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Lrnm
[(Zip  R(ip)>'x(O))' dfl < co
(2.14)
From (2.13) and the Plancherel theorem
IF(t)12 dt I C:2n(lrpII; (2.15) Jom Using (2.9), (2.14) and (2.15) we obtain (2.12). Theorem 2.1 is proved. =
c:2n
Theorem 2.2. Under the assumptions of Theorem 2.1 the trivial solution of (2.1) is asymptotically stable.
Proof: Show that for IIrpl/o < 00 x ( t , rp) + 0,
t
+ co
(2.16)
From (2.11) it follows that x(t, rp) may be considered as a Fourier coefficient for X ( i f l ) . Hence, by virtue of one theorem on Fourier series ([61], p. 529), relation (2.16) holds if each component X,(i/?)may be presented as a sum of two functions. The first of them is absolutely integrable on ( 00, co) and the and tends to second one is bounded, monotonic for all sufficiently large zero for I/?/ + co. On the basis of the Plancherel theorem, (2.6) and (2.13), Jm
I(ipZ  K(ifl))'F(ip)I d p
a,
n
IlKll=
1 i,j=l
IKijl
52
2. Stability of Retarded Equations
(2.17) Here A,,(ifi) = a,,(ifi)"' matrix (ifiI  K(ifi)). If
I4,(fi)I If large
+ O(JfiI"2),
( j , 1 = l , . .., n) are cofactors of the aljx,(0)= 0, then because of (2.17),
C,IBI2,
Jmm
I4,@)12 @ <
aljx,(0)# 0, then from (2.17) it follows that 4j(fi) is bounded and for
I fll:
(2.18) Hence 1 &?) I is expanded in the sum of two functions. One of them for large IfiI is equal to aljxr(O)I and another is absolutely integrable. So (2.16) follows from the work of Fichtengol'tz [ 6 1 ] . Show now that the trivial solution is stable. Define the operator sequence { A , } mapping the space of the initial function cp(B) with norm (2.6) into the space of continuous and bounded functions: A,cp = x(t, cp) for 0 It I n and A,cp = x(n, cp) for t 2 n. Every operator A , is continuous on the basis of solution continuous dependence on initial data. From (2.16) we get that every solution x(t, cp) of the problem (2.1), (2.2) with JJcpJJo< co is bounded. Hence, the sequence { A , } is bounded on every element cp; i.e., sup, A,cp < co.Using now the concentration of singularity principle ([96], p. 269), we obtain that the sequence { A , } is uniformly bounded in n:
Ifil'Ic;=o
IIA,rpllB = sup Ix(t9 cp)l I YIIcpllo nrtrO
(2.19)
53. Methods of Stability Investigation of Linear Autonomous RFDEs
53
where the constant y is independent of n. In the linear case, stability of the trivial solution follows from (2.19). Theorem 2.2 is proved.
Theorem 2.3. Let f o r some y > 0 r m
and also let the characteristic function (2.10) have no zeros in the halfplane R e z 2 y. Then I x ( t , cp)l IC , exp( yt).
Proof: Consider the function y ( t ) = x ( t ) exp(yt). Substituting y ( t ) in (2.1) we obtain the characteristic function AJz) = det[(z  y ) I  R(z  y ) ] of the e q u 6 n for y(t). This and the assumptions of Thorem 2.3 imply that Ay(z) has no zeros in the halfplane Re z 2 0. So by virtue of Theorem 2.2 we get limt+my ( t ) = 0. Theorem 2.3 is proved.
$3. METHODS OF STABILITY INVESTIGATION OF LINEAR AUTONOMOUS RFDEs By virtue of Theorem 2.2 the stability of RFDE (2.1) depends on the presence of zeros of the characteristic functions (2.1 1) in the halfplane Re z 2 0. We shall call “pzeros” the zeros with positive real parts. For ordinary differential equations, function (2.11) is reduced to a polynomial. Conditions for absence of pzeros for polynomials are given by the wellknown RouthHurwitz theorem. For general characteristic function (2.11) such complete results are not obtained.
3.1. Analytical Methods (Pontriagin’s and Chebotarev’s Theorems) Consider the quasipolynomial
c m
D(z) =
r
aljz exp(bjz)
I=O j = 1
Quasipolynomials (3.1) are characteristic functions for differentialdifference equations. The RouthHurwitz problem for quasipolynomials (3.1) may be studied by different methods. Let us examine some of them. Quasipolynomial with commensurable delays may be presented as
54
2. Stability of Retarded Equations
Consider the quasipolynomial D,(z) for z D,(iw) = g(w) + 80).
= io,
where w is a real number:
Pontriagin’s Theorem [175]. I f quasipolynomial (3.2) has no pzeros then all the zeros of the functions g(w) and f (0) are real, simple and alternating and g(w) f (0)f(w)g(w)> 0,
 cg
< 0 < co
(3.3)
For absence of pzeros of D,(z) one of the following conditions is suflcient: (1) All the zeros of the functions g(w) and f ( o ) are real, simple and alternating and inequality (3.3) isfuljilled for at least one real w ; ( 2 ) all the zeros of thefunction g(w) [or f (w)] are real and simple and for each zero relation (3.3) is satisjied.
The simple proof of Pontriagin’s theorem is given by Postnikov [177]. Consider, as example the equation, i ( t ) = bx(t

h),
b >0
(3.4)
The quasipolynomial D(z) = z exp(hz)  b = 0 corresponds to Eq. (3.4). By setting z = iw we get D(iw) = o sin wh  b
+ iw cos o h = g ( o ) + i f ( w )
g(w) = w sin o h  b,
f ( w ) = o cos o h .
Zeros of the function f ( w ) = o cos wh are equal to w o = 0, o k + , = + n/2)/h;k = 0, 1,. . . . These zeros are real and simple. Relation (3.3) for = wk has the form g(0)f (0)  j’(o)g(o) = b > 0, g(wk+i > f(%+ 1) f(o,+ l)g(o, + 1)= wk+ ,h(wk+  b ) > 0. The last inequality will be fulfilled under the conditions (kz
b < W 1 < ... < wk+l
So the trivial solution of Eq. (3.4) is asymptotically stable for 2bh < n. By analogy we may prove that the equation i ( t ) = ax(t) + bx(t  h ) is asymptotically stable if a < h’, a <  b < (a2 + p2h2)1/2,where p is the root of equation p = ah’tgp such that 0 < p < n. If a = 0 we set p = 4 2 . One of the ways of studying quasipolynomials (3.1) with incommensurable delays is generalization of the RouthHurwitz conditions. Expand the quasipolynomial D(z) in the series D(z) = a, + a,z + a 2 z 2 + . . . and define the functions u(z) and u(z): D(iz) = u(z) + iu(z), u(2) =
u(z) = a.
a,z  a3z3 + a5z5  .

a 2 z 2 + a4z4
+.
$3. Methods of Stability Investigation of Linear Autonomous RFDEs
55
Let us introduce the determinants Qm:
Chebotarev’s Theorem [37]. Assume that thefunctions u(z) and u(z) have no common zeros. Then quasipolynomial (3.1) has no pzeros ifand only i f Qm > 0,
1,2,.
..
(3.5) Applications of this theorem are not effective because an infinite number of inequalities (3.5) must be verified. 3.2. Method of D Subdivision. Vyshnegradskii Diagrams For determination of conditions under which quasipolynomial (3.1) has no pzeros, the method of D subdivision is widely applied [37,59, 160(1), 160(2)]. The method of D subdivision is based on the fact that zeros of quasipolynomial (3.1) are continuous functions of parameters a,j and b j . Construct the subdivision of the coefficient’s space by hypersurfaces, the points of which are quasipolynomials with at least one imaginary root. Such a procedure is called a D subdivision. For continuous variation of quasipolynomial parameters the number of pzeros may change only by passage of some zeros through an imaginary axis. Therefore points of every domain of D subdivision correspond to quasipolynomials with the same number of pzeros counted according to their multiplicities. Define now the number of pzeros at least in one point of every domain of D subdivision of the corresponding quasipolynomial. Then for all points of every domain the number of quasipolynomial pzeros will be just the same. D subdivision has a visual form when the quasipolynomial depends on two parameters. Substitute z = io in characteristic equation (3.1), linearly depending on two parameters a and b. Equating to zero the real and imaginary parts, we have
+ bQ,(o) + R,(o) = 0 q w ) = aP,(w) + b Q 2 ( 0 ) + R 2 ( 0 )= 0
U ( w )= a P , ( w )
(3.6)
where P l ( w ) ,. . . , R2(w)are some continuous functions. From Eqs. (3.6) we obtain
a = AJA, b = AJA, A = P,Q,  P2Q, (3.7) The formulae (3.7) determine the point a , = a(@,), b = b(wl) of D curve for w = wl. If o varies from a to 00 we obtain some curves. Besides these
56
2. Stability of Retarded Equations
Fig.2.1. D subdivision for quasipolynomial (3.9).
Fig. 2.2. Stability domain for the equation i ( t )
+ ax(t  1 ) + bx(t  1) = 0.
$3. Methods of Stability Investigation of Linear Autonomous RFDEs
57
curves, D subdivision contains singular straight lines such that either A = 0, A, = 0 or A = 0, A2 = 0 or A = 00, A, = 00. Consider for example, D subdivision of the equation i(t)
+ U X ( ~ +) bx(t  h) = 0
(3.8)
Characteristic quasipolynomial of Eq. (3.8)
D(z) = z
+ a + b exp(  z h )
(3.9)
+
has the root z = 0 for a b = 0. Quasipolynomial (3.9) has an imaginary b cos wh = 0 and y  b sin o h = 0. So the root z = iw if and only if a i boundaries of D subdivision are singular straight line a b = 0 and parametric curves a = (a cos wh)/(sin oh),b = o/(sin oh), 00 < o < 00. D subdivision for quasipolynomial (3.9) is represented in Fig. 2.1. Quasipolynomial (3.9) has no pzeros for a > 0 and b = 0. Hence Eq. (3.8) is asymptotically stable in the domain shaded in Fig. 2.1. The number of pzeros in other domains of D subdivision is indicated in Fig. 2.1. The stability domain for the equation % ( t )+ a i ( t  1) bx(t  1) = 0 is represented in Fig. 2.2. D subdivision for the equation x ( t ) a i ( t  1) bx(t) = 0 is given
+
+ +
+
b
\
P=2
P=2
P.2 P=l
Fig. 2.3. D subdivision for the equation , t ( t ) + nx(t

1)
+ bx(r) = 0.
58
2. Stability of Retarded Equations
in Fig. 2.3. The stability domain of this equation consists of an infinite number of triangles adjoining the axis a = 0. The stability domain for the equation x ( t ) a i ( t ) bx(t  1) = 0 is represented in Fig. 2.4.
+
+
Fig. 2.4. Stability domain for the equation x(t) + ax(t) + bx(t  1)
= 0.
The formulated results enable us to find the stability domains of the simplest feedback systems from Table 1.2. In the theory of automatic control, these stability domains are often called Vyshnegradskii diagrams [219(2), 241(2)]. Consider for example the system with P controller i(t)
+ T  'x(t) + K T  l x ( t

t) =
K T  l u ( t  t)
0'JI
Fig. 2.5. Vyshnegradskii diagram for system (3.10) in parameters a = T  ' , b = K T  '
(3.10)
59
$3. Methods of Stability Investigation of Linear Autonomous RFDEs
100 
50

20 10

52
Q 0 5 1 2 71
001002 0 0 5 0 1 0 2
Fig. 2.6. Vyshnegradskii diagram for system (3.10) in parameters TI = T', K.
10.5 
0.20.1 0.050.02
0 01
L
5
10
20
50
100
Tl
Fig. 2.7. Vyshnegradskii diagram for a system closed by I controller.
The Vyshnegradskii diagram of system (3.10) for a = T  ' and b = KT' is given in Fig. 2.1. But physically sensible feedback systems are possible only for K > 0 and T > 0. Consequently, the system (3.10) stability domain is less than in Fig. 2.1. The Vyshnegradskii diagram of system (3.10) in parameters a = T  ' and b = KT' is represented in Fig. 2.5. Logarithmic scales are used in Fig. 2.6,2.7 and 2.8. The Vyshnegradskii diagram for system (1.6) with an I controller is represented in Fig. 2.7 and with a PI controller in Fig. 2.8. For other diagrams for singlecircuit feedback systems see Gorecki [ 6 8 ] .
60
2. Stability of Retarded Equations
Fig. 2.8. Vyshnegradskii diagram for a system closed by a PI controller.
3.3. Michailov, Nyquist and Integral Frequency Criteria Frequency methods are used in the theory of automatic control for investigation of the stability of autonomous systems. For RFDEs, frequency methods were developed by Kabakov [93] and Tzypkin [232( 1)232(3)]. These methods are based on the argument principle from the complex analysis [123]. The Michailov and Nyquist criteria are the most frequently used. Consider the characteristic functions (2.11) A(z) = det[lr  ~ o w e  z s d K ( s )=] Z"
+ fi(z)z"' + ... + fn(z)
(3.11)
Here all the functions &
+
A ( i o ) = V ( w ) iV(w)
(3.12)
The hodograph of function (3.12) in the complex plane is called a Michailov one.
Michailov Criterion. For asymptotic stability of linear nth order equation with characteristicfunction (3.1l), it is necessary and su.cient that variation of the function A(iw) will be equal to nx/2 when w varies from 0 to co; i.e., arg A(ico)l 0
= nx/2
$3. Methods of Stability Investigation of Linear Autonomous RFDEs
61
The Michailov criterion for RFDEs was proved by Kabakov [93] and Tzypkin [232( l)]. But for RFDEs, Michailov hodographs are more complicated than for ordinary differential equations. Typical Michailov hodographs for stable systems without delays are represented in Fig. 2.9. Michailov hodographs for D(z) = 2z2
+ 0 . 5 ~+ 2 + exp( +zz)
(3.13)
are shown in Fig. 2.10 for z = 1, 5, 10. System (3.13) is unstable for z = 1, 10 and it is stable if T = 5.
I Fig.2.9. Typical Michailov hodographs for nthorder systems without delay.
Let the transfer function of a linear RFDE be equal to W(s).The function W ( i o )is called afrequency response of this system. Nyquist Criterion. Let the openloop system be stable. Then for asymptotic stability of the closedloop system it is necessary and sufJicient that frequency response of the openloop system does not envelope the point ( 1) (see Fig. 2.1 1).
The use of Michailov and Nyquist criteria for RFDEs is difficult because of the complexity of the corresponding curves.
62
2. Stability of Retarded Equations
I
r=10
Fig.2.10. Michailov hodographs for quasipolynomials (3.13).
Fig. 2.11. Nyquist criterion: I, stable system; 11, unstable system.
Integral criteria are more effective. Define the function of logarithmic derivative R(o)for a system with characteristic function A(z) R(w) = Re 
A’(z) A(z)
~
=
d [arg F(iw)] dw ~
U ( w ) V ( w ) V ( w ) U ( w ) U2(w) V 2 ( w )
+
(3.14)
$3. Methods of Stability Investigation of Linear Autonomous RFDEs
63
where A ( i o ) = U(o)+ i V ( o ) . From the argument principle the integral criterion of stability follows. Integral Criterion of Stability [ 1441. For asymptotic stability of RFDE with characteristic function A(z), it is necessary and suficient that
Iom nn 2
R(o)d o = 
(3.15)
Here R(o)is dejned by formula (3.14) The integral stability criterion is more convenient for computer calculations. Choose a number s such that
Ism
R(o) d o < 1
Then instead of condition (3.15) it is sufficient to verify the inequality I s m R ( w )d o >
( n  1)n 2
~
Consider, for example, the system with characteristic function
+
A(z) = 0 . 1 + ~ 0.32 ~ 0.5 + (0.12 + 0.2) exp( zzl) + (0.22 + 0.3) exp(  z t 2 ) We get
+ 0.5 + 0.10 sin zlo + 0.2 cos zlw + 0.20 sin + 0.3 cos z 2 0 V =0 . 3 ~ + 0.10 cos zlw  0.2 sin zlw + 0.20 cos z t o
U
=
0.10’
t20
 0.3 sin z2w
The function R(w) from (3.14) is represented in Fig. 2.12 for z1 = 3.0, z2 = 1.5 and in Fig. 2.13 for t l = 2.5, z2 = 1.7. Computing the integrals we have
[
20
R(w)d o = 3.0455;
z1 = 3.0, z2 = 1.5
At the first case the system is stable and it is unstable at the second one. Remark that studying the function R(o) we obtain some estimations of the transient regimes (see Melkumjan [144] and 44). The wave criterion for RFDEs is developed in Lekus and Rovinskii [124].
64
2. Stability of Retarded Equations
0 5
1 5 _  _ _ _ _ _ _ _ _ _ _ _   _  _ _               
Fig. 2.13. Logarithmic derivative R(w) of unstable system.
3.4. Stability for Arbitrary Delays. Tsypkin Criterion In some problems it is interesting to have the conditions under which an RFDE is stable for arbitrary delays. Consider an openloop system with transfer function W ( s ) = [ R ,  l(s)/Qn(s)] exp( ST), where R, l(s) and Qn(s)
65
$3. Methods of Stability Investigation of Linear Autonomous RFDEs
are polynomials of degrees ( n  1) and n. The closedloop system (Fig. 2.14) has the following transfer function Rn i(s) exp(sz)/CRn i(s) e x ~ (  s z ) + QAs)l
(3.17)
I
I
Fig.2.14. System with separated delay unit.
Tsypkin Criterion [232(2)]. Let Qn(s) be a stable polynomial. Then the closedloop system (3.17) is stable for arbitrary delays 0 T < co ifund only if IQ,,(io)l > l R n  l ( i o ) l ,
co < o < co
(3.18)
For the proof let us remark that the frequency response of system (3.17) may be constructed in two steps. First, one constructs the curve R, l(io)/Q,,(io). By virtue of (3.18) this curve is situated entirely within the unit circle. But the action of the factor exp(  izw) is reduced to the rotation of the vector So the frequency response curve of R,l(iw)/Qn(iw) on the angle (m). system (3.16) is situated entirely within the unit circle also and cannot envelope the point ( 1). By the Nyquist criterion, the closedloop system (3.17) is stable. r ) ~some o,,. Then it is always possible Now let lQn(io)cr)l= ~ R n  l ( i o cfor to choose the rotation angle (zcrocr) such that the frequency response passes through the point (1). In addition, system (3.17) is stable for and unstable for z, < z. 0
(3.19) Q ( s ) = S"
+ ulsh' + . . . + a,,,
+ +
Rj(s) = bojs" + bljs"'
+ . . . + bnj
Assume that \boll .. [born/ < 1 and that the polynomial Q(s) is stable. Then for asymptotic stability of the closedloop system (3.19) for arbitrary delays zi it is necessary and sufficient that m
66
2. Stability of Retarded Equations
The equivalent form of this statement is the following: Under formulated conditions, the quasipolynomial rn
WS)= Q(z>+ C
Rfz) ex~(jz)
j= 0
has no pzeros for arbitrary delays z j 2 0. The problem of stability for arbitrary delays of the system i ( t ) = Ax(t)
+ Bx(t  z),
x ( t ) E R,
(3.20)
is investigated in Lewis and Anderson [126], and Repin [183(1)]. It is proved that if the matrix A is stable and all the roots A j of the equation det[
AA B APE
APE ]=O AA  B
lie inside the unit circle for all real p, then system (3.20) is stable for all delays T 2 0 [183(1)].
3.5. General Function (2.11) [108(5)] Consider the scalar equation XYt) =
j:x"'(t

