mapping
e : V -- Rd x ... x Rd, e(Po, Pi) =
(ei(Po,P1),...,ed(Po,Pl))
e1(Po,Pi)=Po(0), ei(Po,P1)=ei, 2 0 (resp. Dl < 0) each singular trajectories in V (v) n {D1 < 0} (resp. Dl > 0) cross the plane Dl = 0 at v, the singular controls and the adjoint vectors being all distinct.
2. Let v 0 0, on the lines L3, L4. Then there exists exactly one singular trajectory crossing Dl = 0 at v. 3. We have the following blowing-up phenomenon. If Q(b) points towards D1 > 0 (resp. Dl < 0) there exists a conic neigborhood U of Ll such that each singular trajectory with initial condition in U and Dl < 0 (resp. D1 > 0) hits in finite time and with an infinite control the plane Dl = 0 at infinite distance and on a line parallel to L1. We represents on Fig. 8.3 the corresponding singularities of the projection of Z on the sphere. The classification is given with all the details in [22]. It is more precise than the one in [90] where the phase portrait is not known. The situation of codimension one where the lines Li collide is described in [21].
Notes and Sources The first result of genericity in system theory is due to Lobry [81] which shows
that the ad-condition is generic. All the results of genericity concerning the single-input affine systems of this chapter are coming from [26]. The same results are conjectured to be true in the multi-inputs case but the proof is not straightforward. The classification of the singularities of the singular flow has been initialized in [22] and the global classification concerning the singular flow in Euler equations is given in [21]. It can be applied to the attitude control problem.
8.8 Singularities of the Singular Flow of Minimal Order
231
W
E
L1: node
L2 : node
L3L4 :saddle
Behaviors of the projected system near D TO
Fig. 8.3.
Exercises 8.1 Consider the multi-input control system on M: de (t) = Fo(x(t)) +
E`
1
u;(t)F; (x(t)) where m > 2. A singular extremal (z, u) defined on [0, T]
is said to satisfy Goh condition if for each F, G E Span{Fl, , Fm}, y H {HF, HG) (z) = 0 where Hp, He are the Hamiltonian lifts. Show that if -
m > 3 there exists an open dense set of (m + 1)-tuple {Fo, there exists no singular extremals satisfying Goh condition.
,
Fn} such that
8.2 Let el, e2, e3 be R3 canonical basis, Q = (Q1, Q2, Q3) be a quadratic vector field, where Q1 = alx2+a2y2+a3z2+aaxy+a5xz+asyz and Q2, Q3 are respectively defined by changing a2 into b; and c; and let b the constant vector field normalized to b = e3. Let v1 = b, v2 = [(Q, VI 1, VI 1, v3 = [[Q, v1], v2] and assume that the constant vector field VI, v2, v3 are linearly independent.
1. Show that there exists a linear change of coordinates and a feedback u = a(x) + /3v, a(x) quadratic, /3 nonzero constant such that (Q, b) can be taken generically with the following normalization
Q3=0,b=e3,a3=a5=b5=0,b3= ,b6=0,b4 = 1 2. Show that in the above representation we have
(i) D1 =y
8 Generic Properties of Singular Trajectories
232
(ii) D2 = -y3+ 2 -2b1xy2-b1xz2+a4x2z+(a4-3b2)y2z+2(al-1)xyz (iii) D3 = -b2y3 - blx2y - xy2 + alx2z + a2y2z + yz2 + a4xyz 3. Show that the behaviors near L1, L2, L3, L4 of the2projected system are given by (i) near L2 = IRe1: node with two eigenvalues equal to b1 (ii) near L3, L4, L3 = -a4 + f, L4 = -a4 6 = a4 + 2b1 > 0: saddle with two eigenvalues (A, -A) where A 1=1 2b1 - a4L L = L3 or L4. (iii) near L1 = lRb: node with two eigenvalues 1 and Z.
-
8.3 Let M be a a-compact C°°-manifold. Prove that there exists an open dense set of pairs (F1, F2) of COO-vector fields such that for all q E M we have:
Span{adkFiF2i k = 0,
,
+oo} = TqM
(8.2)
8.4 Let M be a a-compact C°° connected manifold of dimension 5 or 6.
F3
Consider the following control system on M: q(t) _ 1 u;(t)F1(q(t)) where F1, F2, F3 are smooth vector fields. Prove that there exists an open dense set of triples (F1, F2, F3) such that every two pairs of points can be joined by a piecewise singular and C°°-curve.
9 Singular Trajectories in Sub-Riemannian Geometry
9.1 Introduction The problem posed by the existence of singular trajectories in the Lagrange problem of the calculus of variations, which can be formulated as the optimal control problem T
min U() 10
subject to
d (t) = f (x(t), u(t)) and the end-points conditions: x(0) E MO, x(T) E M1, was known since a long time, see for instance the discussion in Bliss [18]. It has in some extend slopped the post-war research in this area. It was rediscovered by R. Montgomery in the context of sub-Riemannian geometry (SR-geometry) which has attracted very recently a lot of researchers.
The aim of this chapter is to explain precisely the role of singular trajectories in SR geometry. In our opinion it must be understood as an example of the more general time minimal control problem. In this example the
singular trajectories are exceptional and have the C°-rigidity property analyzed in Chap. 7, if we consider the SR-problem with rank 2-distribution. In SR-geometry the value function is called the SR distance. The key contribution of our work is to make a precise analysis concerning the behaviors
of extremals and optimal trajectories near the singular trajectories. A consequence will be a precise description of the level sets of the distance
function in the abnormal directions showing in particular that it is not sub-analytic. This phenomenon was already pointed out by Lojasiewicz and Sussmann in their article [82) but it covers a lot of different phenomena. Here we give a precise analysis in SR-geometry. Also we show the connection
with the computation of Poincare-Dulac mapping in the problem of limit cycles for planar differential equations. This will provide a connection with recent branches of real analytic geometry concerning in particular the exp-log category and pfaffian mappings.
234
9 Singular Trajectories in Sub-Riemannian Geometry
9.2 Generalities About SR-Geometry (In all this section we are working in the C'-category)
9.2.1 Definition A SR-manifold is defined as a n-dimensional manifold together with a distribution D of constant rank m < n and a Riemannian metric g on D. An q(t), 0 < t < T is an absolutely continuous curve such admissible curve t that q(t) E D(q(t))\{0} for almost every t. The length of q(.) is e(9) =
f
'
T
(9(t), 9(t)) dt
where (,) is the scalar product defined by g on D. The SR-distance between p, q E M denoted dSR(p, q) is the infimum of the lengths of the admissible curves joining p to q.
9.2.2 Optimal Control Theory Formulation The problem can be locally formulated as the following optimal control problem. Let qo E M and choose a coordinate system (U, q) centered at qo such that there exist m (smooth) vector fields {F1,
,
Fm} on U which form
an orthonormal basis of D. q(t) on U is solution of the control
Hence each admissible curve t system TY&
4(t) = Eui(t)Fi(q(t))
(9.1)
i=1
and the length of q(.) is given by fT
e(q) _
(n uq (t)) i dt.
(9.2)
io 1
The length of a curve is not depending on its parametrization. Hence every
admissible curve t '- q(t) with finite length can be reparametrized into a lipschitzian curve s - q(s) parametrized by arc-length: (4(s), 4(s)) = 1
a.e.
If an admissible curve q : [0, T] - U is parametrized by arc-length we have almost everywhere m
m
9(t) = Eui(t)Fi(q(t)), Eu?(t) = 1 i=1
i=1
9.2 Generalities About SR-Geometry
235
and
e(q)=
f
(u(t))dt = T i=1
and the length minimization problem is equivalent to a time minimization problem for the symmetric system (9.1) where the control domain is defined by: Ein 1 u? (t) = 1. This problem is not convex and it is worth to observe that the time optimal control problem with the constraint Ei=1 u? = 1 is equivalent to the time optimal control problem with the u? < 1. Indeed if q(.) is an admissible curve such that the constraint associated control satisfied E' 1 u, < 1 at a Lebesgue time then it can be reparametrized into a curve parametrized by arc-length and with shortest EL`_`
1
length. We can resume this into a proposition.
Proposition 74. Let (U, q) be a chart on which D is generated by an orthonormal basis {F1, , Fm} then the SR-problem (U, D, g) is equivalent to the time optimal control problem for the system m
ui(t)F2(q(t))
4(t) i=1
and the control domain Ei i 1 u? < 1. This result is useful to apply Filippov existence theorem. To save computations when computing optimal trajectories we use the following result whose proof is based on Cauchy-Schwartz inequality and Maupertuis principle.
Proposition 75. Assume that the admissible curve are defined on the same interval [0, T] (e.g. T = 1). Then the length minimization problem is equivalent to the energy minimization problem where the energy of a curve is defined T
by E(q) = f (4(t), 4(t)) dt.
9.2.3 Computations of the Extremals and Exponential mapping Consider the symmetric system on U C 1Rn
4(t) _
i.1
ui(t)Fi(q(t)) = F(q(t))u(t)
where rank{ F, (q), , Fn(q)} = Tn for every q E U. Let T > 0 be fixed and consider the energy minimization problem
Tm
u?(t )dt.
min E(q), E(q) = j i=1
236
9 Singular Trajectories in Sub-Riemannian Geometry
The Hamiltonian associated to the problem is m
H(q, p, u) _
ui (P, Fi(q)) - Po
u?
where po is a constant > 0 which can be normalized as po = 0 or po = 1 The equations of the maximum principle are the following:
4(t) =
OH OP
OH
(q(t),P(t), u(t)), P = - 8q (q(t),P(t), u(t)), (9.3)
8 (q(t),P(t),u(t)) = 0. We shall adopt the terminology used in SR-geometry.