s) dkj(S),
t20
j=O
Assume that the integral uIj= joms'\dkj(s)l,
Blj = jomsl dkj(s)
are absolutely convergent. Formulate some conditions for the absence of pzeros for the characteristic function
in the cases n
=
1, 2.
(a) Let the conditions Boo < 0, a l o < 1 be fulfilled. Then equation Al,(z) = 0 has no pzeros. (b) Let ko(s) be nonincreasing and k,(s) = const for s 2 h 2 0. If Po, < 0 and ha,, < 742, then the equation A1(z) = 0 has no pzeros. The estimation haOO< 7c/2 is exact (see Fig. 2.1).
67
$4. Stabilization System of the TwoReflector Antenna
REMARK3.1. The condition Boo < 0 is necessary for the absence of pzeros for the equation A1(z) = 0. Otherwise, there exists a nonnegative real root of this equation. REMARK 3.2. Let the function ko(s)have a jump a, < 0 in the point s Then the equation A1(z) = 0 has no pzeros if
= 0.
(3.2 1) If, in addition, the function ko(s) is nondecreasing for s > 0, then by virtue of Remark 3.1 condition (3.21) will be necessary. (c) Let the function kl(s) have a jump a, < 0 in the point s = 0 and
Then the function A2(z) has no pzeros. (d) Assume that Boo < 0, ctl1 + ~ ~ < 1~ and / 2 ( 2/?00)"2 > 0. Then the function A2(z) has no pzeros.
Blo Pol
>
For the proofs of statements (a)(d) see Kolmanovskii [108(5)]. As an example consider the equation X(t) = ax(t)
+ bx(t  h),
t20
(3.22)
Because of (d), the trivial solution of Eq. (3.22) is asymptotically stable if 2b))'I2. This example shows that delay may stabilize the system. The system which is unstable for b = 0 becomes asymptotically stable for some b > 0.
2a > 2b > a > 0 and 0 < h < (b/(a
+
$4. STABILIZATION SYSTEM OF THE
TWOREFLECTOR ANTENNA The tworeflector (or Cassegrainian) antenna is used to form a narrow directional source of radiofrequency radiation. The tworeflector antenna consists of a big immovable reflector and of a little movable one, furnished with a control system. The big reflector is a hemispherical metal surface approximately 50 m in diameter, with the axis forming a certain angle with the horizon. The little reflector has a specially calculated complex surface with a radiator situated in its focus. The radiator is placed on a massive platform having a diameter of 7 m. To reach high guidance accuracy the control system includes a platform stabilization system, the precision of which is about 5" to 6". The antenna has to function outdoors continuously
68
2. Stability of Retarded Equations
by day and night under considerable wind and temperature disturbances. Platform deflections are measured by optical pickups. The actuator of the stabilization system contains three screws which can move three points of the platform in the vertical direction. Hence the stabilization system is multidimensional, more exactly threedimensional, including three separate channels for each screw servomechanism. The position errors in the screw servomechanism are measured by hydrostatic pickups which have considerable lags, until 1 to 2 sec.
Fig.2.15. Separate channel of screw servomechanism.
Fig.2.16. Scheme of multidimensional stabilization system
69
& Stabilization I. System of the TwoReflector Antenna
Fig. 2.17. Matrix structural scheme.
A separate channel of the screw servomechanism is represented in Fig. 2.15. Figure 2.16 shows a general scheme of a multidimensional stabilization system, and Fig. 2.17 gives a matrix structural scheme. The transfermatrix function of the openloop multidimensional system is
G(s) = W(s)CC(s)+ D(4l
(4.1)
From the symmetry of the multidimensional system we have Wll
K, D,,
=
W,,
=
W,,
= K,K,oK,K,, =
D,,
=
+ K2)'
= K,s'(T,s K2 =
1
+ K,oK,K, (4.2)
D , , = K , exp(sz)
Cij = K , exp(sz),
i , j = 1, 2, 3,
i #j
Numerical values for parameters are received experimentally [1441 and are as follows: for the engine T = T, = 0.11 sec; K , = 17.58 rad/V sec; reduction coefficient K , = 9 x mm/rad; pickup gain K , = 0.4 V/mm; interconnection gain K , = 0.05 V/mm; voltage gain K , = 250; power gain K,, = 10; local feedback gain K , = 0.03 V sec/rad. The transfermatrix function of the closedloop system is equal to @(s) = G(s)[I G(s)]  l . The characteristic function of the closedloop multidimensional system is
+
+ G(s)] = P,(s) + Pl(s) exp( sz) + P 2 ( s )exp(  2 sz) + P3(s) exp( 3s.r)
D(s) = det[I

(4.3)
71
§5. Liapunov Direct Method for Equations with Delay
R(wl
w
~v =500 Fig. 2.19. Family of R(w) for different K,.
For Kr ~ 0.02 V/sec rad the system is stable. The number of oscillations N decreases with the increase of Kr. Fig. 2.19 shows a family of graphs R(w) for r = 0.3 sec, Kr = 0.02 V/sec rad and K. = 200, 300, 500. Together with the increase of K. the transient time t, also increases, system stability decreases and finally forK.= 500 the system becomes unstable. For K 1 = 15, K 2 = 6.3, T = 0.11 one can evaluate critical lag ren which is equal to 0.5 sec. If 0 :::;; r :::;; rer• then the system is stable; for r > re., the system is unstable. These results have been confirmed by experiments [144].
§5. LIAPUNOV DIRECT METHOD FOR EQUATIONS WITH DELAY Stability theory of RFDEs is one of the important directions in the investigations of these equations. Stability theory is developed in several directions. First, stability of concrete 'types of equations are studied (for example, linear autonomous equations, linear periodic equations, quasilinear equations and so on). Further, general methods similar to the Liapunov direct method are created. In this section some fundamental ideas and results relating to the Liapunov direct method for RFDEs are presented.
72
2. Stability of Retarded Equations
5.1. Application of Liapunov Functions. RazumikhinType Theorems Consider the initialvalue problem for the RFDE q t ) = f(t,X J , x,,(e)
x,(e) = x(t
t 2 to,
+ e)
h Ie Io
= q(e),
(5.1)
Assume as in $1 that cp(6) E C [  h, 01, the operator f (t, cp), f : R , x QH+R, is continuous and Lipschitzian in cp E Q H and f ( t , 0)
(5.2)
=0
Investigate the stability of the trivial solution x(t) = 0 of Eq. (5.1). This is possible by the method of the Liapunov function [3,65,180(1), 180(2), 1973. Formulate a Razumikhintype theorem.
Theorem 5.1. Let there exist, for Eq. (5.1), the continuous positivedejnite function V(t,x), V : R , x R, +R,, whose derivative computed along the trajectories of Eq. (5.1) is nonpositiue for any solution x ( t ) satisfying the inequality V(s,x(s)) IV(t,x(t)),
s I t, t 2 to
(5.3)
Then the trivial solution of(5.1) is stable. If;in addition thefunction V(t,x) has an infinitesimal upper limit and its derivative V is negativedefinite along any solution x(t) of Eq. (5.1) such that
V(s,4 s ) ) I f ( V ( t , x(t)),
s
I t, t 2 to
(5.4)
where f is a continuous function and also f (u) > u for u > 0. Then the trivial solution of (5.1) is unformly asymptotically stable.
For the proof of Theorem 5.1 see Hale [81(1)], and Razumikhin [180(1), 180(2)].
5.2. Method of LiapunovKrasovskii Functionals for Equations with Bounded Delay Krasovskii has proposed to use for stability investigation the functionals defined on C [  h , O ] . Theorems 5.2 and 5.3 are proved by Krasovskii [114(1)114(5)]. Let K R , x Q H+ R , be some continuous functional such that V(t,0 ) = 0
(5.5)
Denote by o , ( r ) , r 2 0, some scalar, continuous, nondecreasing functions
73
$5. Liapunov Direct Method for Equations with Delay
such that o,(O) = 0 and o i ( r ) > 0 for r > 0. The functional V(t,cp) is positivedefinite if there exists a function o1such that V(t, CP) 2 wi(ldO)l),
~PEQH,
~ E R I
(5.6)
The functional V(t,cp) has an infinitesimal upper limit if there exists a function o2such that U t , cp) 5 f%(lIcp(~)lO
(5.7)
Substitute the trajectory xr(8), h I 8I 0 of (5.1), instead of cp(B), into the functional V(t, cp) and denote V ( t ) by V(t, x J . The derivative $' of the functional V ( t ,x,) computed along the trajectories of (5.1) is equal by definition to
P=
lim sup[V(t
+ At)kft)]/(At)
Ar + O
In other words the derivative $' is the righthand upper derivative of the functional V on the trajectories of (5.1). The derivative Pis called negativedejinite if there exists such a function o3that $'< 03(1x(t)l). Theorem 5.2. Let there exist continuous positivedefinite functional V(t,cp) such that VI 0. Then the trivial solution of Eq. (5.1) is stable. Proof: Assume any that
E E (0, H ) .
By virtue of ( 5 . 5 ) we can take d ( ~ to) , such
From condition $'I 0 it follows that V is nonincreasing along the trajectories of (5.1). Hence using (5.6) and (5.8) we have w,(lx(t)l) 5
v,x,) Iv t o , =
XtJ
t 2 to
V(t0,cp) I W1(E),
The monotonicity of w1 implies that Ix(t)l
E,
t 2 to. Theorem 5.2 is proved.
REMARK. If, in addition to the assumptions of Theorem 5.2, the functional V(t, cp) satisfies (5.7), then the trivial solution is uniformly stable. Theorem 5.3. Let there exist a continuous functional V(t, cp) such that
PI
(5.9) Then the trivial solution of (5.1) is uniformly asymptotically stable. Conversely, ifthe trivial solution of (5.1) is uniformly asymptotically stable then there exists ~l(lcp(0)l) V ( t , cp) I co2(llcp(~)ll),
03(IX(Ql)
74
2. Stability of Retarded Equations
a continuousfunctional V ( t ,cp) satisfying (5.9) and a local Lipschitz condition in cp.
For the proof of Theorem 5.3 see Krasovskii [114(5)] EXAMPLE 5.1. Consider a scalar equation i ( t ) = ax(t)
+ b(t)x(t

a > 0, h > 0, t 2 to
h),
(5.10)
where b(t) is a continuous function. Introduce the functional V ( t ,X J
The derivative
= x2(t)
+ a [hx2(s) ds
i. is such that
f'= 2x(t)[ax(t) I v[x2(t)
+ b(t)x(t
+ x2(t