Definition 90. A solution (q, p, u) of the maximum principle is an abnormal (resp. normal) extremal if po = 0 (resp. po = The projection of an extremal in the state space is called a geodesic. A (normal or abnormal) geodesic is called strict if it is the projection of an unique extremal (q, (p, po), u) where 2).
(P,Po) E 1Pn+1.
Abnormal extremals. They correspond to po = 0 and are in fact the singular extremals, solutions of the maximum principle with m
HO(q,P,u) _
ui(P,Fi(q)) =
Hl,o.
They satisfy the constraints (P(t), Fi(q(t))) = 0, i = 1, ... m.
In particular they are exceptional. The singular extremals of minimal order can be computed with the algorithm of Chap. 4. They are smooth curves and their projections are smooth geodesics.
Normal extremals. The normal extremals are the solutions of the maximum principle corresponding to po = z . They are solutions of the following equations
4(t) _
(q(t), p(t), u(t)),
j(t) _ -
a
4 (q(t), p(t), u(t))
(9.4)
8a 1 (q(t),P(t), u(t)) = 0
(9.5)
9.2 Generalities About SR-Geometry
where H j. (q, p, u) _ m 1 ui (p, F1(q)) - 2 m t
237
.
The previous constrained Hamiltonian equations can be easily solved. We
introduce Pi = (p, F; (q)), i = 1,
Solving the linear equation (9.5)
, m.
the control is given by
ui(q,p)=P,, i=1,.m and plugging fi = (fit,
,u,n) into Hi we define the Hamilton function m
Hn (q, p) = Hj (q, p, u.) = 2
P?
(9.6)
,.=1
and the normal extremals are solutions of the Hamiltonian differential equation H 8H (9.7) q(t) = pn(q(t),p(t)), p(t) = --.gn(q(t),p(t))
They are smooth curves. If we use the parametrization by arc-length: Em 1 u, = 1 then the curves are contained on the level set: H,, = 2
Poincard equations. To compute easily the normal extremals it is convenient to use the follwing coordinate system on T'U. On U we can complete , F,,} of TU. the m-vector fields F1, , Fm to form a smooth basis {F1, The SR-metric g on U can be extended into a Riemannian metric on U by tak,n ing the F= as orthonormal vector fields. We set Pi = (p, Fj (q)) for i = 1, and let P = ( P i ,--- , Pn). In the coordinates system on T*U : (q, P) (in gen-
eral not symplectic for the canonical structure) the normal extremals are solutions of the equations m
m
i=1
i=1
Ed]
{Pi, H2 } _ E{Pi, Pj}P?. i=1
We observe that { Pi, P, } = (p, [F, F.jI (q))
and since the Fj are a basis of TU we can write n
c (q)Fk(q)
[Fi, F,[(q) _ k=1
where the ck.7 are smooth functions.
They are several choices to complete the distribution Die = Span {F1,
,
F,n} and some are canonical. This will be discussed later.
238
9 Singular Trajectories in Sub-Riemannian Geometry
Exponential mapping. Consider arc-length parametrized curves. We fix qo E U and let q(., qo, po), p(., qo, po) be the solution of (9.7) starting at t = 0
from (qo, po). It is contained in the level set H = 2. The exponential mapping is the map (9.10)
expgo : (po, t) '-' q(t, qo, po).
Observe that its domain is the set C x R where C : F,i I P? = 1 with q = qo. The important remark is the following. In the Riemannian case m = n and C is a sphere but in the SR-case with m < n, C is a cylinder and in particular
is non compact, see Fig. 9.1. As in the Riemannian case the exponential
cylinder C
C=fin
m=n (Riemannian case)
m 0 be a point where the exponential mapping exp, is not an immersion. Then ti is called a conjugate time along the geodesic and the image is called a conjugate point. The conjugate locus C(qo) is the set of first conjugate points when we consider all the normal geodesics starting from qo. If (q(.), p(.)) is a normal or abnormal extremal with q(0) = qo, the point where q(.) ceases to be minimizing is called the cut point and the set of such points when we consider all the extremals with q(0) = qo will form the cut locus L(qo).
- The sub-Riemannian sphere with radius r > 0 is the set S(qo, r) of points which are at SR-distance r from qo. The wave front of length r is the set W(qo, r) of end-points of geodesics with length r starting from qo.
9.3 Research Program in SR-Geometry
239
9.3 Research Program in SR-Geometry 9.3.1 Classification Given a local SR-geometry (U, D, g) which can be represented as the optimal control problem m
q(t) _
ui(t)Fi(q(t)), min icl
u0 Jo
u?(t)dt
/
i=1
there exists a pseudo-group of transfomations called the gauge group G which is a sub-group of the feedback group defined by the following tranformations: (i) local diffeomorphisms p : q p--' Q of the state space U, preserving qo; (ii) feedback transformations u = (3(q)v preserving the metric g i.e., /3(q) E O(m, R) (orthogonal group). The classification of local SR-geometries is
- Find a complete set of invariants. - Compute normal forms. It is worth to observe that a preliminary step in this classification is the
classification of distributions. In particular the singular trajectories are gauge-invariants. Other invariants are found using the normal extremals.
9.3.2 Singularity Theory of the Exponential Mapping and of the Distance Function In the Riemannian case the local situation is rather simple: there exists a neighborhood V of qo on which each geodesic starting from qo is globally minimizing. In particular the sphere of small radius is smooth. In the SRgeometry with m < n the local situation is intrincated: - the cut locus C(qo) accumulates at qo; - the sphere of small radius have singularities. Hence the remaining of this chapter will be devoted to analyze this singularity problem. We shall restrict our study to the problems where U C ii and D is a rank 2 distribution. More precisely we shall consider in details two situations.
Contact situation: D is a contact distribution. It is the situation for a generic point qo E R3 where the system has no singular trajectory near qo (see Chap. 4). Martinet situation: D is a Martinet type distribution and through qo passes a singular trajectory.
240
9 Singular Trajectories in Sub-Riemannian Geometry
The main difference between both cases is the following. In the contact situation the local situation is intrincated but can be handle with the tools of standard singularity theory; the generic situation can be analyzed and the models of the singularities are polynomial mappings. In the Martinet case this is no longer true and we need much more transcendence because the singularities are not sub-analytic. This phenomenon is due to the existence of abnormal (or singular) extremals.
9.4 Privileged Coordinates and Graded Normal Forms Lemma 52 (Chow theorem). Let (M, D, G) be a SR-structure. Assume that M is connected and D satisfies the rank condition, that is DAL(q) is of dimension n for each q E M. Then for each qo, q1 there exists an admissible curve associated to a piecewise constant control and joining qo to q1. Corollary 18. If (M, D, g) is a SR-structure satisfying the previous assumption one can define a distance dSR on M by setting: dsR(go, q1) = min{e(q(.)) where q is an admissible curve joining qo to q1 }.
Proposition 76 (Filippov theorem). Let (M, D, g) be a SR-structure and assume that D satisfies the rank condition. Then sufficiently near points can be joined by a minimizing geodesic. Hence for r > 0 small enough, S(qo, r) C W (qo, r)
Proof. Locally the problem can be written as a time minimal control problem
for a symmetric system, and the control satisfies the convex constraint: z" 1 u? < 1. Hence condition (iii) of Theorem 10 is satisfied. Since the system satisfies the rank condition, it is controllable and condition (i) is satisfied. The uniform bound condition (ii): I q(t) 1< b, 0 < t < T is satisfied for T small enough and the results follows by a local application of Filipov theorem.
9.4.1 Regular and Singular Points Let (U, q) be a coordinate system centered at qo. Let D = Span{F1 i , Fm} and assume that D satisfies the rank condition on U. We defined reccursively:
D1 = D and for p > 2
D' = Span {D" ' + (D', DP-1] }. Hence DP is generated by Lie brackets of vector fields IF,, , Fn} with length < p. At q E U we have an increasing sequence of vector subspaces {0} = D°(q) C D'(q) C ... C D*(9)(q) = TqM where r(q) is the smallest integer such that Dr(4) (q) = TQM.
9.4 Privileged Coordinates and Graded Normal Forms
241
Definition 91. We say that qo is a regular point if the integer np(q) = dim DP(q) remains constant for q in some neighborhood of qo. Otherwise we say that qo is a singular point. Example 6.
- Contact case: D = ker a, a = ydx + dz; D is generated by F1 = 0
and F2 = a -yes We have [F1, F2] = e and all the Lie brackets of .
length > 3 are 0. Hence DAL is a nilpotent algebra and it is isomorphic to the Heisenberg Lie algebra. The point 0 is a regular point and we have
n1(q)=2,n2(q)=3forallgER3. - Martinet case: D = ker w, w = dz - 2dx; D is generated by Fl = and F2 = 3x + 2 e. The point 0 is a singular point and we have n'(0) _ n2(0) = 2, n3(0) = 3.
Remark 24. At a regular point the sequence np(qo) is strictly increasing.
9.4.2 Adapted and Privileged Coordinates Definition 92. Assume that D satisfies the rank condition at qo where qo is a fixed (regular or singular point). Consider a system of coordinates (U, q) centered at qo such that dql, , dqn form a basis of Tqo M adapted to the flag {0} = D°(qo) C ... C D''(9o)(go) = TgoM.
The weight wj of the coordinate q, is the integer such that dq3 vanishes on Dw-, -1(qo) and does not vanishes identically on Dw-, (go).
Definition 93. Let (U, q) be a chart centered at qo and (D, g) be the SRstructure represented locally by the orthonormal vector fields {F1 i , Fm}. If f is a germ, of smooth function at qo, the order µ(f) of f at qo is: (i) if f (qo) 0 0, p(f) = 0, t(0) = +00(ii) µ(f) = inf p such that there exists v1 i
...oLva(f)(go)00.
, vp E {F1,
,
F.} with L,,, o
The germ f is called privileged if µ(f) = min{p; dfp,, (DP(qo)) # 0}. A coordinate system (qj, , q,) : U IR is said to be a privileged if all the coordinates q; are privileged at qo. We have the following result (See [16]).
Proposition 77. There exists a privileged coordinates system q at every point qo of M. If w; is the order (=weight) of the coordinate qi we have the following estimate for the SR-distance: dSR(0,(gl,...,gn))'
1 glIW
+...+gnI
9 Singular Trajectories in Sub-Riemannian Geometry
242
9.4.3 Nilpotent Approximation Definition 94. Let (U, q) be a privileged coordinate system for the SR structure given locally by the m orthonormal vector fields {Fl, , Fm }. If wj is
the weight of qj, the weight of -4 is taken by convention as -wi. Every vector field F. can be expanded into a Taylor series using the previous gradation and we denote by Ft the homogeneous term with lowest order -1. The , polysystem {Fl, is called the principal part of the SR-structure. We have the following, see [16].
Proposition 78. The vector fields Pi, i = 1,
, m generate a nilpotent Lie algebra which satisfies the rank condition. This Lie algebra is independent of the privileged system.
9.4.4 Graded Approximation Definition 95. Let (M, D, g) be a SR-structure represented locally in an adapted coordinate system (U, q) by m-orthonormal vector fields IF,,
,
Fm}.
The graded approximation of order p > -1 is the polysystem {FP, . , F, } obtained by truncating the vector fields Fi at order p using the weight system defined by the adapted coordinate system.
We shall now discuss the contact and the Martinet case using graded normal forms.
9.5 The Contact Case of Order -1 or the Heisenberg-Brockett Example 9.5.1 The Contact Case in 1R3 We consider
9(t) = ui(t)Fi (q(t)) + u2(t)F2(q(t)) where
Fl=a +y8z, F2=a T -xa
(9.11)
.
If we set F3 = , we get [Fl, F2] = F3. Moreover all the Lie brackets of lengths > 3 are zero. Hence DAL is a nilpotent Lie algebra of dimension 3 and is isomorphic to the Heisenberg Lie algebra h3. We observe that (x, y, z) is a coordinate system adapted to DAL(0) and the weight of x or y is one and the weight of z is 2. The local SR-contact case in R3 is: (U, D, g) where U is an open set containing 0, q = (x, y, z) are the coordinates, D is a contact distribution and g is a smooth SR-metric on
9.5 The Contact Case of Order -1 or the Heisenberg-Brockett Example
243
D. The contact distribution can be written in a smooth coordinate system as D = ker a, where a is the one form
a = dz + (xdy - ydx) and the metric on ker a can be written as g = a(q)dx2 + 2b(q)dxdy + c(q)dy2 where a, b, c are smooth functions. By making a change of coordinates Q = cp(q), preserving D = ker a, with ep = (tp1) cp2i 0. To compute
S(0,1) and W (O, 1) we use the parametrization of proposition 79. The sphere
is obtained for A # 0 by imposing t < tl,.
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9 Singular Trajectories in Sub-Riemannian Geometry
- Sphere S(O, 1). Its intersection with the plane y = 0 is represented on Fig. 9.2 in the domain z > 0. It is smooth except at the intersection with the axis Oz where it admits an algebraic singularity.
- Wave front W(0,1). Its intersection with the plane y = 0 is represented on Fig. 9.3 in the domain z _> 0. The wave front has two types of analytic singularities which are the singularities corresponding to conjugate points.
They are denoted M N,. The two series of both sequences M, and Ni accumulate at 0. 0.4
0.31
Z a2
0.1j
-1
-0.6-0.6 04-0.2
0.2
0.6
0.4
0.8
x -01
Fig. 9.2.
9.5.7 Conclusion About the SR-Heisenberg Geometry This example exhibits one of the main difference between the Riemannian geometry and the SR-Riemannian geometry with rank D = m < dim M.
When A - oo, the conjugate points tend to 0 and the small SR-sphere has singularities. This is due to the noncompactness of the domain of the exponential mapping. Hence one of the main research program is to investigate those singularities. In the Heisenberg case the singularities are analytic for the sphere and for the wave front we have an accumulation of analytic singularities. Clearly we have the following.
9.6 The Generic Contact Case
247
0.41
-1
-0.8 -0.6 -0.4 -0.2
0.2
0 6
0.4
0.8
1
x
Fig. 9.3.
Proposition 82. The Heisenberg sphere S(0, r) is semi-analytic but the wave front is not sub-analytic.
9.6 The Generic Contact Case The Heisenberg case is not a model to study the generic contact situation. Indeed its corresponds to a left invariant SR-structure on the Heisenberg group. The structure admits a nondiscrete symmetry group and the cut locus is precisely the conjugate locus. This situation is similar to the Euclidean structure induced on the sphere S2. The investigation of the general case is the object of several recent papers [6, 51. They contain the description of the small SR-sphere at a generic contact point and also the much more complex classification of all the small SR-spheres for all the generic contact SR-geometries in R3. We shall present briefly the small sphere at a generic point. The analysis and computations are based on graded normal forms.
9.6.1 Normal Forms in the Contact Case To compute a normal form we can use two approaches. First given a contact SR-metric (U, D, g) then D can be identified to ker a, a = dz + (xdy - ydx). The stabilizer of D, that is the set of germs of diffeomorphisms co preserving 0 and the distribution, that is cp * a = ha where h is a germ of smooth
248
9 Singular Trajectories in Sub-Riemannian Geometry
nonvanishing functions has been computed in the litterature devoted to contact geometry, see e.g. [14, 93], and can be used to compute a normal form by normalizing the metric g restricting to diffeomorphisms in the stabilizer. This point of view will be used in the Martinet situation to compute isothermal normal forms where g is a sum of squares. Here we shall present the classical approach to compute formal normal forms in singularity theory, see Martinet [83]. Recall that the gauge group is defined by the following transformations: 1. X = 0 because changing A into -A we exchange only the role of the singularities.
20 Flat case. In this case the foliation is defined by d77 + sin B = 0. It is a pendulum which admits a global analytic first integral. The equation has two equilibrium points which are: a center if 0 = 0' = 0 and a saddle if 0 = ir,0' = 0. On the cylinder all the trajectories are periodic except two
separatrices El, E2 which are saddle connections, see Fig. 9.6. We have two types of periodic trajectories: those which are homotopic to 0 and those which are not homotopic to 0. They admit different parametrization using elliptic functions on R.
9 Singular Trajectories in Sub-Riemannian Geometry
258
Case /3 54 0. In this case neglecting the terms of order e2 we get the equation d$2
+sin0+e(3cosO- = 0. ds
(9.17)
One of the main property of equation (9.17) is the following.
Lemma 63. If /3 # 0, the asymptotic foliation has no global C°-first integral.
Proof. For 0 # 0, the origin becomes a focus and such a singular point has no local CO first integral.
Case 3 = 0. In this case the asymptotic foliation is given by d29
ds2
+ sin B(1 + e 2 a 2 cos 0) = 0 .
(9 . 18)
The main property is the following.
Lemma 64. The equation (9.18) is integrable using elliptic integrals of the first kind. We shall prove this result in the next section.
9.7.9 Integrable Case of Order 0 and Elliptic Integrals
Parametrization and properties. If the metric is of the form g = a(y)dx2 + c(y)dy2, the geodesics equations are integrable by quadratures
9.7 The Martinet Case
259
and the exponential mapping can be computed. In particular it will allow to describe analytically the behaviors of the geodesics near the Martinet plane y = 0 containing the abnormal geodesics. We proceed as follows. Using H,, (P? + P2) = 2 with Pl (y) = I (pz + p= z ), P2(y) = CL where p. and pz are first integrals because the metric doesn't depend upon x and z. We get
f
(f4)2+(Ps+P:2
)2=1
(9.19)
Definition 97. The previous equation is called the characteristic equation. Using the time parametrization dij = it describes the evolution of a one dimensional mechanical system with position y and in a potential field given by V(y)
P12(y)_
Let e(.) _ (x(.), y(.), z(.)) be a normal geodesic defined on [0, T], starting from 0 and parametrized by arc-length. If y is not zero, we denote by 0 < tl < < tN < T the successive times defined by y(t1) = 0. We introduce o
f sign(y(0)) if y(0) # 0 sign(y(0)) if y(0) = 0
Under mild assumptions on V, the motion of y is periodic with period P and we set
y+ =
tmaxx
y(t), y- = 9e
y(t)
The geodesics can be easily computed if we parametrize by y and they are solutions of the following equations cPl dz
y2 VC-P1
dy f P2 dy
2 f P2
dx
(9.20)
with
dt - fZdy
(9.21)
P2
andP2(y)=a 1-P1(y)fortE[0,t1]. If y(T) = 0, T = tN we obtain the following formulas:
- N odd. fP1(y)dy VcP1(y) x(T)=2v./.dy+(N-1) a 1-P y) fl-P (y) J y+
v°
JO
z(T)=2
V_
1yoaf2PPy(y)dy+(N-1)JY+ 0
V-
(9.22)
2f1P1Pi)(y)dy
260
9 Singular Trajectories in Sub-Riemannian Geometry
- N even. 1+
x(T)=NJ
cP1(y)
1-Pl(y)dy
-
(9.23)
y+
z(T) = N
cy2Pi (y)
Jy- 2 f 1 - Pl (y)
dy
and the period is given by y+
P=2
c
1 - PP (y)
y_
(9.24)
dy.