h)],
h)] + ax2(t)  ax2(t  h )
v
= asup
Ib(t)l
ttto
By Theorem 5.3 the trivial solution of Eq. (5.10) is uniformly asymptotically stable if v > 0.
5.3. Equations with Unbounded Delay Consider an RFDE with unbounded delay (5.11) where the initial function cp(8) is continuous. As in $1 denote M a metric space of continuous functions q(8) cp: (00,01 + R, with a metric p . Denote Q H a ~ , p(q,O) I H ) . Let the map sphere in the space M , i.e., QH= ( c p M f : [0, co) x M + R, satisfy conditions sufficient for existence and uniqueness of the problem (5.1 1) solution. Assume also f ( t 7
0 ) = 0, I f ( t , q)l I F,,
t 2 to,
PEQH
(5.12)
Notice that Theorems 5.2, 5.3 are not directly applied to Eq. (5.11) with unbounded delay. Theorem 5.4. Let there exist a continuous functional V ( t , cp), I/: [0, co) x Q H + R , such that Ol(l
cp(0)I) I v ( t , cp) I 02(P(Cp, 0)) i.II 03(Ix(t)l)
Then the trivial solution of Eq. (5.1 1) is asymptotically stable.
(5.13) (5.14)
75
$5. Liapunov Direct Method for Equations with Delay
Proof: Show first that the solution x ( t ) = 0 of (5.11) is stable. Assume any > 0. Choose 6 > 0 such that w , ( E )= 0,(6). Using (5.13) and (5.14), we obtain for any q E Qa that
E
l . 0 I(
I x ( t )I)
I V ( t ,Xr> I
w o
1
cp)
I o,(P(cp? 0))
I0,(6) = 0 1 ( E )
(5.15)
The monotonicity of o1 implies that Ix(t, t o , cp(0))l s E for cp(0)E Qa. Persuade oneself that ~ ( t o, , q(@) + 0 for t + co and for any q(0) E QHl. Here H I is such that w,(H) = o,(H,). Reasoning as in (5.15) we get that Ix(t, t o ,cp(6))l I H for t 2 0 and cpeQH1.Assume now that the solution x(t, t o , cp(B)) does not tend to zero for some q(8) E QH1.Then for some E > 0 there exists a sequence { t i }such that Ix(ti, t o , cp(6))l 2 E and ti co.By (5.12) F,. Therefore, for all s such the derivative i ( t , t o ,cp(8)) is bounded: ii(t)l I that Is1 I . (5.14) it A = &/(2F0) we have Ix(t, s, t o , cp(6))l 2 ~ / 2 From follows PI  o 3 ( & / 2 )= v < 0 for Is1 2 A. So +
+
V ( t )  V(t0)=
1
Pdt 2
ti+A
1
P(t) dt
= 2 VA N ( t )
fist i i  A
where N ( t ) is a number of t i such that t i It. But N ( t ) * 00 for t + 00. Consequently, V ( t )  V(t,) +  co for t + 00.This contradiction with (5.13) shows that x(t, t o , q(8)) + 0 for t + co and cp E QHl. Theorem 5.4 is proved.
5.4. Stability in the First Approximation and under Steady Acting Perturbations Consider the equation
+
q t ) = ~ ( x , ) ~ ( tx,), ,
x, = x(t
+ e),
h I 8I o
(5.16)
Here L(cp):C[ h, 01 + R, is a continuous linear mapping and R(t, cp): R , x C[  h, 01 + R, is a continuous mapping. Along with (5.16) consider the equation of j r s t approximation q t ) = L(XJ =
L
[dqe)lx(t
+ e),
h Ie I o
(5.17)
where K(8) is a matrix whose elements are functions with bounded variations. The characteristic function of Eq. (5.17) is D(z), D(z) = det[
Iho
[dK(0)l]eze I z ]
(5.18)
Assume that IRfa 9)l
Pllcp(~)ll
(5.19)
76
2. Stability of Retarded Equations
Theorem 5.5. Let zeros zi of function (5.18) satisfy the condition Re ziI  v < 0. Then there exists a constant > 0 such that the trivial solution of (5.16) is uniformly asymptotically stable for any mapping R ( t , cp) complying with (5.19) [114(5)]. Requirement (5.19) in Theorem 5.5 may be weakened until the following: there is a number T > 0 and a continuous function $(s) 2 0 such that
+ jt
t+T
IR(4 cp)l I $(t)llcp(~)II,
*(s)
ds 5 B
For nonautonomous or nonlinear equations of first approximation, complete results are not derived, although there are some theorems in this direction [79(4), 8 1(4)]. In particular, if the trivial solution of the first approximation equation is exponentially stable, then the trivial solution of the full equation is asymptotically stable [108(5), 114(5)]. Let us investigate stability under steadyacting disturbances. Along with Eq. (5.1) consider the disturbed systems 4 t ) = f ( 4 x,)
+ N t , x,)
(5.20)
The trivial solution of (5.1) is called stable under steadyacting disturbances if for any E > 0 there exist 6 > 0 and 'I > 0 such that all solutions x(t, to, cp) satisfy the inequality Ix(t,to,cp)l I E for llcpll I 6 and IR(t,$)I I q, where 11$11 IE. Krasovskii [114(5)] has proved that the trivial solution of (5.1) is stable under steadyacting disturbances if it is uniformly asymptotically stable. It is possible to replace the condition of smallness of the disturbance's norm by the assumption of its smallness in the average [63,128].
5.5. Case of a Nonpositive Derivative Construction of LiapunovKrasovskji functionals satisfying the conditions of Theorems 5.2 and 5.3 involve some difficulties. Sometimes it is easier to construct the functionals with a nonpositive derivative. The corresponding theorems follow. First we formulate a statement which extends the BarbashinKrasovskii theorem. Consider an autonomous RFDE of the form (5.1): q t ) = jyx,),
t 2 0, x t
= x(
+ e),
 h I e Io
(5.21)
By definition, the element $ E C [  h, 01 belongs to an wlimit set corresponding to the initial function cp if x(t, 0, cp) is defined on [0, co) and there exists a sequence (t,,}, t, + 00 for n + co such that IIx,,(cp)  $11 + 0, n ,00, and xt,(q) = x(t, O,O,cp). The set Q c C [  h, OJ is called invariant if x,(cp) E Q, t E [0, 0 0 ) for any cp E Q.
+
77
$5. Liapunov Direct Method for Equations with Delay
It is possible to prove that the wlimit set corresponding to any initial function is situated in the invariant set of Eq. (5.21).
Theorem 5.6 [81(1), 81(4)]. Let there exist the continuous functional V(cp):C [  h, 01 + R , such that w1ClCp(O)l1 IUcp)I
wl(t)+ 00,
~zCllcp(e)lll and
t + 00
ex,) 5 0
Let Z be the set of those elements from C [  h, 07 for which V = 0 and Q is the greatest invariant set situated in 2. Then all solutions of(5.21) tend to Q for t * co. In particular, i f the set Q has the only zero element then the trivial solution of (5.21) is asymptotically stable.
Set forth the theorem extending the Matrosov stability criterion founded on the use of two Liapunov functions [108(5), 162(10)]. Consider problem (5.1). Let V(t,cp) and W(t,cp) be continuous functionals defined on I x QH and denote V ( t ) by V(t,x,) and W ( t ) by W ( t ,x,). The derivative W is called integrally unbounded in a set G E Q H if for any number B > 0 there exist a number T ( B ) > 0 and a continuous function ( ( t ) such that uniformly in X, = x(t 0) E G for t 2 to
+
W I
w,
6"
B ((s)dsI
(5.22)
T(B)
Denote d((p(O), C) the distance between the element cp(Q set G c C [  h, 01.
E C [  h,
01 and the
Theorem 5.7 [l08(5), 162(10)]. It is necessary and suficient for uniform asymptotic stability of the trivial solution of (5.1) that there exist thefunctionals V(t,cp) and W ( t ,cp) such that
ol(icp(o)i) w,
I cp) 4 w2(iicp(e)ii), t 2 t o , N O E Q ~ , ~ , E G (2) V I 0 3 ( x , )I 0, where W3(cp) is a continuousfunctional dejined on Q H ; ( 3 ) I W ( t , ~ ) Il L, t 2 to, d o ) E Q H , to E R i ; (4) for any p E (0, H ) there exists a p > 0 such chat the derivative W is integrally unbounded in the set E(p, p ) c C [  h, 01, where
(1)
Ebu, P ) = { v ( @ E Q H , d ( ( ~ ( e )Q(Gj ,
= 0 ) )5 P,
PI llc~(e)III H)
Here Q(W3 = 0) = (p E Q H , W3(cp) = 0). I n addition, the attraction domain of the trivial solution is the sphere S,, where K < H and w 2 ( K )I w,(H).
A general outline of the proof follows. Conditions (1) and (2) ensure the hit of any trajectory x ( t ) in the neighbourhood E(p, p ) of the set Q. By virtue of ( 3 ) and (4) the trajectory necessarily leaves E(p, p ) after a finite time. Then vwill
78
2. Stability of Retarded Equations
be negative. So the function V decreases and the trajectory x ( t ) approaches the origin. For a full proof of Theorem 5.7 see Kolmanovskii and Nosov [108(5)] and Nosov [162(10)].
EXAMPLE 5.2. Consider the equation
a(t) =  a x ( t )
+ b(t)x(t  k ) + c(t)y(t)
j(t>= yc(t)x(t),
t 2 0,
o
h2
(5.23)
Show that sufficient conditions of uniform asymptotic stability are (a Ib(t)l) I c1 < 0, c3 2 Ic(t)l 2 c2 > 0, y > 0. Introduce the functional
+
V ( t ,x , , y,)
= yx2(t)
+ y2(t) + ya
bih
x 2 ( s ) ds
We have I y(a
+ lb1)(x2(t)+ x2(t

k)) I 0
The set Q(ih3 = 0) s C [  k , 01 is, in our case, the set of functions (q,$) E C [  h, 01 such that q(0) = q(  h) = 0. The set Q(o3= 0) does not coincide with the origin. As a second functional we take
w(4 X t Y J = y(t)xtt>
sgn c(t>
(5.24)
This functional is bounded in any sphere Q Hand also
ri/=
 x(t)(  y c ( t ) x 2 ( t ) ) sgn c(t)  y ( t ) sgn c(t) .[ax(t) b(t)x(t  k ) c(t)y(t)]
+
+
The set E ( p , p ) is defined by the inequalities Ix(t)l I p, Ix(t  k)l I p, p2 I y 2 ( t ) 5 H 2 . In the set E(p, p ) we have
W=
c2p2
+ apH + Jb(t)JpH+ y c 3 p 2 I c 2 p 2 / 2
where p is such that apH
+ Ib(t)IpH + yc,p2
I 2apH
+ y c 3 p 2 I c,p2/2.
Hence the derivative Wis integrally unbounded in the set E(p, p). Asymptotic stability of the trivial solution of (5.23) follows now from Theorem 5.7.
5.6. Global Stability Let the map f(t, q),f : R x C[  h, 01 iR, be continuous and Lipschitzian in q E Q H for arbitrary H . Further, I f ( t , q)l I M , , t E R,, q ( 0 )E Q H .
Definition 5.1. T k e trivial solution of (5.1) is globally uniformly asymptotically stable i f it is uniformly stable and iffor any y > 0 and a sphere Qx there
79
55. Liapunov Direct Method for Equations with Delay
exists an l(y,K) such that l(y, K ) . t 2 6,
+
Ix(t,to,cp)l Iy for any
cp(B)EQ,
and
Theorem 5.8 [108(5)]. Let there exist two functionals V ( t , cp) and Wft, cp) satisfying the conditions of Theorem 5.7 in any Q H and q(s)
~1(1cp(O)I) I V ( t ,cp),
+
m,
s
(5.25)
+
Then the trivial solution of (5.1) is globally uniformly asymptotically stable. Notice that under condition (5.25) and other assumptions sufficient for asymptotic stability (see Theorems 5.1, 5.3 and 5.4) the solution is also globally asymptotically stable.
5.7. Survey of Other Results. Exponential Stability The stability problems for RFDEs are studied in different ways. Stability in the first approximation in various senses is studied by Halanay [79(4)], Hale [8l(l)] and Kolmanovskii and Nosov [108(5)]. Stability under steady acting disturbances is considered by Germaidze and Krasovskii [63]. Instability theorems are proved by Shimanov [205(1)]. Critical cases (i.e., the cases in which some roots of the characteristic function lie on the imaginary axis) are also studied [167,178,205(2)]. The Klimushev papers [ l o l l are devoted to the study of the stability of singular perturbed RFDEs. For applications of the vector Liapunovfunctional to the stability of connected systems see Gromova and Markos [73]. Also available is a survey devoted to the Liapunov direct method for RFDEs [3,108(5)]. Mention some results about exponential stability. The trivial solution of (5.1) is exponentially stable if any solution x(t, to, cp) of (5.1) satisfies the inequality
I x(t, t o , cp)l
I Bllcp(e)II exp(  a(t  to))
B > o, a > o, t 2 t o , iiqo(e)ii I H, IH
(5.26)
A necessary and sufficient condition [114(5)] of exponential stability of the trivial solution of (5.1) is the existence of a functional V ( t ,cp) such that
c,
CI /I (P(e>ll 5 v ( t ?cp) I II cp(@ i.5

c, 11% I/>
I V ( t , CP)

/I
v(t,11/11 I c, [email protected] 11/(e>II
(5.27)

where C iare some positive constants. Note that for a linear RFDE the exponential stability is equivalent to the uniform asymptotic stability. Further, for the linear autonomous RFDE (5.17), asymptotic stability implies uniform asymptotic stability and, hence, exponential stability. It is interesting
80
2. Stability of Retarded Equations
to notice that for linear autonomous NFDEs, asymptotic stability does not, in general, imply uniform stability. For example [78(1)], the equation i ( t ) + i ( t  h) x ( t ) = 0, h > 0, has a solution tending to zero for t ,00 as t” where v > 0 (see Chapter 1 45). This means that the trivial solution is not exponentially stable. Hence, asymptotic stability may not be uniform in the initial function.
+
96. CONSTRUCTION OF LIAPUNOVKRASOVSKII FUNCTIONALS
6.1. Linear Autonomous Equations Krasovskii has considered equations i ( t ) = Cx(t)
+ Bx(t  h) +
1;
D(0)x(t
+ 0) do,
XER,
(6.1)
h
A quadratic functional is constructed as [114(6), 183(2)]
+
V(X,,)= x’(O)C~X(O)
x’(s)~(s)x(s) ds j;h
(6.2)
Here a is a constant (n x n) matrix and p, y are (n x n) matrices with piecewise differentiable elements. The derivative P of functional (6.2) along the paths of Eq. (6.1) equals
P = x’(O)ox(O)
+ x’(  h);lx(O)
 h) ds + ~ ~ h ~ ’ ( s ) ~d s( +~ ) x x’(s)cp(s)x( (~) 1;h
(6.3)
Theorem 6.1 [114(6)]. Let the solutions of Eq. (6.1) be asymptotically stable. Then for any functional W of the form (6.3) there exists the uniquefunctional V of the form (6.2) such that V = W. Besides, if the functional W is negative dejinite, then the functional V is positive dejinite. Equations for matrices a, p, y depending on w , A, v, cp and E are derived by Repin [183(2)].
81
$6. Construction of LiapunovKrasovskii Functionals
Quadratic Liapunov functionals giving necessary and sujicient stability conditions for the matrix equation i ( t ) = Ax(t)
+ BX(t  h),
x E R,
(6.4)
are constructed by Castelan [34,35]. These functionals are obtained by passage to the limit in Liapunov functions derived in the usual way for difference equations which approximate RFDE (6.4). Notice that this algorithm is complicated enough and the problem of construction of more simple functionals remains actual. Tzar’kov and Engel’son study the problem of construction of a LiapunovKrasovskii functional for linear autonomous and periodic systems [23 1( l), 232(2)]. The functional is constructed in the form
v,cp) =
11
cp(S)cp(Sl)
dPt
Q
Here pt is a symmetric measure defined on the square Q = [  h, 01 x [ h, 01 for every t. A system of equations is derived for this measure. The equation i ( t ) = + a x ( t  1) is considered as an example by Tzar’kov [231]. LiapunovKrasovskii functional giving the necessary and sufficient condition of asymptotic stability (i.e., the condition n/2 < a < 0, see $3) is
+
cp2(s) ds
V(cp)= ~ ’ ( 0 ) a’
1’
a ( + 1 ‘OS +sina
+ (a2 + 2)
g(s) sin as ds 
1
~
2a
where
6.2. Stability of a Chemical Reactor Closed by a P Controller Investigate the stability of the chemical reactor described in Chapter 1, $1. Assume that the controller is a P controller and the actuator has dead zone, saturatim and hysteresis loops. A structural scheme of this feedback system is represented in Fig. 2.20. The nonlinear element describing the actuator may be defined in different ways. The output u(t) may be considered as a onevalued functional depending on previous history x(s), 0 I s I t ; i.e., u(t) = F(x(t + e)), t 2 8 I0. The distinctive feature of such a description is that
82
2. Stability of Retarded Equations
Fig. 2.20. Structural scheme of feedback system for chemical reactor.
the functional has unbounded delay for t + co.It is obvious that F satisfies the Lipschitz condition:
IF ( d t +

F($(t
+
I
r II cp(t
+ 0)  rL(t + 0) IIC[O,t ]
or IF(cp(t + 6))  F($(t
+ Wl'
[om
jorndsR(r,cp> $)
I&
 s)  $(t  s)l' dsR(s9 cp,
$)
(6.5)
r2
+
The system represented in Fig. 2.20 is described by the equation T i ( t ) x(t) = K,K,F(x(t  z + 0)) + K,K,y(t  7). Let j ( t ) be a fixed input signal, x ( t ) corresponding to j ( t ) be a disturbed solution and ( x ( t ) z(t)) be a disturbed solution. The disturbance z ( t ) satisfies the equation
+
~ i ( t+) z ( t ) = K,K,F,(z(t

+ 8))
(6.6)
where ~,(z(t
+ 8)) = q x ( t  7 + e) + z(t  T + e))  F(x(t  5 + el) z2(t z + e) d , ~ ( s x,, , zS) F:(z(t  + 8)) I 
lom
Consider the functional rr
r m
V
=
T z 2 ( t )+ K , K , Y  ~J
[d,R(s, X,, z,)] 0
J
z2(sl) d s , tr+s
83
96. Construction of LiapunovKrasovskii Functionals
The derivative of this functional is
P = 2 z ( t ) [  z ( t )  K,K,F,(z(t
+ K,K,r'z2(t)
 7
+ 6))]
d,R(s, X,, z,)  K , K p r  ' Jorn
. Iornz2(t T
+ s) d,R(s, X,, z,) I
+ K,K,rlF:(z(t
T
2z2(t)
+ K,K,rz2(t)
+ 6 ) ) + K,K,r1z2(t)r2
. {omz2(t  7 + s) d,R(s, X,,z,)
 K,KpF1
2 z Z ( t ) [  1 + K,K,r]
If K,K,r < 1, then all hypothesis of Theorem 5.4 are fulfilled. Hence, the trivial solution of Eq. (6.6) will be asymptotically stable. In other words, the closed system of a reactor regulation will be stable.
6.3. Scalar Nonlinear Equations Consider the RFDE
Kernel ko(s) has bounded variation on [0, a]. The initial function rp(6) E CB[  co,01; i.e., it is continuous and bounded on (  co,01. The continuous functional a(t, rp): [0, a)x CB[  co,01 + R , is such that a(t, 0) = 0 and
(6.9)

Let the kernel ko(s) have a jump b > 0 in zero and also
Idko(s))+ r,
b> +O
cll0
+
Jams
dR(s) < 00
(6.10)
84
2. Stability of Retarded Equations
Then the trivial solution of Eq. (6.8) is asymptotically stable. For the proof, consider the functional V(t,x,)
= X2(t)
I’,
+ j+mo dko(s)
+I*
ds,
X2(Sl)
jOrndR(.)
6’,
x2(s1) ds,
(6.1 1)
The derivative of the functional (6.1 1) is
v = x2(t)(  2b +
+ 2x(t)
c jm
[
a(t, x,) 
dk,(s)
x ( t  s)
< Ix’(+ 
Idko(s)l 
+O
.]
1 (6.12)
Hence, by virtue of Theorem 5.4,the trivial solution is asymptotically stable under conditions (6.10). Investigate the similarly nonlinear equation i(t) = 
~ ( t S)
d,k,(t,
S)
+ ~ ( tx,),,
t 2
0 (6.13)
lorn
xo(0) =
do)
Here k,(t, s) = 0 for t 2 0, s = 0 and
ko(t, s) =
s:
f ( t , sl) ds,
+
m
ai(t)e(s 7i(s)) i= 1
t 2 0, s > 0, sl(t)= 0, al(t) # 0, 2 4 s ) = 1
+ sgn s
The function ko(t, s) has for all t a bounded variation in s E [0, m). Assume that the functions ai(t) are continuous and that zi(t) are continuously differentiable.Function f ( t , s) is continuous and f ( t , s) = 0 for t I 0, s 2 0. There exists a constant L > 0 and a bounded nondecreasing function g(s) such that for all t 2 0 the series lai(t)l converge uniformly and 1 Iai(t)I < L,
I : ,
I?=
js;lf(t>s)l
ds I g ( 4

g(s,),
Vs,,
s2,
0 < s1 < s 2 a
Let the Eq. (6.12) coefficients meet the formulated conditions and also let
85
$6. Construction of LiapunovKrasovskii Functionals
(a) 0 I ti I q < 1 for all i (hence functions t  t i ( t )have inverse differentiable function ki(t)). (b) r
sup [Iomds
Pm
[+’
m
If(tl, s)l d t ,
+
120
s dR(s) lom
+ f l.l’)/mi(s)Ids]
< co.
i=2
Then the trivial solution of (6.1 3) is asymptotically stable. For the proof, consider the functional r m
+f
prs
kiW
Iai(s)lx2(s zi(s)) ds
i= 2
+
I,
IomdR(s)[;2(tl)
dt,
(6.14) From (6.14) and (a)(c) it follows that x2(t) I
v<
v I c,JJx,lJ;
[
x2(t) 2a,(t) m
 1 Iai(t>I i=2
m
jom
If(t
+ s, s)l
ds  2r 
If(t, s)\ d s om !
1 Iai(ki(t))hi(t)I i=2
1
I P1X2(t)
So, because of Theorem 5.3, the trivial solution of Eq. (6.13) is asymptotically stable. b(t)x(t  h), conditions (a)(c) of For the equation i ( t ) = a(t)x(t) asymptotical stability take the form inf, a(t) > sup,)b(t)/,t 2 0. The cases in which equations of the first approximation have no jump at zero are more complicated. They are studied on the basis of degenerated functionals in Chapter 3, $4.
+

6.4. Nonlinear Equations of Second Order
Some functionals for secondorder equations are constructed by Kolmanovskii [105(1I)] and Kolmanovskii and Nosov [108(5)]. In view of
86
2. Stability of Retarded Equations
their awkwardness we study here only the system i ( t > = Y(tX
P(t) = a (t)y(t)
+ a(t)x(t

(6.15)
h)
Assume that 0 < A I a(t) I C , , 0 < a(t) I C , ,
2a(t) 2 ha(t) + [tt+hu(s)ds,
b(t) I 0
(6.16)
Then the trivial solution of Eq. (6.15) is stable. For the proof, consider the functional
+ y,(t) +
I/ = a(t)xZ(t)
It may be verified that
P I Ci(t)XZ(t)
+ yZ(t)
[
za(t)
+ ha(t)] +
6"[email protected]) ds]
Now stability follows from Theorem 5.3. If both inequalities (6.16) are strict then by Theorem 5.3, we have asymptotic stability. If only the first inequality (6.16) is strict, then asymptotic stability may be proved by means of the Matrosov criterion 5.7. The set Q (G3 = 0) in this criterion for the functional I/ consists of elements y,(B) E C [  h, 01 such that y ( t ) = 0. Define the second functional
w = y(t)x(t

h)
(6.17)
Functional (6.17) is bounded in any sphere Q H and
w = x(t)y(t)x(t

h)  a(t)xZ(t  h )  y(t)y(t  h )
In the set E ( p , p ) (i.e., for pz < llxt112 I H Z and Iy(t)l I p ) we have W I ApZ
+ p H SUP l ~ ( t )+l p H I Apz/2 120
Here p is such that
p H sup la(t)l
+ pH I Ap2/2
t>O
Now the asymptotic stability of the trivial solution of Eq. (6.15) follows from Theorem 5.7.
87
$7. Stability of Nuclear Reactors
In Sinha [210] stability of some third and fourthorder equations is investigated by using LiapunovKrasovskii functionals. For references devoted to applications of the Liapunov direct method to RFDEs see Alekseevskaja and Gromova [3].
57. STABILITY OF NUCLEAR REACTORS 7.1. SingleTemperature Reactor with Convective Feedback
A mathematical model of the singletemperature reactor with convective feedback described in 41.1.6 is [70] N ( t ) = [ aT(t  z)
T ( t ) = r"(t)


& ( N ( t ) NO)]N(t)
(7.1)
N o  T(t)]
Investigate the stability of the stationary solution T(t) = 0, N ( t ) = N o . The system of the first approximation for (7.1) is ii(t) = bn(t)  aT(t

z),
T ( t ) = n(t)  T ( t )
where a = a N O ,b = EN,,, n(t) = N ( t )  N o , r tion for system (7.2) has the form
=
(7.2)
1. The characteristic equa
z 2 + ( 1 b)z+b+aexp(z(z)=O
(7.3)
D subdivision (see Chapter 2, $3.2) for quasipolynomial (7.3) consists of the straight line a + b = 0 and the curve a(o) = (1
+ 02)o/(o cos oz + sin oz)
b ( o ) = o ( w sin 05  cos O T ) / ( O cos oz + sin oz)
In Fig. 2.21, D, denotes the stability domain for z > 0 and D, u D, is the stability domain for z = 0. The delay z has a certain destabilizing effect on system (7.2), and it is necessary to take it into account. According to Theorem 5.5 about stability in the first approximation, system (7.1) is asymptotically stable if its parameters lie in the stability domain D, of linear system (7.2) ~701.

7.2. Stability of Two Interconnected Reactors
The dynamics of two interconnected reactors exchanging energy by neutrons under the assumption that the internal feedback of each reactor
88
2. Stability of Retarded Equations
tb
C={( t TI’,( Fig. 2.21. Stability domain for singletemperature nuclear reactor.
may be characterized by negative reactivity factors is described as = pl(t) 
” N , ( t ) + cxlz N z ( t  zlZ)+
4
11
ml
1 Ail Cil i=l mz
Nz(t)= pz(t)
 P2
N,(t)
+ clzl N , ( t  z21) + 1 AjzCj2 12
12
Cil(t)=
 A ~ ~ c+~Pi, ~( ,~( t)) , 1
j= 1
i = I , . . . ,m,
11
+
ejz(t) = Pl(t) = P l O  El“l(t)
1,
(7.4) ~ ~ ( t ) j , = 1, . . . , m,
 N101,
P Z ( t ) = P z o  &,“At)  N201 Here N i , pi(i = 1,2) are neutron density and reactivity in the reactor number i, respectively; li is neutron lifetime; C i k ,I , and P i k are concentrations, decay constants and fractions of group number i of retarded neutron emitters in reactor number k, respectively; m iis a number of groups of retarded neutrons in reactor number i; ei is a negative power reactivity factor; and PI, = x ? L l B i k Investigate the stability of the stationary solution = 1,. . . ,m,
Cil = Cilo > 0,
i
C j 2 = C j z O> 0,
j = 1,. . . ,m 2
(7.5)
89
97. Stability of Nuclear Reactors
Let us introduce the new variables
Equations (7.4) can be written in the form
= xl(t)  yil(t),
i = 1, . . . ,m,
A,
j = 1,. . . ,m,
A;'$il(t)
The parameters of system (7.6) are expressed in an obvious way from the parameters of system (7.4). Consider the LiapunovKrasovskii functional Y2 v = Y1 x: + x: + c pilA;ly;l + 2 2 1 ml
i=
Jtr12
m2
pLj,Ajz1yj22 j =1
Jt121
The derivative of functional (7.7) along the trajectories of system (7.6) is
v = b,x;(t)[l + Xl(t)]  b,x:(t)[1 + x2(t)] i= 1

j=I
[Xl(t  T12)  xz(t)I2/2  CxAt  z21)  x1(t)I2/2
According to physical interpretation, the quantities N , ( t ) and N 2 ( t ) are always positive; this implies that [l + x,(t)] > 0, [l + x2(t)] > 0. In this subspace all assumptions of Theorem 5.3. are verified. Thus, the stationary solution (7.5) is asymptotically stable for all zlz, zZ1 and all positive parameters of system (7.4) [70].
90
2. Stability of Retarded Equations
58. MATHEMATICAL MODELS IN IMMUNOLOGY
Problems of mathematical study and simulation of immune processes have ' been considered by many authors [14(2), 55, 1411. This section deals with some immunological models devoted to the description of virus diseases. These models have been essentially developed and studied by Marchuk [21, 139(1)139(4)], whose papers we follow. 8.1. Model of Virus Disease Let us assume that a small quantity of viruses (antigens) has penetrated a human organism and after a certain time reached an organ which they can affect. The viruses begin to multiply and infect the cells and then hit blood and lymphatic nodes. The viruses that have penetrated the lymphatic nodes have some probability of meeting lymphocytes that react to viruses of a given kind. The lymphocyte divides and transforms into an antibodyproducing plasmacyte. The formation time of plasmacytes is about 18 to 24 hours. We shall denote the number of viruses in an organism as V , the number of plasmacytes producing antibodies as C, the number of antibodies as F , the relative characteristics of the damaged part of tissue as m, and u, p, y,p c , pf,p, q, CT,,urnas different positive constants. The variation of the number of viruses in the organism will be described by the equation
P(t) = p y t )  p F ( t ) V ( t )
(8.1)
The first righthand term of Eq. (8.1) describes the result of virus division. The second term denotes the number of viruses neutralized by antibodies F . Equations for C, F and m are derived in a similar way. Finally, we arrive at the following system of FDEs describing the dynamics of a virus disease:
P(t>= (B  YF(t))V(t) C(t) = <(m)uF(t  7)V(t  z)  pc(C  C*)
m
(8.2) =P
W )

W F ( t ) V ( t ) P,F(t)
k(t)= dV(t)

pL,m(t)
Here C* denotes the constant level of plasmacytes in an organism. The delay in the second equation (8.2) represents the time interval from the beginning of stimulation of lymphocytes to the beginning of a mass antibody synthesis. The factor ((m) takes into account the fact that antibody production falls if the vitally important organs are seriously damaged. The typical curve of ((m) is represented on Fig. 2.22. There is a segment [O, m*] such that ( ( m ) = 1 on it; i.e., the functioning of the immunological system does not depend on the 7
91
$8. Mathematical Models in Immunology
Fig. 2.22. Typical curve of c(m).
dynamics of the disease. For a bigger damage (i.e., on the segment [m*,11) antibody production falls rapidly. The values of m* and of the decreasing rates are different for different diseases. According to the biological interpretation, we consider only the initial data V ( t ) = 0,
t < 0, V(0)=
F(0) = F,,
v,,
C(0) = c,
m(0) = m,
(8.3)
8.2. Analysis of Model (8.2), (8.3)
Theorem 8.1. Let the initial data (8.3) be positive. Then the solutions of system (8.2) are also positive for all t 2 0. From the first equation (8.2) we have
[sd(B
V ( t ) = V, exp
 yF(s)) ds
1
2 0,
t20
(8.4)
On the segment [0, z] the second equation (8.2) has a form C ( t ) =  p c ( C  C*), C(0) = c,.
The solution of this equation is C ( t ) = C* + ( C ,  C*) exp(  p , t ) 2 0. Analogously from the third and fourth equations (8.2) we may derive that F(t) 26 and m(t) 2 0 on [0, r ] . In the similar way we treat the segments [z, 251, etc. The result of Theorem 8.1 coincides with the biological interpretation of the model. The functions V ( t ) , C ( t ) , F ( t ) represent the concentrations of different substances and must be positive, and also 0 Im(t) I1.
92
2. Stability of Retarded Equations
Theorem 8.1 implies that C(t) > C* for t 2 0 if co> C*. Really, from the second equation of system (8.2) using the variationofconstants formula and the inequality t(m(s))F(s z)V(s  t) 2 0, we have c ( t >= c *
+ (Co  C*) exp(pct) + a
<(rn(s))F(s z) J O
. V ( s  t) ds 2 C* System (8.2) has two stationary solutions. Solution 1 (V, = 0, C , = C*, F , state of a healthy organism.
= pC*/pf = F*,
rn,
= 0)
describes the
Solution 2 V2 = Pc(PfB  YPc*)/B(aP  Pc?r> c2 = (WfP
 9Y2PcC*)/P(aP P J Y )
F 2 = B / y , m2 = SV2/pm.If these values are positive, they can be interpreted as a state of chronic disease.
Theorem 8.2. Stationary solution 1 is asymptotically stable if
B < YF*
(8.5)
Proof: If system (8.2) is linearized in the neighbourhood of the stationary solution y , Ci, F i , mi, i = 1, 2, we have lXt) = (B  YFi)U(t) Y V f ( t )  ~ f ( t ) u ( t )
i ( t ) = CtFiU(t  5 )
+ ffi/,f(t
 t)  pcs(t)
+ au(t  z)f(t
f ( t >= ~ s ( t) (VYY+ ~ f ) f ( t) ?yFiu(t)  ~ f ( t ) o ( t ) b(t>= 04t)

 t)
(8.6)
PmAt)
Here we denote u(t) = V ( t )  6 , s(t) = C(t)  Ci, f ( t ) = F(t)  F i , p ( t ) = m(t)  mi, i = 1, 2. The characteristic equation for the linear part of (8.6) in the neighbourhood of solution 1 can be written in the form
D(4 =

Z)(P(,
+z) =0
Condition (8.5) implies that all zeros of D(z) are negative. Thus, applying Theorem 5.5 on stability in the first approximation to system (8.6) we complete the proof. Let us investigate now the infection dynamics of a healthy organism. Assume that a small quantity Voof viruses has penetrated it at initial moment
93
$8. Mathematical Models in Immunology
t = 0. Consider the component V ( t ) of the solution of system (8.2) corresponding to the initial data V, > 0, C , = C*, F , = F*, m, = 0.
Theorem 8.3. Let condition (8.5) hold and also let 0 < V, < IB = P V  ' ~  ~ C*P f Y  l V 
(8.7)
Then the component V ( t ) decreases on the interval [O, a).
At t = 0 we have v(0) = (B  yF*) V, < 0. Consequently, V ( t ) decreases on some interval [O, t , ) . If t , # a,then there is a first moment t , such that P(t,) = 0 and v(t,+ A) 2 0, A > 0. Therefore, it follows from the first equation (8.2) that yF(t,) = B, yF(t, + A) IB and P ( t , ) S O . On the other hand, from the third equation (8.2), Theorem 8.1 and Eq. (8.7), we have =m
&Z)
t , >  w F ( t , ) V ( t z )  PfF(t2)> PC*  rB%  P f P Y  > 0
This contradiction proves Theorem 8.3. The value IB is called an immunological barrier. Theorem 8.3 states that if at the initial moment the immunological barrier has not been passed, then the disease is mild and the quantity of viruses penetrating an organism decreases. Consider now the stationary solution 2. In this case the characteristic equation of the linear part of system (8.6) has the form
W )= (Pm + Z ) D , ( Z ) D,(z)=(z3az2bz+d+gzexp(zz)fexp(zz)) a = Pc
d
+ VYV, + Pf > 0,
= YPCVB&
> 0,
b
= Pc(?YV,
9 = WV,,
f
+ P f )  VYBVz
(8.8)
= Bg
Theorem 8.4. Zf pu,t s 1,
0 < (f  d)(a  gz)' < ( b  g  f z )
(8.9)
then stationary solution 2 is asymptotically stable.
Proof. The proof of Theorem 8.4 will follow from Theorem 5.5 about the first approximation stability if we show that Eq. (8.8) has no zeros in the halfplane Re z 2 0. Applying the Michailov criterion (see Chapter 2,§3.4) to the quasipolynomial Dl(z), we obtain

+ d  f cos o~+ g o sin w t V ( o )= Im D,(w) = o3 bw + gw cos o z + f sin wz
U ( w ) = Re D,(w) = ao'
Since U(0) = d  f < 0, V(0)= 0. The inequalities B(a

g z ) > ypC,(l  p c z ) 2 0,
p C a> b 2 br
94
2. Stability of Retarded Equations
imply for w > 0 dU/do >~
+
4  2( f~~ g)s  SZ] > o [ ( u  gz)
+~
( b g  ft)] > 0
Consequently, there is a value o2such that U ( 0 2 )= 0
and
V(0,) < w2[o:  b
+ g + fzl
<0
On the interval (0,w 2 ) we have U(o)< 0 and V(o)< 0. The vector Dl(iw) turns to the angle n/2 when o increases from 0 to w 2 .The function V ( w )+ co if w + co.Thus, there is a value w 3 such that V(wJ = 0 and the vector D l ( i o ) turns to 742 when o increases from o2to w g .When w increases from o3to co,the vector D,(iw) also turns to 71/2 because V ( o )grows more rapidly than does U(o).Finally, the vector Dl(io) turns to 3n/2 when o increases from 0 to 00. According to the Michailov criterion, the quasipolynomial D,(z) has no zeros in the halfplane Re z 2 0. Theorem 8.4 is proved.
Fig. 2.23. Typical Michailov hodograph for D,(iw).
A typical Michailov hodograph for D , ( i o ) is shown on Fig. 2.23. Notice that conditions (8.9) depend on the value of delay z.
8.3. Discussion Consider now some biological results. On the basis of the previously stated mathematical model we can get typical pictures of virus disease dynamics. If there is a sufficient number of functioning antibodies, the viruses that invade
95
58. Mathematical Models in Immunology
6
16