Hence we get explicit formulas to evaluate the exponential mappings.
Characteristic equation in normal form and elliptic integrals. The behaviors of normal geodesics near the abnormal direction in SR Martinet geometry can be understood as a property of elliptic integrals. Such standard objects in mechanics appear in the problem as follows. Assume that the metric is given by g = (1 + ay)2dx2 + (1 + ryy)2dy2 where a, ry are real parameters. We have P, = cos 0, P2 = sin 9, px = cos 9(0) and P3 = ps = A. By symmetry
we can assume \ > 0 and the characteristic equation can be written 2
(v v' )2+(P:+2)=a with a = (1 + ay)2, c = (1 +'yy)2. Using the parametrization dT = 73-7 we get l2
(
L
\ dT /
F(y),
where F(y) = (1 +ay)2 - (Ps +pz 2 )2. The analysis of the motion is related to the roots of F(y) = 0. The function F is a quartic which can be factorized into F = F1F2 where
Fl =(1+ay)-(px+pz 2 ), F2=(1+ay)+(px+\Y!). We can write
F(y) = where
(2m2
- 2 (y - 0)2) (2m" - A2 (y + A )2) a2
2m"=1+pz-
a2 2a
9.7 The Martinet Case
261
and m2+m"=1,m2>0ifa0 0 and m2>0if a#0and 00nrr.If we set y
and we get
v y+ a
a
2m/'
2m
'1
2m
2mf
F(y) = 4m2(1 - 772)(m" + m2Tj2).
(9.25)
The roots of F on C are 17 = f 1 and r = f mW7 . Definition 98. The geodesics are said critical when m" = 0. (The quartic F admits double roots) Lemma 65. If a # 0 in the graded normal form of order 0 then there exists (in the integrable case) critical geodesics starting from 0.
Geometric interpretation. If a 54 0, the abnormal geodesic is strict. The previous lemma gives us a clear geometric interpretation of the role of a. If
a = 0, the situation is similar to the flat case; in this case there exists no critical geodesics starting from 0, but when px -- -1, m" - 0 and we have geodesics starting from 0 which are nearby the critical case. The role of a
is to push critical geodesics has admissible geodesics starting from 0.
Normal form. The characteristic equation can be normalized as follows. Indeed if a # 0, there exists two distinct reals v1, v2 such that the pencil F1 + vF2 is a perfect square: K, (y - p)2 and K2 (y - q)2. Using the homographic
transformation
u=(y-p)(y-q)-1
(9.26)
the characteristic equation takes the normal form dy
F(y)
-
(p - q)-'du
(9.27)
(A1u2 + Bl)(A2u2 + B2)
and the trajectory u(.) can be computed using an integral of the form
f J
du
(Alu2 + B1)(A2u2 + B2)
which is called an elliptic integral of the first kind, see [76]. If a = 0, the result is still true. Indeed in this case 77 = 1 and F(y) is normalized as before using a simple dilatation: q =
Hence we get a neat interpretation of our graded normal form of order 0.
262
9 Singular Trajectories in Sub-Riemannian Geometry
Proposition 85. The graded form of order 0 is a coordinate system in which the oscillation of a normal geodesic with respect to the Martinet plane can be parametrized in the integrable case using elliptic integral of the first kind.
We shall now parametrize the geodesics in the flat case using elliptic integrals of the first and second kind. The previous reduction will show that the integrable case of order 0 has the same transcendence.
9.7.10 The Martinet Flat Case If g = dx2 + dye the geodesic equations take the form
Pi(t) = yP2(t)P3 P2(t) = -yPi(t)P3 P3(t) = 0
x(t) = P1(t)
y(t) = NO z(t) = z Pl(t)
Notations. We compute the geodesics starting at t = 0 from 0. We set Pl (0) = sin cp, P2(0) = cos p (i.e. cp = 0(0) + and A = P3(t) = P3(0). Let S 2) be the group generated by the two diffeomorphisms S1 : (x, y, z) i - (x, -y, z)
and S2 : (x,y,z)'-' (-x,y,-z).
Symmetries. In the flat case the SR-geometry is left invariant by S. Indeed
it is the symmetry group of the geometry. Observe it is a discrete group and the situation is different from the Heisenberg case where the symmetry
group is not discrete and the sphere is a surface of revolution. The symmetry group is related to the geodesic equations as follows. If we change p into ir - cp, the solution (x, y, z, P1, P2, P3) is changed into (x, -y, z, P1, -P2, P3) and if we change p into -cp and A into -A it is changed into (-x) y, -z, -P1i P2, -P3). Hence when computing the geodesics we can assume
Assumption (H1). V El - 2, +2[, A >- 0.
Quasi homogeneity. In the flat case the geodesic equations are invariant for the following transformations
x=eX Pi=Q1
y=EY P2=Q2
z=63Z P3=
and the length is changed as follows L(X, Y, Z) = s-1L(x, y, z).
9.7 The Martinet Case
263
Parametrization. We have the following particular solutions. When cp = x(t) = t, y(t) = z(t) = 0 and it corresponds to the abnormal geodesic. If A = 0, x(t) = t sin cp, y(t) = t cos cp, z(t) = sin cp cos2 cp and their projections on the plane (x, y) are straight lines (they are the geodesics of the Riemannian
s
metric gR = dx2 + dy2). The others solutions are computed using elliptic
integrals of the first and second kind. The characteristic equation is equivalent to
y2=(1-P1)(1+P1)=(1-Px- )(1 +Px + ipx). We introduce 0 < k, k' < 1, k2 + k'2 = 1 defined by k2 = sine
" (4
-
1 - sin cp
cp
2) =
k'2 _
1 + Sin cp
2
2
where cp E] - 2 , + [. Hence y satisfies 2
y2 = (2k2 - 2 A)(2k'2 + 2 A). Assume A > 0 and set 71 = y2k , we get the equation 2
,
=(1-r12)(k2+k2ii2) which has to be integrated with the initial condition 71(0) = y(O) = 0 and using the initial branch f1(0) > 0 since y(0) = P2(0) > 0. The solution can be written using the Jacobi elliptic function cn whose inverse mapping is defined by
cn-1(x,k) _
d71
x'
(1 -,i2)(k'2 + k2ii2)
and we set
r1(t) = -cn(K(k) + tf, k)
Vr-
where the period is 4K, K being the complete integral of the first kind 11
K(k)=J
o
d'1
(1 - r12)(k'2 + k2712)
f
=J (1-k2sin9)-4dO. o
77 cos(K(k)+t ' , k) and represents a periodic motion whose period is 4K and with amplitude 2. We have two limit behaviors. When cp - 2 and K(k) --+ 2 and cp -' 7r+ - - , then k' - 0 and K(k) ti Ink, . Hence K(k) --+ +oo, when k -+ 1 Hence y(t)
and admits a logarithmic singularity. In the pendulum representation we have the system
264
9 Singular Trajectories in Sub-Riemannian Geometry
y=sing, B= -Ay. Since y(O) = 0, we have 9(0) = 0 and we have only the oscillating solutions of the pendulum. The two limit behaviors k -b 0 corresponds to the linearized pendulum and k - 1 corresponds to the oscillating solutions tending to the separatrix realizing the saddles connections. The others components can be computed similarly. The only additional transcendence we use is the Jacobi epsilon function, see [76] defined by rt.
E(t, k) =
J0
dn2u du =
rent [1-- k2u2
J0
1-u2
du.
Endly we get the following parametrization.
Proposition 86. Arc-length parametrized geodesics starting from 0 are given by
x(t) = -t +
3(E(u) - E(K))
y(t) = -- 2k Z(t)
=3
((2k2 - 1)(e(u) - E(K)) +
k'2tV,-\)
+2k2snu cnu dnu)
where u = K + t \/'A-, A > 0, lP E] - 2 , + 2 [, snu , cnu dnu and E(u) being the Jacobi elliptic functions, K and E(K) being the complete integrals of the first and second kind, or the particular solutions when A = 0 x(t) = tsin ip y(t) = t cos ca z(t) = 11 sinwcos2
s
where P E) - M' j [ and the curves deduced from the previous ones using the symmetries Sl2 : (x, y, z) '- (x, -y, z) and S2 : (x, y, z) "- (-x, y, -z).