&A2 +
Fig. 2.24. Typical curves of virus disease dynamics: 1, mild case of disease; 2, chronic form of disease; 3, sharp form of disease; 4, mortal form of disease.
an organism will meet with a powerful response and their number will decrease and approach zero. This is a mild case of a disease described by Theorems 8.2, 8.3 and shown by curve 1 on Fig. 2.24. In this case the immunological barrier is not passed by the viruses. If the immunological barrier is passed, the number of viruses grows. If, in addition, the number of viruses grows more rapidly than does the number of antibodies neutralizing the viruses, the curve of virus concentration begins to grow exponentially. However, after plasmacytes have formed and mass antibody production has begun the growth of the virus concentration slows down and some time later it falls rapidly. At the same time, there occurs a reproduction of new antibodies whose total number decreases exponentially until the normal immunological level is reached. The damaged part of the organ in which an evolution of the virus population occurred begins to recover exponentially. This is a sharp form of a disease (curve 3, Fig. 2.24). If the immunological system is weak and cannot neutralize the viruses, then the number of viruses grows without restriction and the organism dies (curve 4, Fig. 2.24). It may happen, however, that a process of virus multiplication goes on in the organism. The viruses bind with antibodies present in blood plasma. Thus, a balance is established between the number of viruses generated every second and those neutralized by antibodies. Here we deal with a stationary process which can be interpreted as a chronic or persistent form of a disease (curve 2, Fig. 2.24). These four types of virus disease dynamics are illustrated by Table 2.1. The existence of these types of disease dynamics has been confirmed by numerical simulation of model (8.2). Marchuk [139(3)] also considers more complicated models containing 8 or 11 equations, but qualitative analysis of these models is more difficult.
96
2. Stability of Retarded Equations
Table 2.1
Arise conditions
Disease form
Pecularities
B < YF* B > YF*
1. Mild 2. Sharp
V(t)
+ 0,
t
t
co
t ( t ) is great
/9  yF* > 0.33 PS > rlYf
3. Chronic 4. Mortal
0 < B  F*y < 0.33
V(t)
+
B > YF* SP < rlrf
V(t)
+
V, > 0, t + co co, f + co
$9. DESIGN OF ADAPTIVE CONTROLLER FOR RETARDED SYSTEMS Adaptive controllers, and in particular model references adaptive controllers (MRAC), are often used for quality control of plants with a wide range of variation of dynamical properties or with considerable noise [I108(5), 1221. An MRAC principal scheme is represented in Fig. 2.25. The aim of MRAC design is the construction of an adaptation circuit which provides the ) x ( t )  y(t). Here x ( t ) is system output and y(t) stability in phase error ~ ( t = is model output. 9.1. Scalar Equations
Let the system be described by the scalar RFDE i ( t ) = (a
.~
+ AU + &(t))x(t) + ( b + Ab + 6b(t)) ( t h + Ah + 6h(t)) + f ( t ) , t 2 to
(9.1)
Here Aa, Ab, Ah are a priori unknown quantities from some domains: IAal 5 A , , IAbl I B,, IAhl I H , < h. Parameters 6a(t), 6b(t), 6h(t) are changed by adaptation circuit and f ( t ) is an input, I f ( t ) l I F o . Along with RFDE (9.1) consider the reference model j ( t ) = ay(t)
+ by(t  h) + f ( t ) ,
t 2 to
(9.2)
with zero initial data x ( t ) = y ( t ) = ~ ( t=) 0, t I to. For a > Ibl the trivial solution of Eq. (9.2) is uniformly asymptotically stable (see $3). At first assume that delays in the system and reference model are equal, i.e., Ah 6h(t) = 0. Then equation for ~ ( t=) x ( t )  y ( t ) is
+
i(t) = ae(t)
+ bE(t

h)
+ cr(t)x(t) + B(t)x(t
t 2 t o , a(t) = AU
+ du(t),

h)
P ( t ) = Ab
+ 6b(t)
(9.3)
97
59. Design of Adaptive Controller for Retarded Systems
Y
Fig. 2.25. Principal scheme of adaptive controller.
The design of an adaptation circuit may be done on the basis of the LiapunovKrasovskii functionals method [108(5), 122,163(2)].Consider the functional V t , E,, 4
8) = y(t)E2(t) + U 2 ( t ) + B2(Q
;
+ (4) 
7 ( t ) ) l  h & 2 ( s ) ds
The derivative of functional (9.4) along the paths of Eq. (9.3) is equal:
P = (7  2ay)c2(t) + 2 y b ~ ( t ) ~( th)
+ 2 y x ( t ) ~ ( t ) a ( +t ) 2yx(t  h ) ~ ( t ) B ( t ) Take the adaptation circuits of parameters a and
as
(9.4)
98
2. Stability of Retarded Equations
Model
I
I Multiplier
Plant
L) I
Delay unit
iMultiplier
Multiplier
Fig. 2.26. Scheme of adaptive controller for system with delay.
The principal scheme of an adaptive system using algorithm (9.9, (9.6) is represented in Fig. 2.26. The derivative of functional (9.4) along the paths of Eqs. (9.3), (9.9, (9.6) is
i. = (j  2ay)&Z(t) + 2Yb&(t)&(t h)
99
$9. Design of Adaptive Controller for Retarded Systems
For all t 2 0 let
Theorem 9.1 [108(5), 163(2)].
C, I ay
+ Ibl I  C
a
0 I C, I y(t) I C,,
+ (bly + 37 I C4
<0
<0
(9.8)
<0
a?  + y
Then in adaptive system (9.3), (9.9, (9.6) we will have s(t) +0 t + co. If, in addition, for any number p , q, p 2 + q2 2 p 2 > 0 and B > 0 there exists a number T(B) such that
Itf+
(py(t)
+ qY(t

h))' dt 2 3 8
(9.9)
T(B'
then the trivial solution of Eqs. (9.3), (9.5), (9.6) is globally uniformly asymptotically stable and stable under steady acting disturbances. Proof: According to Theorem 9.1 we get C5(E2(t)+ a2W
+ P2(t))Iv,
Et,
I C,
a,8)
max (~'(s)
+ a 2 ( s ) + /?'(s>)
thjsSt
Relation (9.7) yields
V
I (a
+ 1 b 1 + iv)y(t)s2(t)+ ( ay + I b 1 y + f j ) s 2 ( t

h)
I  C4(&2(t)+ &2(t h)) I 0 Assume now that &(t)does not tend to zero. Then the integral from f' is unbounded but the difference V ( t )  V(0) is bounded. Hence s(t) + 0, t + co. For the proof of the second part of Theorem 9.1 let us use the Matrosov criterion (Theorem 5.7). The equality = 0 is possible only for s(t) = s(t  h) = 0. So functional (9.4) satisfies the first two conditions of Theorem 5.7, As a second functional consider
w,st, a,P)
= (a(t>x(t)
+ P(t)x(t  h))E(t)
(9.10)
Denote E ( p , p ) the domain E(p, p)
B)
CCh, 01 x R,,I4t)l
I P? Is(t  h)l I p, 0 < p 2 < a2(t) + B2(t) < a:},
= ((4 a,
a, < a
The derivative W of the functional W along the paths of Eqs. (9.3), (9.9, (9.6) equals W = (a(t)x(t) p(t)x(t  h))[as(t) bs(t  h) a(t)x(t)

+
+
+
+ P(t)x(t  h)I + s(t)(d/dt)(a(t)x(t)+ B(t)x(t  h)) = (a(t)Y(t)+ bYt)y(t  h))' + c(t)f(a, b, x, i,~ ( t )~, ( t h)) (9.1 1)
100
2. Stability of Retarded Equations
For W in the domain E ( p , p ) the estimate holds
s,
s,
s+ T(B)
s+ T ( B )
W t )dt
2
+ qy(t  h))’ dt  21 B 2 2.5B, p z + q2 2 pz > 0
(py(t)
p = tL(s), q = P(s),
~
Statement of Theorem 9.1 now follows from Theorems 5.7, 5.8. 9.2. Delay Adjustment [162(1 l)]
Let the reference model be described by (9.2) and let the plant be described by the equation i(t)=
ax(t)
+ bx(t  h + Ah + 6h(t))+ f(t) t 2 to
If x(t) is a twice continuously differentiable function and IA all t > t o , then ~ ( t=) ~ ( t) y(t) satisfies the RFDE
+ bs(t  h) + b[x(t  h + z(t))x(t = [email protected]) + bs(t h) + bo(t)i(t  h) + b(zZ(t)/2) X(t  h + [email protected])) where z ( t ) = Ah + 6h(t), 0 < O < 1. i(t) = as(t)
(9.12)
+ 6h(t)(< h for  h)]

(9.13)
Consider the following algorithm of delay selfadjustment: t ( t ) = (d/dt)(bh(t)) =
by(t, x)W  h1.m  V(t, x)C4t)/u(t)I
(9.14)
Here u(t) is a sensitivity function of solution x(t) relative to the following bu(t  h) + b i ( t  h), u(t) = 0, t 2 t o . delay variations: u(t) = au(t) Assume that lu(t)l > 0. Then algorithm (9.14) is physically realizable. Algorithm (9.14) is more complicated than (9.5) and (9.16) since, in order for it to be realized, we must know not only x(t) or x(t  h), but also the derivative i ( t  h). Notice that if a system is described by Eq. (9.1), and the variables x(t), x(t  h), i ( t ) may be measured exactly, then Aa and Ab may be determined algebraically from the equation, and we may then set 6a(t) = Aa, 6b(t) = Ab. Otherwise, under these assumptions the problem of parameter adjustments a(t) and B(t) is trivial. But for Eq. (9.12) the knowledge of x(t), x(t  h), i ( t ) and i ( t  h) is not sufficient in general for the determination of z(t) by an elementary method. Therefore, it is meaningful to take into account i ( t  h) in adaptation algorithm (9.14) for z(t).
+
101
$9. Design of Adaptive Controller for Retarded Systems
Theorem 9.2 [l08(5)]. Let thefirst condition of Theorem 9.1 be fuljilled, let the function x ( t ) be twice continuously differentiable and let IX(t)l Ic,
0I c, Iq(t, x) I c,,
lu(t)l > 0
(9.15)
Then the triuial solution of system (9.13), (9.14) will be uniformly asymptotically stable and stable under steadyacting disturbances for all suficiently small Ah such that Iz(t)l < hfor t 2 to.
9.3. Multidimensional Systems Let the reference model have the form j ( t ) = A(qO>y(t) + B(qO)y(t h) + f ( t ) ,
(9.16)
y(t)ERn
The system equation is i ( t ) = A(q)x(t)
+ B(q)x(t  h) + @(t)+ f ( t ) ,
x E Rn
(9.17)
Here matrices A(q) and B(q) are linearly dependent on the parameters q = ( q l , . . . ,qm), and @(t) is the action on the system from the adaptation circuit. We are given 4 = qo + Aq>
Aq
= (Aq,, . . ., Aqm)
(9.18)
Then Eq. (9.17) has the form
rA
+ B(qo)x(t  h)
i ( t ) = A(qO)x(t)
+f i= 1
aqi
Aqi x(t)
+ aB Aqi x(t aqi 

1+
h)
+
@(t) f ( t )
The problem is to choose a function @(t) such that E(t) + 0 for t
(9.19)
co. Put
Equation (9.20) is an adaptation signal for which the functions Gq,(t), i = 1,. . . , m, must be determined. For E = x  y, we get
+
rd
i ( t ) = A(qO)&(t) B(qO)E(t h)
+
f i= 1
aqi
x(t)
+ x(t aB aqi

1
h) ai(t)
(9.21)
(9.22)
102
2. Stability of Retarded Equations
Find functions vi by using the Liapunov direct method. There exists a quadratic functional vl(t,E ( t ) ,
E,) =
r' = r > o
+
&yt)r(t)E(t) T/z(&,),
(9.23)
such that the derivative of functional (9.23) along the paths of the system i(t)
=
+
(9.24)
A(qO)&(t) B(qO)&(t  h)
5 wg( le(t)/), where the subscript indiwill be negativedefinite ( pl)(9.24) cates that the derivative is taken along the path of Eq. (9.24). The existence of such a functional follows from Krasovskii Theorems 5.3, 6.1. Take now the functional rn
V ( t ,4% &,, u(t>,K t ) ) = vI(4
40,E,) +
1
(9.25)
i= 1
The derivative P of functional (9.25) along the trajectories of equations (9.21), (9.22) is equal to
Assume now that
Then the derivative V will be negativedefinite relative to
'
8,:
0
= (p1)(9.24) 5  w 3 ( 1 E ( t ) l )
Finally we get the equations of the adaptation circuit
It may be proved that E ( t )  + O as t  + co (See Kolmanovskii and Nosov [108(5)], $6.1). For the proof of the relation a ( t ) 0 some further assumptions must be made as in Theorem 9.1. +
$10. Stability of Viscoelastic Bodies
103
910. STABILITY OF VISCOELASTIC BODIES
This section is devoted to the study of applications of Liapunov direct methods to the investigation of the stability of viscoelastic bodies. General stability statements formulated in terms of the existence of a Liapunov functional could be easily obtained by adaptation of the theorems from $5. But the greatest interest is in constructing new Liapunov functionals and obtaining the effective stability conditions for concrete problems.
10.1. Constitutive Equations of Viscoelastic Bodies Viscoelastic bodies are characterized by the need to consider their memory effect, i.e., the influence of all past states of bodies from the initial to the current instant. Such phenomena are called hereditary. General features of phenomena under consideration are creep and relaxation. Notice that for a viscoelastic material which has been unstrained at all times prior to t o , the strain c(t) at time t is determined by the stress history o(z), to 5 z It through the constitutive equation. In accord with Reiner’s second axiom of rheology [6], viscoelastic materials are those whose behavior is partly fluidlike and partly solidlike. The principle phenomenon distinguishing viscoelastic substances from purely viscous fluids is the occurrence of strain recovery and viscoelastic substances are distinguished from perfectly elastic solids by the occurrence of creep. It appears that in viscoelastic substances stress produced by strain should relax as in purely viscous fluids, but at a finite rate. In order to obtain the constitutive equation, the Boltzmann superposition principle was used. The evolution of stress and strain are governed by the Constitutive equation
Here E(t) is a strain, o ( t ) a stress, E the Young’s modulus and K ( t , s) a creep kernel. The constitutive equation of nonhomogeneous ageing bodies has the form C6l
Here p(x) is a function characterizing the heterogeneity of an age. The model of nonhomogeneous ageing bodies generalizes the foregoing models of classical viscoelasticity.
104
2. Stability of Retarded Equations
Fig.2.27. Viscoelastic hingedhinged bar on viscoelastic foundation.
10.2. Dynamic Stability of a Viscoelastic Bar Let us derive the stability conditions of a viscoelastic hingedhinged bar on a viscoelastic foundation (Fig. 2.27). The bar is subjected to an axial load P . Denote y ( t , x ) as the lateral deflection of the bar from its straight position. The foundation action on the bar is equal to q =  C y  C l j , where j = ay(t, x)/at and constants C > 0, C , 2 0. The bar stability is investigated with respect to the initial disturbances of its deflection.
Definition 10.1. The bar is called stable iffor any E > 0 there exists a S ( E ) > 0 such that W ( t ) = yo(y(t,x)’ + j ( t , x)’) dx < E as soon as W(to)< 6 ( ~ ) . The bending moment M is given by M ( t , x ) = EJ(Z

R)y” = E J
Here J is the moment of inertia of the cross section and r(t  s) a relaxation kernel. The equation of equilibrium is
M” + Py”  q
=
(10.1)
pj
where p is the density of the bar material. Represent the deflection as a Fourier series m
y(t,x ) =
1 a,(t) sin d 1 x , n= 1
I,
71
=
1
(10.2)
105
$10. Stability of Viscoelastic Bodies
Substituting (10.2) in (lO.l), we obtain an RFDE for a,(t) a,
+ Blnhn+ BOnan+ B2,
I:
r(t  s)a,(s) ds
bOn= [c ~ i f n + 2 ~ ~ i ; ' ~ 4 1 ~  1fiIn , =c l p pzn = pL1(n11)4EJ, an(td = an07
=
0 (10.3)
hn(t0)
= ano.
Hence the investigation of the stability of the trivial solution of Eq. (10.1) is reduced to the investigation of the stability of Eq. (10.3), studied earlier in this chapter. It is convenient to put a,(t) = 0 for t < t o . Then a solution of (10.3) may be considered as a solution of
ii,
+ Blnhn+ POnan+ Bz,,
61
r(s)a,(t  S) ds = 0
(10.4)
Consider the Liapunov functionals V,
2 vn
+ 2a,Z(t)B3n +
= hi(t>
+ l:r(s) B3n
= Bon
ds l  s d t l
+ I,rW
[
hn(t)
+ Blnan(t)  Ilr(s) ds jt:san(tl)
jt:ld(f2) + B3,a;(tdI
dtl]
d t ~
(10.5)
ds
of functionals (10.5) along the trajectories of Eq. (10.4) The derivatives satisfy the inequality
From the positivedefiniteness of functionals (10.5) and the negativedefiniteness of functionals (10.6), we arrive at the following stability conditions
P < max(EJn2lF2,2 m ) C1pL'>
I:
sr(s) ds
(10.7)
Actually, relations (10.5) and (10.7) yield

m
1 (a,2(t)+ h i @ ) )+ 0,
t
+ co
n= 1
Hence, by virtue of the Parseval equality, W ( t )+ 0 for t + 00. But for any finite time interval, W ( t )= O(W(to)).So the bar is stable under conditions (10.7).
106
2. Stability of Retarded Equations
511. ABSOLUTE STABILITY OF SYSTEMS WITH DELAY Absolute (in the LurieAizerman sense) stability has been investigated by Popov and Halanay [176]. Razvan’s work [ l 8 l ] and the appendix to the Russian translation of it by Lihtarnikov and Jakubovich [130] contain the most important results in this domain. We limit our presentation to the statement of the problem and the formulation of two theorems.
a Summer
Fig. 2.28. Feedback system with nonlinear actuator.
Consider a feedback system with delay shown in Fig. 2.28. System phase variables are measured by pickups. The adder generates a control signal (r = c l x l . . . + c,x, = c’x, which governs an actuator. The actuator output 5: is equal to f(a) in the case of direct automatic control without delay, i.e., ( ( t ) = f(a(t)). When there is a delay in the actuator, we assume that t(t>= f ( 4 t  4). Suppose that plant dynamics is described by the RFDE
+
i ( t ) = Ax(t)
+ Bx(t  h) + b((t),
~ ( tE )R,
Here the matrices A, B and vector b are constant. The system closed by an adder and an actuator with delay is described by the equation i(t) =
Ax(t)
+ Bx(t

h) + bf[.(t  h ) ] ,
C’X
(11.1)
107
$11: Absolute Stability of Systems with Delay
If the actuator has no delay, then we get k(t) = Ax(t)
+ Bx(t

h)
+ bf(a(t)),
(11.2)
= C'X
As a rule, the characteristic f(a) is nonlinear and its exact determination is difficult. Assume that a continuous characteristic f ( a ) is defined for all af(0) = 0 and also satisfies the following conditions: (a) There exist constants k,, k, > 0 such that k , 0 2 (b) J:f(a) do = CO.
< of(o)
k,a2.
Such characteristics are called admissible.
Definition 11.1. Control system (11.1) [or (11.2)] is said to be absolutely stable in the angle [k,, k,], ifthe trivial solution ofEquation ( 1 1.1) [or (1 1.2)] is globally asymptotically stable for all admissible characteristics f (a). For determination of absolute stability conditions of concrete systems one uses the ordinary frequency domain criteria of PopovHalanay [1761. We now state these criteria.
Theorem 11.1. Let all the roots of characteristic equation det[A lie in the halfplane Re z that
(l/K)
+ B exp( zh)

<  c i < 0. Also let
zZ]
=0
(11.3)
there exist a number q > 0 such
+ Re(1 + j o q ) exp(jwh)c'(A + B exp(joh) .b>O,
joZ)'
k,
j z = 1
Then system (11.1) i s absolutely stable in the angle [k,, k,].
Theorem 11.2. Let all the roots of characteristic Eq. (11.3) lie in the halfplane Re z I CI < 0 and also let there exist a number q > 0 such that (l/W
+ R(l + jwq)c'(A + B exp( jwh)
jwI)'b 2 0
The system (11.2) is absolutely stable in the angle [k,, k,].
108
2. Stability of Retarded Equations
912. STABILITY OF LINEAR PERIODIC EQUATIONS 12.1. General Stability Theorem
Consider the equation (12.1) Here L(t, cp), L : R , x C [  h, 01 ,R , is a continuous map linear in cp and w periodic in t : u t U t , CiCP
+ 0, cp)
=u
t , cp)
+ C2$) = C i U t , CP)+ C2L(t, $1,
t ER,,
~ p ,$ E CCh,
01
Basic results about the stability Eq. (12.1) were obtained by Halanay [79(2), 79(4)], Shimanov [205(3)], Stokes [223( l)], Hale [81(4)] and Zverkin [253(3)]. See also El'sgol'tz and Norkin [59], Halanay [79(4)], Kolmanovskii and Nosov [108(5)]. Introduce the shift operator U ( t )of period w on the solution of Eq. (12.1) U(t)cp = x , + ~ ,
U(t): C [  h, 01 + C [  h, 01
The operator U ( t ) maps any function cp(B) E C [  h, 01 into the part x,+, of the solution of Eq. (12.1) satisfying the initial condition x, = cp. The shift operator U ( t ) is called also a monodromy operator [223]. It may be proved [79(2), 2231 that for w > h and any t the monodromy operator U ( t ) is completely continuous and its spectrum is at most countable and does not depend on t : a(U(s)) = a(U(t))for each s and t. The eigenvalues of operator U ( t ) also called characteristic (or Floquet) multipliers of (12.1). Theorem 12.1 [223]. For asymptotic stability ofthe solutions ofEq. (12.1) it is necessary and suficient that the spectrum a( U ( t ) )of the monodromy operator U ( t ) lies inside the unit circle. I f all Floquet multipliers p j ofEq. (12.1) are such that lpjl I 1 and the multipliers p k with lpkl = 1 have simple elementary divisors, then the trivial solution of Eq. (12.1) is stable.
12.2. Particular Classes of RFDEs
Theorem 12.1 reduces the problem of the stability of RFDE (12.1) to the investigation of spectrum a( U(t)). Usually the problem of determination or estimation of a(U(t))is very complicated and now there are no significant results in it. But for some particular classes of Eq. (12.1) this problem is
I09
$12. Stability of Linear Periodic Equations
studied better. Consider a scalar equation with delays divisible by period [253(3), 253(5)]: i(t)
+ b,(t)x(t) + b,(t)x(t  mlo)+ b,(t)x(t  rn2w) = 0 bi(t + = bi(t), i = 0, 1, 2, x(t)E R , W)
(12.2)
where rn, and m, are positive integer numbers. Let us look for a solution of the form x(t> = P(t> exp(W,
P(t
+ W ) = P(t>
(12.3)
Such solutions are usually called Floquet solutions. For p ( t ) we get the equation @(t>+ g(4 M t ) = 0 g(t,
A) = b,(t)
+ b , ( t ) exp( m,Aw) + b,(t) exp( m,Aw)
Integrating this equation we conclude that the function
i
p ( t ) = C exp At 
c I g(s,
A) ds
must be w periodic. Hence we have the following equation for A: AW
+ jomg(s,A) ds = 0
(12.4)
Equation (12.4) is a quasipolynomial. We know (see Chapter 1,95) that Eq. (12.4) has at most a countable number of roots and each of them has finite multiplicity. The Floquet solution (12.3) corresponds to every zero of quasipolynomial (12.4). By Theorem 12.1 for asymptotic stability of Eq. (12.2), it is necessary and sufficient that all the roots A j of Eq. (12.4) satisfy the condition Re Aj < 0. For systems of equations with delays divisible by period, it is also possible to write out a characteristic equation, but its coefficients depend on the elements of the fundamental matrix of an ordinary differential equation. In this case, simple assertions are not derived [253(5)]. Nevertheless there are some results for equations with quasiconstant coefficients [115]. Thus for a system
110
2. Stability of Retarded Equations
the particular solution has the form PI = [email protected])tlP(t, 111,
4 0 ) = 1, p(t, p) = P(t
+ w, PI
Consequently, if all the roots I j of Eq. (12.6) satisfy the condition Re < 0, then system (12.5) is asymptotically stable for sufficiently small p. Some stability criteria for periodic equations with small parameters are obtained by a Laplace transform. For example, in Valeev [233] it is proved that the trivial solution of the scalar equation i(t) = 2p(cos t)x(t  h), p > 0, is stable if sin h > 0 and unstable if sin h < 0.
12.3. Floquet Theory Floquet solution of Eq. (12.1) is a solution x(t) such that X,+w =
pxt,
x, # 0, p # 0, a < t < 02
The number p called a Floquet multiplier is also an eigenvalue of the monodromy operator U ( t ) .The Floquet exponent I equals I = o In p. If p is a simple eigenvalue, then the Floquet solution $(t) = p(t) exp(At), p(t o)= p(t) corresponds to p. Let p be an rmultiple eigenvalue of the operator U(t). Then a finitedimensional subspace E,(t) c C [  h, 01 corresponding to p is invariant relative to U(t). The action of the operator U(t) in subspace E,(t) is described by a matrix. This matrix has Jordan form for appropriate basis. The following Y Floquet solutions correspond to an rmultiple eigenvalue
+
ICljl(t) = Pjl(t>ex~(At) = [{l/[m(j> t+hjmcj,(t)

l]!} t m ( j )  lpjl(t) + . . + ~jm(j)(t)Ie~*(12.7) j = 1, . . ., q,
m(1) + . . .
+ m(q) = r
Here q is the number of Jordan cells of the matrix describing the action of operator U(t) in an invariant subspace E,(t) c C [ h, 01 and m ( j ) are dimensions of these cells. All the functions pjl(t), . . . ,pjmcj,(t)in formulae (12.7) are o periodic.
Theorem 12.2. Any solution x(t) of Eq. (12.1) may be expanded in an asymptotic series over Floquet solutions (12.7)for t + 00. I n other words, for every solution x(t) and a number cc > 0 there exists thefinite sum %(t)of Floquet solutions (12.7): (12.8)
111
512. Stability of Linear Periodic Equations
such that
Ix(t)  %(t)l < C exp([email protected]),
fl > a,
t +
00
Coefficients of the expansion (12.8) are determined by the aid of a socalled conjugate system [253(5)]. As an addition to Theorem 12.2 it may be proved that X(t) is a solution of some ordinary differential equation. This equation may be written out if Floquet multipliers are known. Hence, solutions of periodic RFDEs are approximated by solutions of ordinary differential equations with precision to functions vanishing faster than any exponential ones. It is possible to develop the perturbation theory of RFDEs similar to those for ordinary differential equations. Notice that in general Theorem 12.2 may not be made more precise; i.e., asymptotic convergence of a series over Floquet solutions cannot be changed by convergence in the usual sense. Hence, a system of Floquet solutions may be incomplete. EXAMPLE 12.1 [253(5)].
Consider a scalar equation
i(t)=
27c(cos(2nt))x(t  1)
(1 2.9)
Since the mean value of the coefficients equals zero, Eq. (12.4) takes the form A = 0. The Floquet solution $l(t) = exp(sin 27ct) corresponds to the unique root A1 = 0 of Eq. (12.4). Equation (12.9) has no other Floquet solutions, and so the system of Floquet solutions is not complete. Nevertheless, Theorem 12.2 is valid for Eq. (12.9). For any solution x(t) of this equation and arbitrary p > 0, we have x ( t ) = C,$l(t) + O[exp( Pt)]. Consider, for example, the solution x1 of Eq. (12.9) defined by the initial condition xl(t) = I , 0 < t _< 1. By the step method (see Chapter 1, $3) we get
Expanding the solution $l(t) in a Taylor series, we obtain
O<[
r(t)is gamma function growing quicker than any exponent. Therefore,
$l(t) is an asymptotic representation for the solution xl(t). Notice that x l ( t )  t+hl(t) is a solution of Eq. (12.9) decreasing more quickly than any
exponent. Equations with constant coefficients have no solutions that decrease so quickly. Hence, an RFDE with periodic coefficients may not be reduced to stationary equation by change of variables, unlike ordinary differential equations.
112
2. Stability of Retarded Equations
It is interesting to define the classes of equations for which the system of Floquet solutions is complete (or incomplete). Equations with delays divisible by period are being investigated in detail [116, 253(5)]. Consider, for example, the scalar equation m
i(t) =
1 a , ( t ) x ( t jw), j=O
aj(t
+ o)= a,(t)
(12.10)
A system of Floquet solutions of this equation is complete in C[O, mw] if the function am(t)is of constant sign and does not equal zero in entire segments. For the equation i ( t ) = a(t)x(t  w ) with alternating signs of a(t), the Floquet system is incomplete.
Chapter 3
Stability of Neutral Functional Differential Equations
$1. STABIILITY OF LINEAR AUTONOMOUS NFDEs 1.1. General Stability Theorems
Consider an NFDE with arbitrary aftereffect ( x ( t ) E R,)
+
[dK,(s)]x(t  s)
i(t) =
Jorn
[ d K , ( s ) i ( t  s),
t20
(1.1)
JOrn
with initial data x,(e)
e Io
io(e) = @(el,
= V(e),
(1.2)
Assume that all elements KY of the (n x n) matrices K Oand K , are functions with bounded variation on [0, a)and also jornslldKl(s)ll < a,
i
i, j = 1
jornlldKl(s)ll
=
1 Y,
n 1 1 ~ 1 1 1=
1 IKYl
Y >0
(1.3)
i, j = 1
The initial function q(e) is continuously differentiable and IIcpII1
=1401
Fj(t) =
+ [email protected](O)l +
(
JornlFo(t~l~dr)l'z + (JomlFl(t)l~dt)l'z< a
(1.4)
i"
[dK,(~)]cp"'(t s),
j = 0, 1
Under these assumptions problem (Ll), (1.2) has the unique solution x(t, cp) such that Ix(t, cp)l
+ [email protected],
cp)l 5 C,IIcpII, exp(Czt)
113
114
3. Stability of Neutral Functional Differential Equations
where C,,C , > 0 are some constants. Hence there exists a Laplace transform X(z) of the solution x ( t , cp). Using relations (2.2.7)(2.2.10), we get LIZ 
R,(Z)Z

Ro(z)lX(z)
Rj(z)=
= (I  R,(z))cp(O)
e" d K j ( s ) ,
+ Fo(z) + F , ( z ) e"Fj(t) dt,
Fj(z)= Jom
JOm
X(z)
(1.5)
e'"x(t, cp) dt
= !om
The function A(z) = det[Iz  R,(z)z  Ro(z)]
= z"[l  det
R,(z)]
+ O(lzl"')
(1.6)
is called a characteristic function corresponding to Eq. (1.1). The L,stability and asymptoticstability definitions for Eq. (1.1) coincide with Definitions 2.1.3 and 2.1.4 (Chapter 2, Definitions 1.3 and 1.4) with (IcpII, instead of p .
Theorem 1.1. Let function (1.6) have no zeros in the halfplane Re z 2 0 and kernels K j ( s )satisfy requirements (1.3). Then the trivial solution of Eq. (1.1) is L , stable and asymptotically stable. ProoK The proof of Theorem 1.1 is similar to the proofs of Theorems 2.2.1 and 2.2.2. On the basis of conditions (1.3) we obtain for Re z 2 0
From (1.7) follows the existence of a bounded matrix [I  R , ]  l such that ll(I  R,(~))~11 I y',Rez 2 0. MultiplyingEq.(1.5)on(f  R,(z))l,we have
[Iz  (I  R,(z))'RO(Z)X(Z) The matrix R,(z)
=
= cp(0)
+ (I

R1(z))'(Fo(z)
+ (F,(z))
(1.8)
(I  R1(z))lRo(z) and the function
P,(z)
= ( I  R,(z))l(Fo(z)
+ P,(z))
are bounded for Re z 2 0. Hence Eq. (1.8) takes the form (fz  R,(z))X(z) = cp(0)
+ F2(Z)
(1.9)
Equation (1.9) is just similar to Eq. (2.2.9). So by virtue of the Plancherel theorem and boundedness of the matrix ( I  R,(z))' we have
1
m
m
IF2(ifi)12dfi 5 2ny,
115
$1. Stability of Linear Autonomous NFDEs
As in Theorem 2.2.1 we can prove that
J"
IX(iP>12 dP <
II'pII1
<
m
Hence, L , stability is proved. The asymptotic stability proof is just the same as in Theorem 2.2.2. Theorem 1.1 is proved. For some NFDEs requirements (1.3) may be weakened. Consider the NFDE 'x
[ d K O ( ~ ) ] x (t S)
i(t)=
+ m1= K m i ( t  h,) 1
JOrn
+ [ o m i ( s ) i ( t  s ) ds,
t 2 0, X E R ,
(1.10)
Here K , are (n x n) matrices, and the (n x n ) matrix A(s) has absolute integrable elements and m
C IIKmII < 1,
Jomlll~s)ll ds <
(1.11)
m= 1
Conditions (1.11) are weaker than (1.3). The Laplace transform X(z) of Eq. (1.10) solution satisfies the relation [B(z)z