Return mapping and intersection of S(0, r) and W (O, r) with the Martinet plane. Definition 99. Let e(., gyp, A) be a geodesic parametrized by arc-length. We the map which associates to e(., cp, A) its first (resp. denote by Rl (resp.
n-th) intersection with the Martinet plane. Its domain is the set cp # n!2 , A # 0. They are called respectively the first and nd return mapping. Image of R1. We may assume by symmetry W E) - 2 , + [. Let r > 0, using 2 the parametrization of Proposition 86 we represent the image of Rl for t < r on Fig. 9.7. The important property is the following: RI maps the curve r,
defined by r/ = 2K onto the curve cl and preserves the orientation of the curves. The interior of the shaded domain is sent onto the interior of the shaded image.
9.7 The Martinet Case
265
+1
-1
Image R. , t 0 and setting ti, = r =', which corresponds to the first intersection with the Martinet plane, in the parametrization of the geodesics one gets
x(k) = -r + 2 (E(3K) - E(K)) (9.28)
z(k) = 3ai ((2k2 - 1)(E(3K) - E(K)) + 2Kk'2) (9.29)
Using the relation E(3K) = 3E(K) and the notation E for E(K), we get that the intersection of the cut locus with the sphere S(0, r) is a curve denoted k c(k) contained in the Martinet y = 0 which admits the following parametric representation
x(k) _ -r + 2ry z(k) - s ((2k2 - 1)E + k'2K) where k E]0,1[ and the curve deduced from c using the symmetry (x, z) H
(-x, -z). In order to understand the properties of this curve, the following properties are fundamental. First, by definition
K=
ri
J
o
Vfd
(1 - rtz)(k'2 + k2g2)
and
E=
rx
J
=
(1 - k2sin2o)-4d9
Jo
fI
dn2udu=(1-k2sin20)1do
0
z[1+(2)z
K
z
k4+...J k2+(2a) z 2 E=2[1-(2) k2-3(24) k4+...1
9.8 Estimates of the Sphere and of the Wave Front
267
In particular, when k -* 0+, the cut locus in the domain z > 0 is of the graph of an analytic function which can be represented by 3
2r3(x - r) + o(x - r)
Z
Hence it is a semi-analytic object. When k -+ 1, the situation is quite different. This is due to the following fact. We can extend the mapping k H K(k) to an analytic function on C\[1, +oo[ which presents a logarithmic singularity when k 1. More precisely we have a representation with k' = 1 - k K(k') = ul (k')in
+ u2(k')
where u1i u2 are analytic function of k' which can be easily computed by reccurence k42
ul = 1 +
+
o(k'3),
k42 + o(k 3)
u2(k')
The function E has a similar property and can be decomposed into
E(k') = u3(k')ln
+ u4(k')
where u3, u4 are analytic at 0 and moreover u3(k') =
k4+ o(k 3), u4(k') = 1 -
k82
+ o(k '3).
Introducing
Xi=k',
X2 =
14 In k,
we have X1 = 4e-3k. Such a graph is not analytic but is pfafflan because solution of X2dX1 - X1dX2 = 0.
Hence near k' - 0, the cut locus is the image of a pfafflan set and in general we cannot expect such an object to be semi-analytic. Indeed precise computations give the following.
Theorem 29. When k' - 0, the graph of c is given by
r3 x+rl3 F( x+r) 6 \ 2r / + 2r / where F is a flat function of the form F(X) = -4r3X 3e- * + o(X3e- 1). In particular the sphere is not sub-analytic.
268
9 Singular Trajectories in Sub-Riemannian Geometry
This is not a new result in optimal control, see [82], but here we can be more
precise using the exp -In category in real analytic geometry [78]. Indeed to compute the graph we must eliminate the parameter k' occuring with logarithmic complexity in the parametric equation. Using an implict function theorem in this category we can prove the following theorem, see [4].
Theorem 30. The intersection of the sphere S(0, r), r > 0 with the Martinet plane y = 0 in the domain z > 0 is near X = 0 with X = 2r' a graph of the form
z=F(X,eX%1 where X > 0 and F is an analytic mapping from a neigborhood of OR2 onto
Wave front. We can make the same analysis and compute the wave front represented in the domain y = 0, Z > 0 on Fig. 9.8. It contains an infinite
Fig. 9.8. where the curve cn corresponds to the n-th intersection of the geodesics with y = 0 and the curve cl belongs to the sphere. In particular it has an infinite number of intersections with the axis Oz and number of curves cl, c2i
belongs to no reasonable category of analytic geometry. Nevertheless each curve has the same properties than the curve cl, that it is not suban-
alytic but belongs to the exp-ln category.
9.9 Conclusion Deduced from the Martinet SR-Flat Geometry Concerning the role of Abnormal Geodesics in SR-Geometry 9.9.1 Behaviors of the Normal Geodesics near the Abnormal Direction - Geodesic C°-Rigidity We represent on Fig. 9.9 the behavior of a geodesic e(.) = (x(.), y(.), z(.)) near the abnormal direction (i.e. p - -1). ). The coordinate y oscillates pe-
9.9 Conclusion Deduced from the Martinet SR-Flat Geometry
x
Y
t
269
z
X
t -> y(t)
t
t -> x(t)
t t -> z(t)
Fig. 9.9.
riodically with period 4K and for t ' - x(t), z(t) there exists a shift. The average behavior can be obtained by taking the respective lines joining 0 to (2K, x(2K)), (2K, z(2K)). This shift is a property of the Jacobi epsilon function E. In particular at the first intersection z(k) =
_± ((2k2 - 1)E + k '2 K) 3A 22
and when o -+ -L,
k
1 the shift is given by
lim z(k) =
k-.1
4 3A
The numbers is a basic invariant and explain the C°-rigidity of the abnormal extremal, discussed in Chap. 5. More precisely we have the following
geodesic rigidity property. Proposition 87. For all M > 0, there exists e(M) such that if e
:
t-
(x(t), y(t), z(t)), t E (0,T]. is a normal geodesic whose image is not contained in the axis 0s corresponding to the abnormal direction and with length less than M then we cannot have simultanously y(T) = z(T) = 0 if I y 1:5 e(M)-
9.9.2 Nonproperness of the First Return Mapping Rl and Geometric Consequence We recall on Fig. 9.10 the first return mapping for p E] - 2 , + [ and with fixed length. We observe that the mapping R1 is not proper near 2to the point (-r, 0) and this is due to the logarithm branch of K(k) when k -, 1. If we use the pendulum representation this nonproperness is due to the behaviors
of the trajectories tending towards the separatrix, see Fig. 9.11. The
270
9 Singular Trajectories in Sub-Riemannian Geometry
a
P
ai x
+r
Fig. 9.10.
Fig. 9.11.
Fig. 9.12.
9.10 Cut Locus in Martinet SR-Geometry
271
phenomenon is similar to the one encountered in the evaluation of PoincareDulac return mapping with respect to a section of a separatrix cycle, see Fig. 9.12. This shows the connection between the SR-sphere in the Martinet case
with the problem of computing the Poincare return mapping in the 16-th Hilbert problem concerning the number of limit cycles for planar differential
equations. All techniques developped to study this mapping can be used to evaluate the SR-distance.
9.10 Cut Locus in Martinet SR-Geometry An important question briefly discussed in this section is to understand the role of abnormal direction in the cut locus. We represent on Fig. 9.13 the cut locus in the Martinet flat case, in the northern hemisphere A > 0 of the sphere. The two points A and -A are the intersections of the abnormal direction with the sphere. The abnormal geodesic is not strict and both points A and -A are contained in the equator A = 0. It can be understood as a deformation of
Fig. 9.13.
the contact case represented on Fig. 9.14. Next we shall conjecture the cut locus in the Liu-Sussmann example (79], where the SR-metric is given by 2
D = ker w, w = (1 + ey)dz - 2 dx 2 (1
y)2 + dye
The model is not generic because the orthonormal frame generates a nilpotent algebra. Moreover the geodesisc equations are integrable. They are the following in cylindric coordinates 2
± = (1 + ey) cos 8, y =sin 8, z = 2 cos 8,
272
9 Singular Trajectories in Sub-Riemannian Geometry
conjugate point
Heisenberg cut locus
generic cut locus in the contact case
Fig. 9.14. 6
= -(Pxe + pzy)
where px and p,z are first integrals and pZ = A. The angle evolution is the pendulum 8 + A sin 8 = 0 and the constraint y = 0 defines the section 6 = -pXE.
If E # 0, the abnormal line is strict and intersects the northern hemisphere A > 0 at a single point A. If e = 0, we are in the flat case and the cut locus is represented on Fig. 9.14: each point of y = 0 minus ±A is a cut point endpoint of two disctinct geodesics corresponding to oscillating trajectories of the pendulum. If E 96 0 the cut locus splits into two distinct branches ramifying at A, a branch Lc corresponding to oscillating solutions of the pendulum and a branch LD corresponding to rotating solutions of the pendulum. The extremity of each branch not ending at A are conjugate points, see Fig. 9.15.
cut locus in 1S-example
Fig. 9.15.
9.10 Cut Locus in Martinet SR-Geometry
273
Notes and Sources A good general presentation of SR-geometry is provided by (69]. See also the articles of Bellaiche and Gromov in [16]. A detailled analysis of the contact case is provided by the series of articles [6, 5] from Agrachev-Chakir-GauthierKupka; see also the article from [9] concerning the generalization of the Dido problem. The description of the Martinet case is given in details in the two articles [4, 23] and see [29] for a generalization.