R,(z)]X(z)
= B(z)cp(O)
+ Fo(z) + P,(z),
m
B(z) = I 
C K,
exp(hmz) 
A(s) exp(zs) ds
(1.12)
m=l
Theorem 1.2. Let the characteristic function A(z) = det[B(z)z  R o ( z ) ]
(1.13)
have no zeros in the halfplane Re z 2 0 and kernels K O ,K m and A(s) satisfy conditions (1.3), (1.11). Then the trivial solution of Eq. (1.10) is L , stable and asymptotically stable. Proof: The function X(z) is analytical in Re z > 0 and continuous in Re z 2 0, consequently, x(t) = 
2n
1"
X(iP)ei8' d/3
(u
By virtue of (1.12)
x(iP) = [B(iP)  R,(ig)I'[B(iP)x(O) + F0(iD) + F,(iP)I
116
3. Stability of Neutral Functional Differential Equations
NOWprove the estimate similar to (2.2.13). Using (1.2) and (l.ll), we have
IIB(iB)ll <
IIRo(iP)II < 0O
(1.14)
= O(IPIn')/lN$)I
(1.15)
Therefore
II&iP)iD
 &,(iB))'Il
Evaluate A($). By the assumption made, A($) # 0 for all p. Further, take jomexP((iBs)llA(s)llds
+
0,
181
+
as the Fourier coefficient of an absolutely integrable function. Then there exists a Po such that 111  B(ifl)II I y1 < 1,181 2 Po. For IpI 2 Po the matrix B(iB) is nonsingular and also IIB'(iB)ll I 7:'. But then ldet B(iB)I 2 C > 0, IBI > Po. So for some p1 > Po we obtain
IA(iB>I = IPl"ldet
B(m+ O(lBI"')
2 CIP1"/2
From here and (1.14), (1.15) it follows that
II(B(iB)iB  Ro(iB>>'II5 CzlBI',
Il.S(ib)iB Ko(ifi))' 11 5 C , ,
IBI < p < 00 +
 00
Now for the end of the proof it is sufficient to repeat the proofs of Theorems 2.2.1 and 2.2.2.
REMARK1.1. Let condition (1.3) be invalid. Then the trivial solution may be unstable even if all the zeros of function (1.6) lie to the left of the imaginary axis. For example, the trivial solution of the equation L(L(L(x)))= 0,
L(x) = i ( t )  i ( t  1)
+ 7?X(t)
(1.16)
is unstable in the Liapunov sense. But at the same time all the zeros of the quasipolynomial corresponding to Eq. (1.16) lie to the left of the imaginary axis [216].
REMARK1.2. Let requirement (1.3) be fulfilled for some y > 0 and let (1.17) Further, assume that in the halfplane Re z 2 7, function (1.6) has no zeros. Then, similar to Theorem 2.2.3, it may be proved that Ix(t)l I C , exp(  y t ) . 1.2. Methods of Stability Investigation of Linear Autonomous NFDEs All the methods from Chapter 2, $3 may be applied for investigation of characteristic functions corresponding to linear autonomous NFDE. The
117
$ 1 . Stability of Linear Autonomous NFDEs
Fig. 3.Z. Stability domains for Eq. (1.17).
theorems of Pontrjagin und Chebotarev are valid for NFDEs without any changes and the method of D subdivision is also applicable. The stability domains for the equation i ( t ) = ax(t)
+ bx(t  z) + c i ( t  T )
(1.18)
in the space of parameters (a, b ) for c = 0.9, 0.6,0,0.6,0.9 are shown in Fig. 3.1. Equation (1.1) is always unstable for IcI > 1 because the corresponding quasipolynomial has a chain of asymptotic zeros of the form (1.5.5),with the positive real part equal to lnlcl. For all IcI < 1, Eqs. (1.1) have a common part of the stability domains a > Ibl. It follows from the Tsypkin criterion (see Chapter 2,93). Vyshnegradskii diagrams for control system (1.6.l), having the transfer function W ( s )= K ( T s I)' exp( ST), closed by a PD controller and by a PID controller in parameters Tl = T  ' , I , = I / T , D , D/T, are shown in Figs. 3.2 and 3.3 (consequent equations can be found in Table 1.2).
+
a=
118
3. Stability of Neutral Functional Differential Equations
2Stable
0 2
05
2
1
5
T1
Fig 3.2, Vyshnegradskii diagram for a system closed by a PD controller.
5
2
L Stable
0.2
0.5
1
2
r,
Fig.3.3. Vyshnegradskii diagram for a system closed by a PID controller.
119
$1. Stability of Linear Autonomous NFDEs
Fig. 3.4. Michailov hodograph for a stable NFDE.
The frequency stability criteria of Michailov and Nyquist from Chapter 2,93 are not valid for NFDEs in general. It is connected with oscillations in the Michailov hodograph. Consider, for example, the equation i(t)=
ax(t)
= ci(t

a >0
l),
(1.19)
The Michailov hodograph D(iw) = iw  a + ciw exp(  iw), 0 I w < co,for stable Eq. (1.19) when IcI < 1 is represented in Fig. 3.4 and for unstable Eq. (1.19) when IcI > 1 in Fig. 3.5. From Figs. 3.4 and 3.5 it follows that the functions arg D(iw) have no limit for o + co and that their increments on the interval 0 I w < co are not defined. But Michailov criterion is true [7] for a particular characteristic function
[
det I z  H
J(s)eCZsds

lorn

z
Jl(s)e’S ds Jorn
where
/oml
J(s)lep” ds < co,
J,(s)leps ds < &I,
1
p
>0
120
3. Stability of Neutral Functional Differential Equations
Fig. 3.5. Michailov hodograph for an unstable NFDE.
$2. STABILITY OF AEROAUTOELASTICITY EQUATIONS
2.1. Equations of Unsteady Motion of an Elastic Rigid Body The dynamic model of unsteady motion of the elastic flying vehicle must take into account the interaction of aerodynamic, elastic and inertial forces (i.e., aeroelastic effects) as well as the dependence of aerodynamic characteristics on the past states of flow (or on the aerodynamic trace). One such description is the aeroautoelasticity models [19,201. Practically, one can consider only small perturbations of the motion and deformation parameters. The widely used method of known forms is based on the representation of deformations as a series of known forms which are usually the natural oscillation forms of the vehicle in a vacuum. If one is restricted to a finite number n of the oscillation forms, one can construct an approximate finitedimensional model of flying vehicle motion p(ij
+ 2waeq + 0 2 q )= c
(2.1)
121
$2. Stability of Aeroautoelasticity Equations
Here p, w, z are the diagonal (n x n ) matrices of generalized masses, frequencies and damping factors of natural oscillations; q is a vector of generalized coordinates; and C is a vector of generalized forces. The matrices p, w, ae are defined simultaneously with the calculation of natural oscillation forms on the basis of a masselastic model of the vehicle. The knowledge of a vector q permits us to calculate the displacements of any point (5, I], i)of the vehicle and their derivatives according to the expression
Here
are natural oscillation forms of the vehicle in the coordinate system
(t,I], [) fixed in the principal inertial axes of the vehicle. The vector of the generalized forces C for the known aerodynamic pressure p(5, I], [,t), applied to the vehicle surface cr, is
(2.3) Here E ~ j , = 1, . . .,N , are the kinematic parameters, including vectors q(t) and q(t), vector of wind velocity A(t), and also vectors d(t), &t), describing the states and the rates of states of control surfaces. We write the reaction of component C , of the generalized force vector on the step change of E, (i.e., the unit step function) in the form
H;(e)
= C;
+ z;(e)
(2.4)
Hence C: is a quasistationary aerodynamic derivative. The functions fz(8) are called the unsteady unit step functions. They describe the influence of the aerodynamic trace. For the subsonic speed (Mach number M < 1) the function f;(8) are determined for all moments 8, 0 I 8 < co and for sufficiently great 8, I;(O)zAf?*. For the supersonic speed ( M > 1) f38) = 0 for 8 2 8* > 0 [7, 191. The unsteady unit step functions are calculated theoretically on the base of the discrete vortices theory or determined experimentally. Usually they are represented by tables or by diagrams. Figure 3.6 shows the unit step function of aerodynamic moment m, with respect to the diametrical axis, passing by the choice of a rectangular wing with aspect ratio of A = 5. Curve 1 corresponds to M = 0.7 and curve 2 to M = 0.9. The quasistationary values of moment mz = H;=(co) are indicated by dotted lines on Fig. 3.6. Using the Duhamel integral, we can write any aerodynamic characteristic in the form C,(t) =
1 C,,, J
C,,(t) = C & J+ E~
(2.5)
122
3. Stability of Neutral Functional Differential Equations
Fig. 3.6. Unsteady unit step function of aerodynamic moment m,: 1, M
= 0.7; 2,
M
= 0.9.
Substituting (2.5) in (2.1) we obtain the closed linear equations of Jiying vehicle perturbated motion
144 + oaq + 0 2 q )  C4q  Cqq 
sd
@(0)ij(t  0) d0 = C66
+ [:II(0)6(t
 0) d0
s,'
@(0)q(t  0) d0
+ CA' + CAA
+
Equations (2.6) describe the deterministic model of flying vehicle dynamics. The stochastic model is more adequate. It takes into account the atmospheric turbulences and chaotic noises in the pickups. The stochastic differential equations for the turbulent wind velocity A and the model of pickups are taken in the form
A=F,A+ti,,
i=Mr+i+ti2
Here til, Lj2 are vector stochastic processes of the white noise type with known intensity matrices, r is a vector of pickup states and 2 is a vector of output
123
12. Stability of Aeroautoelasticity Equations
pickup signals. The drives of control surfaces are described by the equation 6 = S6 + Gu where u is a vector of control drives signals. Finally, the linear stochastic aeroautoelastic model has the form
ff
Equation (2.7) is an SFDE of the neutral type. In Chapter 4 we shall formulate the theorem of existence and uniqueness of the solutions of such equations and shall give some methods of stability investigation. 2.2. Stability of Aeroautoelastic Equations Study the stability of deterministic equation (2.6) for the control vector
u
= 0:
q(t)
+ A q ( t ) + Bq(t) =
c
Z,(O)G(t
 0) de
+ S:I,(B)q(t
(2.8)
 d) d9
According to Theorem 1.2 for asymptotic stability of the solution of Eq. (2.8), it is necessary and sufficient that the characteristic function D ( z ) has no zeros in the halfplane Re z 2 0. The characteristic function D ( z ) is defined with the aid of the theorem on the Laplace transform of convolution and is equal to
D(z) = det[Zz2
+ Az + B  z J
Io(s)ezSds  z 2
J
0
Il(s)e’” ds] (2.9) 0
The Michailov criterion holds for function (2.9) if for some y > 0
jom
Ii(s)ey”ds < 00,
i = 0, 1
The kernels I,(s) are usually uncertainties Mi(s), i = 0, 1. It is shown by Astapov [7] that if function (2.9) has no zeros in the halfplane Re z 2 I, I > 0 and the value
+
is sufficiently small then the replacement of kernels I , ( @ by Zi(e) Mi(9), i = 0, 1, does not distort the asymptotic stability. Consider in detail the
124
3. Stability of Neutral Functional Differential Equations
Fig. 3.7. Stability domain of triangular wing for M
=
1.2.
onedimensional oscillation of the control surface, the so called control surface buzz [121]. Let a rigid wing move in gas flow and turn with respect to an axis. The angular wing displacements are restricted by an elastic spring of rigidity k. The motion equation is ij
+ aq + bq = kl a=
Io(0)q(t  0) d0 loW
 k mwz,
b = k,
+ k,
 k,m;,
sd
Il(0)ij(t  0) d0
k,
= ~ S 6  ~ / ( 2 5 , (2.10) )
k , = kb2/(J , u ~ ) If one considers the quasistationary model (i.e., J o ( 0 ) = 0, Jl(0) = 0), then Eq. (2.10) is asymptotically stable for a > 0, b > 0. Investigate the stability of a triangular wing with aspect ratio I = 2.5 for M = 1.2 and M = 2. We shall use the results of Astapov [7], and Belotzerkovskii [19], who give the aerodynamic coefficients and unsteady step functions. Figure 3.7 presents the stability regions obtained with the aid of the Michailov criterion for these cases. Figures 3.8 and 3.9 present the stability region of a rectangular wing with I = 5 and I = 2.5 for M = 0.4. It is clear from these figures that aerodynamic aftereffects increase the stability region for subsonic speed and decrease it for supersonic speed. The new qualitative phenomenon is also revealed. The vertical straight line crosses the stability region boundary twice: on Fig. 3.7 for M = 1.2 and 0.73k1< a < 0.79k1,and on Fig. 3.8 for 0.76kl < a < 0.64k1. It means that the wing is stable if the speed is less than a critical speed ukr or more than the other critical speed u : ~ .For the speeds varying from u,l, to the wing is unstable; it oscillates with increasing amplitude and after some time goes to ruin. This effect is observed in practice.
UK
125
42. Stability of Aeroautoelasticity Equations
Fig. 3.8. Stability domain of rectangular wing for 1 = 5 and M = 0.4.
It can be explained in the models which take into account the aerodynamic aftereffects. The twodimensional flexuretorsion flutter of a rigid rectangular wing is considered by Astapov [7], and Belotzerkosvii [19]. The critical flutter speeds are obtained: in the model with aftereffects it is equal to 290 m/sec, and in the quasistationary model it is equal to 190 m/sec. Thus, allowing for the aftereffects of the aerodynamic trace permits us to obtain more adequate models of unsteady motion (aeroautoelasticity model) which gives more precise quantative and qualitative descriptions of the unsteady motion of an elastic flying vehicle. b
Stable
0.11 Fig. 3.9. Stability domain of rectangular wing for 1 = 2.5 and M = 0.4.
126
3. Stability of Neutral Functional Differential Equations
53. LIAPUNOV DIRECT METHOD FOR NEUTRALTYPE EQUATIONS
3.1. Introduction Application of the Liapunov direct method to NFDEs is a nontrivial problem. Several assertions of this type could be obtained by formal extension of the corresponding theorems for equations with delay. It should be noted however that the results thus obtained assuming the existence of positive definite Liapunov functionals of systems’ paths have limited applications. The reason is that it is very difficult to construct Liapunov positivedefinite functionals for NFDEs. Instead of this, another method of stability investigation is stated below based on the use of only positivesemidefinite Liapunov functionals. In this case the investigation consists of two stages. At the first stage a positivesemidefinite (but not, as usual, positivedefinite!) functional is constructed. The second stage is connected with the study of stability of some functional inequalities generated by the positivesemidefinite functional constructed earlier. Some results of this section are established for equations with arbitrary (finite or infinite) aftereffect. The considered method of stability investigation of NFDE with the help of positivesemidefinite functionals was suggested by Kolmanovskii and Nosov [108( l)]. In Kolmanovskii and Nosov [108(2), 108(5)] positivesemidefinite functionals were built for concrete systems and the stability conditions formulated in terms of the systems’ coefficients were obtained. Some problems connected with the application of the Liapunov direct method to NFDEs are considered in other works [48, 81(4), 150, 1791. 3.2. Some Definitions Denote M as the metric space of continuous functions cp(t), cp: (  GO, 01 + R, and Q H is a sphere in M (see Chapter 1, $3). Let F, G: [0, GO] x QH+ RH be two preassigned continuous maps such that IG(t, CP)~ + IF(t, 9)l I F o ,
~ E C O ,GO),
~ ( Q E Q H
(3.1)
Consider the initialvalue problem for the NFDE (d/dt)[x(t)  G(t, x,)] = F ( t , x,),
xo(e)
=
cp(e),
GO
t 2
< e I 0, x, = x(t
0
+ e)EM
(3.2) (3.3)
Assume hereafter that the solutions of problem (3.2), (3.3) exists, is unique and continuously depends on initial data. Let G(t, 0) = F(t, 0) = 0. Then system (3.2) has a trivial solution corresponding to the initial function CP(e)= 0.
127
53. Liapunov Direct Method for NeutralType Equations
Definition 3.1. The trivial solution of problem (3.2), (3.3) is called ( a ) stable if for any E > 0 there exists 6(&) > 0 such that ( x ( t ,cp)( I & f o rt 2 0 as soon as p(cp, 0 ) I b(&);( b ) asymptotically stable if it is stable and lim x(t, cp)
= 0,
V cp(0) E R c M
rm
The region R is referred as the attraction region of the trivial solution.
Later, a great role will be played by functional inequalities Y,)l
=
IY(t> G(t, Y,)l If ( t ) ,
Y d 6 ) = cp(0)
(3.4)
Here f ( t ) is a nonnegative continuous scalar function and cp(6) E M . Designate by y(t, cp) any solution of inequality (3.4).
Definition 3.2. The trivial solution y(t) = 0 of inequality (3.4) is called ( a )f stable i f f o r any E > 0 there exists 6(&)> 0 such that Iy(t, cp)l I E , t 2 0,for all cp and f such that p(cp, 0 ) I &E), sup, f ( t ) I a(&); (b) asymptotically f stable if it isfstable and moreover lim,+my(t, cp) = Ofor all cp(0)~Rc M and a n y f ( t ) such that f ( t ) + 0 at t + 00.
3.3. General Theorems Let V ( t , x,, Z ( t , x,)) be some continuous functional determined for all QH and t 2 0 and such that its derivative P = dV(t, x,, Z ( t , x,))/dt exists along the trajectory of equation (3.2). Since the derivative dZ(t, x,)/dt exists and the derivative i ( t ) may not exist, the requirement of the existence of the derivative t imposes certain restrictions on the dependence of V on x,. As in Chapter 2, $5, we designate by o i ( u ) some continuous nondecreasing functions such that wi(0)= 0 and wi(u) > 0 for u > 0. X,E
Theorem 3.1. Let the functional V ( t ,x,, Z ( t , x,)) exist and satisfy the previously mentioned assumptions. Further, Wl(IZ(t9 x,)l) 5 v t , x,, a t 7 x,)) I W Z ( P ( X , ? 0 ))
P I 0
(3.5) (3.6)
Let the trivial solution of the inequality (3.4) be f stable. Then the trivial solution of equation (3.2) is stable. ProoJ: Take an arbitrary E E ( O , H ) .Since the inequality (3.4) solution g(t) = 0 is fstable, one can find bl(&)> 0 such that any solution of inequality 6,, p(cp, 0 ) I 6,. (3.4) will satisfy the relation ( x ( t ,cp)l I E , t 2 0, if f ( t ) I Now take 6 , ~ ( 0 ,6,) such that ~~(6,) = ~~(6,). Then p(cp, 0) 5 6, by virtue
128
3. Stability of Neutral Functional Differential Equations
of conditions (3.5) and (3.6) we obtain o l ( I Z ( t rx,)l) I V ( t , x,, Z ( t , x,)) I V(0,cp, Z(0, cp)) I w,(6,) = ~ ~ ( 6 ,This ) . and the monotonocity of o1imply f(t)I 6,. Taking into account the f stability of inequality that IZ(t, x,)l I (3.4), we obtain Ix(t, cp)l I E, t 2 0, for all cp(0) such that p(cp, 0) I 6, 5 6,. Theorem 3.1 is proved.
Theorem 3.2 [108(5)]. Let there exist a functional V ( t ,x,, Z ( t , x,)) satisfying requirements of Theorem 3.1 and also V I  4 1 Z ( t r x,>l>
(3.7)
Let the trivial solution of inequality (3.4) be asymptotically f stable. Then the trivial solution of equation (3.2) is asymptotically stable.
As a corollary, consider the ordinary differential equation i ( t ) = F(t, x(t)),
x(t,)
= xg
(3.8)
Here F : [O, co) x R, + R, is a continuous function satisfying the local Lipschitz condition in the second argument, and F(t, 0) = 0. Along with (3.8) consider an NFDE i ( t , x,) = ~ ( tz (, t , x,)), Z ( t , x,)
= x(t) 
X,v)
=~
(0)
G(t, x,)
(3.9)
Here G(t, x,) satisfies the conditions formulated previously.
Theorem 3.3. Let the trivial solution of Eq. (3.8) be asymptotically stable. Further, let the trivial solution of inequality (3.4) be asymptotically f stable. Then the trivial solution of (3.9) is asymptotically stable. For the proofs of Theorems 3.2 and 3.3 see Kolmanovskii and Nosov c108(5)1.
3.4. Global Stability Investigate the problem of global stability of the trivial solution of Eq. (3.2). Assume that the functionals F(t, cp) and G(t, cp) are continuous and determined on the whole space [0, 00) x M. In addition, IF(t, cp)l I F,, IG(t, cp)l I G, for all H > 0, cp eQH and t 2 0. Suppose also that the conditions of existence and uniqueness are fulfilled for an arbitrary sphere S,.
Definition 3.3. The trivial solution of Eq. (3.2) is called globally asymptotically stable if it is stable and lim x(t, cp) = 0, t + co for any initial function 440) E M . Definition 3.4. The trivial solution of inequality (3.4) is called f bounded if bounded solution y(t, cp) corresponds to every bounded function f ( t ) .
I29
$3. Liapunov Direct Method for NeutralType Equations
Theorem 3.4. Let the continuous functional V(t,x,, Z(t, x,)) exist and satisfy all the conditions of Theorems 3.2. In addition, let the trivial solution of inequality (3.4) be f bounded and asymptotically f stable, and also let ol(u)+
u+
00,
co
(3.10)
Then the trivial solution of Eq. (3.2) is globally asymptotically stable.
The proof of this theorem may be obtained by suitable modification of the results from Kolmanovskii and Nosov [108(5)].
3.5. Stability of the Functional Inequalities Establish some conditions off stability for the case in which the space M coincides with CB[  co,O]. Lemma 3.1. Let the continuous functional G(t, cp), dejined on [0, co) x CB[  co,01, satisfy the Lipschitz condition IG(4 CP)  G(t, $)I
VllV  $ 1 1 ~
(3.1 1)
If; in addition, v
(3.12)
then the trivial solution of inequality (3.4) is f stable and f bounded.
Proof: From (3.4) it follows that
IY(~)I
5 IG(t, Yr)l + f ( t )
vll~t(e)B+ f ( t )
Designate m(t) = ~up,,,~,,ly(s)l. Then m(t>5 vm(t)+ v l l d e ) l l S
+ sup f ( s )
(3.13)
0,SSl
The estimate (3.13) means that the trivial solution of inequality (3.4) is f stable and f bounded. Formulate some assertions [lOS(5)] about asymptotic f stability of the trivial solution of inequality (3.4) in the case of a finite time lag. Lemma 3.2. Let G(t, y,) = g(t, y(t  h)), where g : R , x R, + R , and h > 0. Then $ Ig(t, y(t  h))l 5 v(y(t  h)(,0 < v < 1, then the trivial solution of the inequality Iy(t)  g(t, y(t  h))l 5 f ( t ) is asymptotically f stable and f bounded. Recall some concepts connected with the theory of almostperiodic functions. The spectrum of the almostperiodic function cp(t) is the set A(cp) such that A(cp) = {A:R,:M[e'Arcp(t)] # 0) t
PT
130
3. Stability of Neutral Functional Differential Equations
The module mod(cp) of the almostperiodic function cp is the set mi&, AieA(q),mi an integer, K a natural number i= 1
Lemma 3.3. Let G(t, y,) = g(t, y(t  h)), g : R , x R , + R,, h > 0, Ig(t, y(t  h))l 5 v(t)ly(t  h)l, v(t) > 0. Let the continuous function v(t) be almost periodic and satisfy the conditions V { k 1Emod(v), A # 0, [hA # O(mod 2z)]),
M[lnv(t)] < 0
Either v(t) is o periodic, h is incommensurable with o and
[ouhv(t) dt < 0 Then the trivial solution of (3.4) is asymptotically f stable and f bounded.
Lemma 3.4. Let the functional G(t, cp) satisfy the Lipschitz condition IG(t, cp)