Exercises 9.1 In Heisenberg SR-geometry compute the models of the singularities M; and Ni of the wave front. 9.2 Consider a SR-geometry (U, D, g) in R3 where D = ker w with 3
W = we = dy - (xy + 3 + xz2 + mx3z2)dz or
w = wh = dy - (xy + x2z + mx3z2)dz
and g = a(q)dx2 + 2b(q)dxdz + c(q)dz2. Compute in both cases the graded approximation of order -1. Compute in this approximation the normal and abnormal extremals. 9.3 Consider the flat Engel SR-geometry in R4 with coordinates (x, y) z, u),
D=Span {F1,F2}, F1 = F+yj+'j3U, F2 =
ZOFj
andg=dx2+dy2.
1. Compute the abnormal geodesics. 2. Compute the normal geodesics equations using the Poincare coordinates Pi = (p, F1) where F3 = [Fl, F2] and F4 = [[Fl, F2], F2]. Prove that P4
and C = P1P4 + 2 are first integrals. 3. Integrate the normal geodesics flow using elliptic integrals.
9.4 Consider the local SR-geometry (U, D, g) where D is a rank two COOdistribution and g is a C°°-metric. We can identify the SR-geometry to the set of orthonormal pairs of vectors fields. Show that for an open dense set of pairs (Fo, F1) for the Whithney topology each abnormal or normal extremal is strict.
9.5 Consider a smooth SR-structure (M, D, g) where g denote the scalar product on D. Given a smooth vector field X, we denote by exp tX its corresponding flow. The vector field is called an infinitesimal symmetry of the distribution if (exp tX) *,d = A and of the SR-structure if (exp tX) * 4 = A and (exptX) *g = g. The Lie algebra of the distribution (resp. SR-structure) is denoted SymD (resp. Sym(D, g)).
274
9 Singular Trajectories in Sub-Riemannian Geometry
1. Prove the following:
a) X E SymD if and only if adX (a) C A. b) X E Sym(D,g) if and only if adX E so(D(q)) for all q E M where so(D(q)) is the set of antisymmetric matrices over the space vector D(q) with scalar product gq. 2. Consider the Heisenberg SR-geometry with D = Span{Fi, F2}, F1 & + x a: , and g is defined by taking Fl, F2 orthonormal. F2 = a) Show that
SymD = {X; X = Pox +Qay
+ Rz},
(9.30)
where P = -fq-xfz, Q = f, R = xff- f with f being an arbitrary function.
b) Prove that the symmetries of (D, g) form the four dimensional Diamond Lie algebra: Span{X°, X1, X2, X3} where: 49
X0
a
1
19
+X-
- y2) 8z
(X 2
Xl
9
(9
TX +yOz
X2= 9 X3 8z
3. Consider the flat Engel case with D = Span{F1i F2), F1
+X 28v = ay +X 8z F2 =
8
(9.31)
Compute SymD and prove that Sym(D, g) is isomorphic to the Engel Lie algebra: {Fl, F2}AL.
9.6 [Grusin example] Consider the singular Riemannian problem:
i(t) = u(t) y(t) = x(t)v(t) min
J
tl (u2(t) + v2(t)) i dt
Define the length of a curve y(.) by e(y) = fro (v2(t) + v2(t))dt and the distance between two points qo, ql as d(qo, ql) = infe(y) where y(.) is an admissible curve joining qo to ql.
9.10 Cut Locus in Martinet SR-Geometry
275
1. Prove that outside the line x = 0, the problem is Riemannian and is defined by the metric ds2 = dx2 + 2. Show the following estimates:
x
dy2.
2(IxI+Iyl ) 0 and make the following reparametrization: ds = (1 + 2Q) A, show that 9(s) = s + Bo, 9o =constant.
5. Assume A large enough, e = z small parameter. Set x = EX, Y = EY, z = E2Z. Prove that:
X =Xo+E2Xl+0(62) Y = Yo + E2Y1 + o(E2)
Z = Zo + o(E)
where go = sin(s + Oo) Yo = cos(s + 90)
Z0 = sins + 9o)Yo(s) - cos(s + 9o)Yo(s) 2
X1 = sin(s + 9o)Q(Xo, Yo) X2 = cos(s + 90)Q(X0, YO)
and Q is the quadratic form defined by 1+2
Cy2+...,
= 1+Q = 1+Ax2+2Bxy+
9.9 Consider the flat Engel SR-geometry in R4 with coordinates q = (ql, q2, q3, q4) and defined by the two orthonormal vector fields: a Fl-8ql+92
2 q2 a
a
aq3+ 2aq4 F2 - a aq2
and let L 1i L2 be the two matrices
0000 0010
L1=
0001 0000
_
,
0100 0000
L2- 0000 0000
9.10 Cut Locus in Martinet SR-Geometry
277
1. Show that the flat Engel SR-geometry can be lifted to a right-invariant SR-geometry:
R(t) _ (uI (t)LI + u2(t)L2)R(t),
min Ito (ui(t) + u2(t)) where R belongs to the Engel group Ge represented by the nilpotent 1 q2 q3 q4
0 1 q, 2 0 0 1 ql
matrix
(0
0
1
2. Prove that the Martinet flat case is isometric to (Ge/H, dqi +dq2) where H is the following sub-group of Ge : {exp t[L1, L2); t E R}. 3. Compute the flat Engel SR distance from 0 to the line (ql, q2, *, q4) (resp. (qi, q2, q3, *)) and compare with the flat Martinet distance from 0 to (qi, q2, q4) (resp. with the Heisenberg distance from 0 to (qi, q2, q3)). 4. Deduce that the flat Engel SR-sphere is not sub-analytic. 9.10 Consider the Martinet SR-geometry of order 0 with orthonormal frame
1(0 + 2 ), TO where a = (1 + ay)2, c = (1 + /3z +7y)2, a,,O,-y are real parameters and a > 0. Let X = 2r' , Z = rs, r > 0. Prove that near (-r, 0, 0) the trace of the sphere S(0, r) with the Martinet plane y = 0 in the
domain z < 0, is a graph of the form: Z = - 2 X2 + o(X2). 9.11 Prove that in SR-geometry each CI-local minimizer is a global minimizer if its length is small enough. 9.12 Let U be a neighborhood of qo in R" and consider the SR-problem:
q(t) _
M
min
T
fo
,-1
u?(t))idt
with q(0) = qo. Prove that if m < n, then in every neighborhood of qo there exists a point q qo where the SR-distance function to qo is not continuously differentiable.
9.13 Let U be a neighborhood of qo in R" and consider the following SRproblem: m
Q(t) _
min
u+(t)Fa(q(t))
/ T( u; (t)) i dt
278
9 Singular Trajectories in Sub-Riemannian Geometry
with q(O) = qo. The set of controls is endowed with the L2[0, 11 topology and this induces the Hl -topology on the space of trajectories. Prove that for any
small enough r, the set of minimizers of prescribed length r is compact in the Hl -topology. 9.14 Let D = Span.{Fl, , F,,,) be a m-dimensional distribution of a manifold M and consider the system m 4(t)
ui(t)Fi(q(t)).
(9.33)
1. The distribution D is called fat at a point q E M if for any vector field X with X (q) E D, X (q) 0, then we have: [X, D] (q) = D(q) + Span{ [X, YJ(q); Y E D} = TqM.
(9.34)
Prove that if D is a fat distribution at qo, then there does not exist a nontrivial singular trajectory starting at qo.
2. The distribution is called medium fat if for every vector field X E D, X (q) 54 0 then
[X, D') (q) = D2(q) + Span{(X,YJ(q); Y E D2) = TqM.
(9.35)
Prove that if D is a medium fat distribution at qo then there doest not exist a nontrivial singular trajectory starting at qo and satisfying Goh conditions.
9.15 Consider a smooth control system of the form: 4(t) = f (q(t), u(t)) where q(t) E U neighborhood of 0, u(t) E JIB'" and f (0, 0) = 0.
1. Show that a necessary condition for the existence of a C'-feedback control u(.) such that 0 is asymptotically stable for 4(t) = f (q(t), u(q(t)) is that the mapping (q, u) -- f (q, u) is onto on open set containing 0. 2. prove that the following system in R3: ±(t) = u(t)
W) = v(t)
z(t) _ (u(t)y(t) - v(t)x(t)) cannot be made asymptotically stable at 0 using as smooth feedback.
9.16 Let0 0, Lt = Wt(Lo) is Lagrangian.
3. Along '(.), the vector field H is transverse to Lt for t > 0. 4. The time tc is a conjugate time along y(.) if and only if the projection is singular.
Proof. It is a reformulation of the results of section 2.1 using Lagrangian formalism. Indeed the fiber Lqo is Lagrangian by definition and Lt is a Lagrangian manifold because it is the image of Lqo by a symplectomorphism Wt. By definition the conjugate times are defined as the singularities of the mapping: t'- 7r(exp t H (' (0)), hence we deduce 4. Remark 28. Let 0 < t < tic where tic is the first conjugate point along ry"(.) Then locally near ir('(t)) the Lagrangian manifolds Lt are parametrized by
a generating function t - S(t, q). This is equivalent to solve HamiltonJacobi-Bellman equation. Indeed we integrate the extremal trajectories starting from q(0) = qo to construct a mapping t i- S(t, p) and p is eliminated
using the implicit function theorem by solving at t fixed the equation: S(t, p)
=q
This gives us the generating mapping S(t, q).
Beyond the first conjugate point the Lagrangian manifold Lt is still well defined but we must use singularity theory to analyze the structure of (Lt, ir). This leads to the next section.
10.4 Singularity Theory of the Generating Function Generating Family Locally Lt can be represented by a generating mapping S. Itt was a recent research activity of singularity theory to make a classification of pairs (L, 7r). All the simple singularities are classified in [15], they correspond to singularity up to codimension k = 6. Here we give the classification up to codimension k < 3, see [15] for details.
Proposition 90. Up to codimension k < 3 the generating function S have the following normal forms:
284
10 Resolution of the Singularity near a Singular Trajectory
k > 1: A2 : S = pi (fold) k > 2: Moreover we have, A3 : S = fpi + g2pi (cusp) k > 3: Moreover we have (swallowtail)
A4 : S = pi + g2p3l + gspi
(2 cases)
D4 : S = pi f plp2 + gspi
Remark 29. The two first cases are equivalent to Withney classification for applications from the plane onto the plane. Definition 102. Let L be a Lagrangian manifold. A caustic is the projection on M of the singularities of (L, 7r). Another way to parametrize Lagrangian manifolds is the concept of generating family coming from optics. Definition 103. Let N, M be two smooth manifolds, u E N, q E M being coordinates. The family F(u, q) is called a generating family for the Lagrangian manifold L C T * M if
L={(p,q); 3us.t.
OF(u, q)
au
OF
=O,p= a-
}.
Construction of F. (see [15]) The Lagrangian manifold L is defined locally by its generating function S(qj, pT) by the equations: PI =
as
as
q-I
agj
OPT.
We set
F(u, q) = S(qj, u) + (qT, u)
where (,) is the scalar product and dim u = dim 7 = m (u = pT). We define a Lagrangian manifold V on the extended cotangent bundle T*(IRrn X R'1) by setting
OF y = T U-
OF
OS
au + '
p = aq .
By cutting by y = 0, one gets as 19U
since u = pT.
as OPT,
10.5 Application to Optimal Control Theory and to SR-Geometry
285
10.5 Application to Optimal Control Theory and to SR Geometry Consider the following control problem
rT min J L(q(t), u(t))dt, T fixed, q(O) = qo, q(T) = q1 0
subject to 4(t) = f (q(t), u(t)). The previous theory can be applied to analyze the structure of the optimal synthesis excepted in two situations: 1. When the transversality condition 3 of Proposition 89 is not satisfied; 2. When the reference trajectory is singular.
The case 1 is encountered in SR-geometry. Indeed, the length of a curve does not depend on the parametrization and the optimal control problem is parametric. This induces a symmetry which has to be taken into account when writing Hamilton-Jacobi-Bellman equation. This will be handled in the next section.
10.5.1 The SR-Normal Case We shall restrict our analysis to the SR-case
4(t) = ul(t)F1(q(t)) + u2(t)F2(q(t)), q(t) E M min e(q) = fT (U2 1(t) + ua(t)) #dt
where F1, F2 are two smooth linearly independent vector fields. In this problem the Lagrangian has the following property L(q, A4) = L(q, 4)
for ). > 0
and the cost is not depending upon the parametrization. The normal case fits in classical Lagrangian singularity if we choose a distinguished
parametrization. Lagrangian manifolds in the normal case. Let P, = (p, F; (q)) i = 1, 2, the Hamiltonian associated to normal extremals is given by Hn = (Pi +P2 ). 2 the folLet t'-. -f(t), t E [0, T] be an injective reference geodesic. We assume lowing:
286
10 Resolution of the Singularity near a Singular Trajectory
Hypothesis. We suppose that ry(.) is strict, that is there exists an unique liftingt [5''] of ry in the projective bundle P'(TM). We use the following notations:
- exp,ytol is the exponential mapping. It is defined by t 1--4 Ir(q(t)) where t i-+ q(t) is solution of Hn starting from y(O) at t = O.We assume that the normal extremals are parametrized by arc-length, that is H = 2 - Lt = expt Hn (Ty(O)M). - Qt: intrinsic second-order derivative evaluated along -Y and indexed by t E]O, T].
The following results are standard.
Proposition 91. 1. Lo = T(O)M is a Lagrangian linear manifold and for each t > 0, Lt is a Lagrangian manifold.
2. The time t > 0 is conjugate along y if and only if the projection it : Lt M is singular at ry- (t).
3. Assume that the geodesics are parametrized by arc-length t and let W = Uo 0. 6. Let D be the distribution defined for x, y, z small by dy - zdx = cos 9dz - sin 8dx = 0.
a) Show that b is spanned by
316
10 Resolution of the Singularity near a Singular Trajectory
Xl
=cosOa +zcosOa +sin0az y
X2
= ae
b) Prove the following:
(i) D2 = ker (dy - zdx) (ii) D is an Engel system (iii) The singular trajectories are the leaves of j c) Let 7 : 0' - (0, x(9), y(0), z(0)) be a curve defined on [0, P] tangent to D and (x, y, z) vanishing at 0 and P. (i) Using 0 = cos 0dz - sin Odx show the existence of a smooth function h(0) defined on [0, PI which satisfies the equations:
zcos0-xsinO=h, zsinO+xcos0=-h'. with dy = zdx - xdz prove the relation y(O) = h(0)h'(0) +
fe
(ii) Assume that P < ir, prove that h = 0. (iii) Assume that P > 7r, construct a 1-parameter family of functions ha(0) such that the above formulae are valid. d) Deduce that the singular line y : 01-* (0, 0, 0, 0), 0 E [0, P] is C1-rigid if and only if P < Tr.
e) Compute along -y the differential operator DZ introduced in Sect. 10.10.2 and the corresponding conjugate points.
f) If T is a tubular neigborhood along y prove that DZ can be written as J" + K(y(t))J = 0 where -y(.) is a singular trajectory contained in T. The function K defined on T is called the curvature tensor (in the exceptional case).
10.2 (Goursat systems in dim 5) Consider the two-dimensional distribution D in 1R5 such that
dim D2 = 3, dim D3 = 4, dim D5 = 5. 1. Compute a local coordinates system. 2. Compute the singular trajectories.
10.3 (Cartan system in dim 5) Consider on 1R5 a two dimensional distribution and assume
dim D2 = 3, dim D3 = 5. 1. Compute a local coordinates system. 2. Compute the singular trajectories. 3. Compare Goursat and Cartan case.
10.11 Micro-Local Analysis of SR-Martinet Ball of Small Radius
317
10.4 Consider the problem:
x(t) = 1 - y2(t), y(t) = u(t), min J T u2(t)dt 0
where T is fixed to 1, x(0) = y(0) = 0. 1. Show that the normal extremals are solution of±(t) = 1 - y2(t) px(t) = 0 py(O) = 0
py(t) = 2pxy(t)
y(t) = py(t)
associated to the Hamiltonian Hn(4,p) = px(1 - y2) + Zp2y.
2. Set px(0) = 2 and integrate the previous equations. If A > 0, show that the solutions starting from x(0) = y(O) = 0 are given by
y(t) = Ash(v't)
pxt =
A
2
py(t) = Av5ch(V t) x(t) = t(1 + 22)
-
A2
sh(2v' t)
(10.8)
3. Let A > 0, y(0) = py(0) > 0. Show that the level set r1
Cr = min
u2(t)dt = r > 0
J0
is given by the relations 1
A
2r
(
- Lsn2f 2
y(l) = Ashv5
+1]
a
X(1) = 1 + 22 4. Deduce that near the end-point M = (1, 0) of the abnormal line the union of M with the level set C, is a graph of the form 4
X(1) = 1 - - + 3y2 exp( 4r) + o(y2 exp(
yr) ).
Prove that near M in the sector A > 0 the value function is given for
x ,-
1 by y4
y4
_y2 S(x,y)=41-x+1-xexp(1-x)+o(y
4 IexL) - x ).
318
10 Resolution of the Singularity near a Singular 'Ilajectory
5. Describe near M the stratification of the value function.
6. Compute the Jacobi equation and discuss the existence of conjugate point. 7. Represent the wave front and analyze its singularities.
10.5 Let L be a smooth Lagrangian such that L(x, >d, t) = \L(x, t, t), A E 1R\{0}.
1. Show that
Lz2.i = 0.
(i, L±) = L,
2. If p = e deduce the following:
H=(p,i)-L=0,
detLxx=0.
10.6 Consider the Riemannian problem in the plane
mint
4(t) = ul(t)FI(q(t)) +u2(t)F2(q(t)),
where q = (x, y), F1 =
7X
,
F2 =
TV-
T
I ui(t) + u2(t) I Idt
where q(0) E M: parabola s
(x(s),y(s))
1. Use the maximum principle to write the extremals of the problem. 2. Compute the caustic. 3. Represent the wave front. 4. Analyze the singularities of the caustic and the wave front. 5. Compute the optimal synthesis. 10.7 Consider the following optimal control problem: m ui(t)Fi(q(t))
4(t) = Fo(q(t)) + i=1
jT
min
m
> u?(t)dt i=1
where T is fixed and q(0) = qo.
1. Compute the intrinsic second order derivative of the end-point mapping of the cost extended system: 4(t) = Fo(q(t)) + E ui(t)Fi(q(t)) i=1
M
4°(t) _
u? (t) i=1
with (q(0), q°) = (qo, 0)
10.11 Micro-Local Analysis of SR-Martinet Ball of Small Radius
319
2. Consider the following optimal problem in 1R2:
i(t) = 1 - y2(t) y(t) = u(t) ri min J u2(t)dt 0
with x(0) = y(O) = 0. Compute the intrinsic derivative of the end-point mapping of the cost extended system. 10.8
1. Consider the Hamiltonian function:
H(q,p) = 1(q2 +P2),
(q,p) E R2.
(10.9)
Integrate the following equations:
4(t) = 8p (q(t),p(t)),
P(t)
aq
(q(t),p(t))
(10.10)
with q(0) = qo, p(0) = po.
2. Let Lo be the line p = po and Lt = expt H (Lo), L = Ut>oLt. Represents L in the time extended phase space (q, p, t). Show that L projects diffeomorphically onto the plane (q, t) if and only if t < 2 3. Prove that there exists a function S on the surface L which satisfy the condition
dS = pdq - Hdt.
(10.11)
4. Solve the Hamilton-Jacobi equation:
T + 1(q2+(aq)2' =0 with S(q, 0) = p(O)q.
(10.12)
11 Numerical Computations
11.1 Introduction In Chap. 9 and 10 we study the singularities of the exponential mapping near singular trajectories. You cannot get away from such analysis in order to determine for a given sub-Riemannian structure the analytic properties of the sphere, conjugate and cut loci. As these analytical computations are in general highly nontrivial problems, numerical computations provide a good alternative. In particular, as we have seen along this book the determination of conjugate points in optimal control theory is of crucial importance, but their formal computation is in general not possible. However, the algorithm to compute the conjugate points, and hence the conjugate locus, is simple and very easy to implement numerically. If in this chapter we restrict ourself to the sub-Riemannian case, the generalization to more general problem is straightforward. On the other side, if the definition of cut points is still simpler than the one for conjugate points, their computations is much harder even numerically and is not a topic of this book.
11.2 Numerical Algorithm In this section we describe algorithms and numerical methods used to compute the conjugate points and to obtain the sphere. The purpose of this chapter is not to develop new and sophisticated numerical methods but to use and adapt classical numerical methods to solve ordinary differential equations. In our opinion the results obtained with this simple approach are very satisfactory and the pictures significant. The advantage of the suggested method is that we can use existing and well tested numerical algorithms already implemented, and that the user does not have to make tedious computations but only need to provide the differential Hamiltonian equations governing the flow for the normal extremals as well as its linearization to compute the conjugate points. Let us start by reminding the algorithm to compute the conjugate points along normal extremals. Let U be an open subspace of R's and consider the following local sub-Riemannian problem on U:
322
11 Numerical Computations m
u:(t)F:(q(t))
4(t) _
min J(i(t) , 4(t))y dt
)
where the vector fields Fi are smooth and generate a distribution D of rank m, and (,).g describes the scalar product associated to the metric g. In the sequel we assume the vector fields Fi to be orthonormal for g. Hence we are minimizing:
sn JT(u(t))cit. 1: We have seen in section 9.2 that the normal
i=1
extremals are solutions of the following differential Hamiltonian equations:
q(t) =
P(t) _
n (q(t),P(t)),
-aqn (q(t),P(t))
(11.1)
where the Hamiltonian is given by Hn(q, p) = 1 Ei_1 Pit, Pi being the Poincare coordinates: Pi = (p, Fi(q))2. We assume the geodesics parametrized m
by arc-length, which is E P,2(t) = 1 for all t. Then we have Hn(q(t), p(t)) _
LetgoEUandr>0fixed and expgo:CxR -+U (Po, t) '-' q(t, qo, Po)
be the exponential mapping, the domain C being given by m
C={PoElRn; E(Po,F:(go))2=1}.
(11.2)
Because the possible existence of strictly abnormal geodesics, the wave front
W(qo, r) is the closure of the image of C by expQ0 evaluated at t = r. The sphere S(qo, r) centered at qo and of radius r defined as the set of points at a distance r from qo is a subset of the wave front. A point is said to be conjugate to qo along a normal extremal if it is the image of a point (po, t1) E C x IR such that the exponential mapping is not an immersion. The time ti is called a conjugate time. Numerical computation of conjugate points uses the variational equation corresponding to the normal Hamiltonian flow. Recall that the vertical (respectively horizontal) space are respectively subspace of T*U generated by the vectors of the form ayp- (respectively f3j ). If q(tl) is conjugate to qo = q(O) along a normal extremal, there exists a nontrivial Jacobi field J such that J(O) and J(ti) belongs to the vertical space (see Chap. 6). If z = (q, p) and bz = (Jq, bp), a Jacobi field is solution of the variational equation given by: bz(t) =
ae n(q(t),P(t))bz(t).
(11.3)
11.2 Numerical Algorithm
323
To find the conjugate points to qo, we proceed as follows. Let Ji(.) be the Jacobi fields defined by Ji(0) = ei, i = 1, , n, where the vectors e 1, - , en form a basis of the vertical space. Then, a time t,, is conjugate to 0 if there exists a = ( a1 ,--- , an) E Rn, a # 0 such that: - -
-
n
aiirh(Ji(tc)) = 0
(11.4)
i=1
where 7rh is the projection mapping on the horizontal space: 7rh(Ji(t)) _ bgn(t)). Numerically we apply the following procedure. (bg1(t), bQ2(t), Given an initial starting point qo and an initial value po, we integrate in parallel the flow for the normal extremal corresponding to these initial values
and the flow of the variational equation for the n initial values 6z(0) = ei. We then have n solutions of the variational equation: bzi(t) = (bqi (t), bpi (t)), linearly independent at the starting point qo. At a conjugate time, the projection of these n solutions on the state space are linearly dependent, see (11.4). Hence to find the conjugate points, we must check at each step of integration if the determinant vanishes:
bqi (t) ... bqi (t)
D=
(11.5)
bgn(t) ... bgn(t)
where bqi = (bqi , ... , 6q;).
The conjugate locus is the set of first conjugate points, then it can be computed using the previous method with po varying in C.
To integrate the flow corresponding to the normal extremals and the variational equation, any classical numerical method can be used. The pictures shown in this book have been computed using an explicit Runge-Kutta method of order 5(4), due to Dormand and Prince. Moreover, to increase the efficiency of the computations, the Runge-Kutta method is supplemented by
a dense-output. Indeed, the classical Runge-Kutta methods are not efficient when we need to approximate the solution in a dense set of points, for example to draw a graphic or to study the zeroes of a function involving the approximated solution (as it is the case to determine the conjugate points). By adding a dense output, we then obtain a numerical approximation with regularity properties (continuity, C1, we can impose some values of the function and its derivative, etc) depending on the choice of interpolation used for the dense output method. For our problem, the gain due to the dense-ouput is important as we need to find the zeroes of a determinant, see equation (11.5). Then, instead to refine the step of the Runge-Kutta method and start again the algorithm, we simply use a dichotomie to locate with a very good precision the zeroes. The dense-ouput used by the authors is of order 4 due to Shampine, it provides a globally C1 approximation of the solution. More details about this numerical method can be found in [53].
324
11 Numerical Computations
The wave front is computed using the same numerical method to evaluate
the flow exp Hn (qo) at t = r and by taking its adherence. The sphere S(qo, r) is the exterior envelop.
11.3 Applications In the next few sections, we present the results obtained when applying the algorithm and the numerical method described previously to some subRiemannian structures. We will start with the contact situation, pursue with the Martinet case and finally analyze the elliptic and hyperbolic cases.
11.3.1 Contact Case in Dimension 3 The contact case of order -1 and the generic one are discussed in respectively Sect. 9.5 and 9.6. The picture of the generic conjugate locus can be found in Sect. 9.6.2, Fig. 9.4.
11.3.2 The Martinet Flat Case It is the local sub-Riemannian structure (D, g) defined on a neighborhood U of 0 E R3, where D is the kernel of the Martinet one-form: w = dz - z dx, q = (x, y, z) are the coordinates of R3 and the metric is given by g = dx2+dy2. The Martinet flat case is analyzed in Sect. 9.7. If we take F1 = +I FX- , F2 = a as generators for the distribution D =ker w, then we have [F1, F2J = yWZ_ and the abnormal geodesics are contained in the analytic surface {y = 0} called the Martinet plane. In the sequel we assume the starting point of the geodesics qo to be the origin. The abnormal geodesics starting from 0 and parametrized by arc-length are the straight lines: x(t) = ±t, y =- 0, z = 0. The equations of the normal
geodesics are given in Proposition 86. In particular, as the equations are integrable in the flat situation, we have an analytic expression for the normal geodesics in terms of the elliptic functions. This means we can use softwares as Mathematica or Maple to draw the geodesics and even the sphere (as we know that along a normal geodesic, the cut point corresponds to its first intersection with the Martinet plane). However, two importants remarks have to be made.
First, if we do the computations using the analytical expressions with the elliptic functions it is very time consuming and there is no comparaison with the numerical computations. If succeded, the computations for the sphere would take several hours instead of few minutes. Moreover, the precision of the algorithm is such that we would not see a difference between the two methods of computation. The second remark concerns the conjugate points and the conjugate locus. Using the reduction procedure described in [41, we can compute a parametrization of the conjugate points as follows. If
11.3 Applications
325
,y(t) = (x(t), y(t), z(t)) is a normal geodesic parametrized by Proposition 86, a conjugate time along 'y(.) satisfies the following equation: c(t,'P,
A)
= 8J (t,,P, A)
! (t,