1
4lso  11/11, 0 < < 2
G(t,
and also G(t, cp) is independent from the values of the function q(8), 8 E [A, 01. Here 0 < A < h. Then the trivial solution of (3.4) is asymptotically f stable and f bounded.
3.6. Equations with Bounded Delay Formulate the stability conditions for the case of Eqs. (3.2) with finite delay and for those with infinite delay such that the initial set coincides with the bounded interval [  h, 01. So F is a continuous functional F : [0, co) x C[  h, t ] + R,.
Theorem 3.5. Assume that the functional G(t, cp), G: [O, R, satisjes the condition IG(t + s, cp)  G(t, $)I
cx))
x C[ h, 01 +
s %(S) + ullq  11/11
O < a < 1, V q , l l / € Q H , O s t
< co, s r 0
(3.14)
Let there exist the functional V ( t ,x,, Z ( t , x,)) satisfying condition (3.5) and
i.
dIx(t)l)
Then the trivial solution of (3.1) is asymptotically stable.
(3.15)
131
$3. Liapunov Direct Method for NeutralType Equations
Proof: By virtue of Theorem 3.1 and Lemma 3.1 the trivial solution of (3.2) is stable. Hence x(t, cp) E Q H for cp(#) E Qa, 6 > 0. Show that x(t, cp) + 0, t + co. Suppose the contrary. Then there exist such a number v > 0 and a sequence of points ti + co such that Ix(ti)I > v. From (3.2) it follows that
~ ( t= ) G(t, xt)
+
1:
F(s, xS)ds
Here it is assumed that x(s) = cp(s) for s I0. Then taking into account (3.1), we have for A 2 0
I x(t + 4  x(t) I = I G(t + A, &+A) G(t, Xt) rf+A
(3.16) Designate
From (3.16) it follows that
+
p(r]) 5 (1  c t  f t l
max max j x ( ~ A)

tp(e)j+ o,(q)
+ F,q
1l2A hSOSO
1
So, due to the uniform continuity of the function x(t) on the closed segment [  h, q ] , the righthand side of this inequality tends to zero as r] + 0. Hence there exists a i j > 0 such that p(ij) 5 v/2. In this case for z E [ti  ij, ti + i j ] we have Ix(z)l > v/2 uniformly for all i. From this follows a contradiction: V + co for t + co. Theorem 3.5 is proved REMARK3.1. The assertion of Theorem 3.5 is valid for the functional G(t, xJ, G: [0, co) x C [  h , t ] + R , satisfying the condition IG(t
+ s, x,+,)

+
G(t, x,)l I o 5 ( s )
CI
max Ix(z hsrst
O
s20,
+ s)  x(z)l
Vx,€QH, t 2 0
3.7. Examples EXAMPLE 3.1.
Consider an equation of the type (3.2)
X + CX(t  h ) + g ( x ( t ) + Cx(t  h)) = 0,
h >0
(3.17)
132
3. Stability of Neutral Functional Differential Equations
where xg(x) > 0 for x # 0 and ICI 1. Prove that the trivial solution of (3.17) will be stable. Write (3.17) in the form
+ Cx(t  h),
Z ( t , x,) = x(t)
z
=
w, w =  g ( Z )
Consider the functional
+ Iozg(s)ds
V=
(3.18)
The derivative of functional (3.18) is equal to
P = ww + i g ( Z ) = 0 Hence by virtue of Theorem 3.1 and Lemma 3.1 the trivial solution of Equation (3.17) is asymptotically stable. EXAMPLE 3.2. Derive the stability conditions for the equations X(t)
+ cp(x(t))i(t)+ f ( x ) = 0,
m, + x,)
cp(Z(t,x * > ) z ( tx,) ,
f(0)
=0
+ f(Z(t3 x,))
=0
(3.19) (3.20)
+
where Z(t, x,) = x ( t ) i[exp(sin t ) ] x ( t  1). Under the conditions xf(x) > 0, x # 0, cp(x) > 0, the trivial solution of Eq. (3.19) is uniformly asymptotically stable [lS]. By virtue of Theorem 3.3 and Lemma 3.3 the trivial solution of Eq. (3.20) is also asymptotically stable. If, in addition,
s,’/o)
ds + 00,
x + 00
then the trivial solution of Eq. (3.20) is globally asymptotically stable.
$4. CONSTRUCTION OF DEGENERATED FUNCTIONALS FOR CONCRETE SYSTEMS 4.1. Stability of a Chemical Reactor Closed by a PD Controller
Investigate the stability of the chemical reactor described in Chapter 1, $1 and Chapter 2, $6. Figure 3.10 represents the structural scheme of the investigated system. As in Chapter 2, $6 the output u(t) of a nonlinear actuator is considered as a functional depending on the preceding states of the input x(s), 0 I s I t, i.e., u(t) = F(x(t + e)), t I 8 I 0. Assume also that this functional satisfies Lipschitz condition (2.6.5) from Chapter 2, $6.
W. Construction of Degenerated Functionals for Concrete Systems
133
Fig. 3.10. Structural scheme of a chemical reactor closed by a PD controller.
The functioning of the system illustrated in Fig. 3.10 is described by the equation
Let j ( t ) be some fixed input signal, and let X(t) be the unperturbed solution. Designate the perturbed solution by X(t) + z(t). The perturbation z(t) satisfies the equation
Introduce the following functional
134
3. Stability of Neutral Functional Differential Equations
The derivatives of the functional (4.2) is equal to
V = 2 [ T ~ ( t+) KoK,DF,(z(t
+ yz’(t)
z
+ O))][z(t) + K,K,F,(z(t z2(t  z
jomd,R(s,X,, zs)  y


5
+ e))]
s)d,R(s, X,,z,)
JOrn
I
 Tz2(t)
+ K,K,(D + T ) E1 z2(t) + yr2z2(t) 
+ ~ K ; K , ~ D F : ( ~ (+~ 0)) + K , K , ( D + T)EF:(z(t  y j O r n i i (t 7  s)d,R(s,
Set E
= rl,
y = K,K,(D
VI
[2T

+ 0))
x,,zs)
+ T ) r  l + 2KgK;D. In this case we obtain + 2KoKp(D + T)r + 2KgK,2Dr2]z2(t)
Using Theorem 3.4, we see that the feedback system shown in Fig. 3.10 is asymptotically stable provided that the requirements T > KoKpDr2,
2 T > 2KoKp(D
+ T)r + KgKiDr’
hold. If D = 0, then these requirements coincide with stability conditions of a reactor closed by a P controller, obtained in Chapter 2, $6.2. 4.2. Stability of OneDimensional Nonlinear Systems
Here we investigate the conditions of stability of trivial solutions of scalar equations ~ (t S) dko(s)
+
s:
i(t
 S)
+
t 2 0, X, = ~ ( t 01,
dk,(s)
+ ~ ( tx,),
a < e I o
(4.3)
In some situations system (4.3) is a particular case of equation (3.2). Therefore, in these situations, from the results of Chapter 3,$3 one can extract some conditions of stability of system (4.3). However, owing to the specific features of Eq. (4.3), one can obtain the stability conditions under wider assumptions. These assumptions are connected first of all with the understanding of the solution of Eq. (4.3) and also with the requirement that the derivative of the Liapunov functional be negative only almost everywhere (but not everywhere, as in Chapter 3, 93). At the same time it should be emphasized that the method of stability investigation of Eq. (4.3) is the same as in Chapter 3, $3. Formulate the basic assumptions. The kernel k,(s) has bounded variation
135
&Construction I. of Degenerated Functionals for Concrete Systems
on [O, co), and the corresponding integral in (4.3) is understood in the sense of Stieltjes. The kernel k , ( s ) is determined by kl(s)
=
J
4 t ) dt
+ 1 p,,
0
kl(0)
=0
h,<s
Here summation extends to those values of n for which h, 5 s. The function A(s) is bounded and Riemann integrable on [0, 00). Finally, h, 2 0 and lpl[ Ipzl + ... + Ip,, + ... < co. The functional a(t, cp) is defined and continuous on [0, 00) x CB[  co,01. Also, the functional a(t, cp) satisfies, for any function cp E CB[  co,01, the conditions
+
a(t, 0 ) = 0,
J
r
1 4 4 cp)I2
m
lcp(S)l2
dR,(s, cp)
V cp~CB[co,O]
d R l ( s , cp) I r: < co, 0
Set
It is assumed everywhere below that
The solution x ( t ) of Eq. (4.3) is determined by the initial data
x,(8) = cp(8), i(8) = Q(O), i I0 (4.5) Here cp(8), 8 5 0 is an absolutely continuous bounded function, and the function @ ( O ) is bounded. Under these assumptions there exists the only solution x(t, cp) of problem (4.3),(4.5); i.e., there exists the only function x ( t ) limited and Riemann integrable on each finite interval, which is equal to @(O) at 8 5 0 and such that the function
i(t)
and x ( t ) = cp(0)
will be the solution of problem (4.3), (4.5). Set for each function cp E CB[  co,01 a c p )=
cpm

+
J:
i ( s ) ds, t 2 0,
j0k
dk,(s)
Notice that when requirements (4.4) are met, the trivial solution of the inequality IZ(cp)I 5 Co is f stable (see Chapter 3, 93). By analogy with the proof of Theorem 3.5 we establish Theorem 4.1. Theorem 4.1. The trivial solution of Eq. (4.3) is globally asymptotically stable, i f conditions (4.4) are fuljilled and there exists a fiinctional V(cp), V(cp) =
Wcp) + Z2(cp)
(4.6)
136
3. Stability of Neutral Functional Differential Equations
which satisfies the local Lipschitz conditions, and
0
(4.7)
W(rP) I %(llrPlle)
The derivative of functional (4.6) along the trajectories of system (4.3), exists almost everywhere for all t 2 0 and
vs %(IX(t)l) The proof of Theorem 4.1 can be found in Kolmanovskii and Nosov [108(2), 108(5 ) ] . Obtain on the basis of Theorem 4.1 some concrete stability conditions. Suppose, first, that the kernel ko(s) has a jump in zero, equal to a , > 0, and
Then the trivial solution of Eq. (4.3) is globally asymptotically stable. Consider the functional
W , )= Z’(X1) + (I+ , Moo)
jornIdk1(s)l~  s x 2 ( s lds, )
+O
Estimates (4.7) follow from the inequality j o m ~ d k i ( sl)~, X 2 ( s l ) d s 1 5 ~ 1 i ~ ~ ix= t091 ~ ~ ~ ~
The derivative of functional (4.8) along the trajectories of Eq. (4.3) is
p I2x2(t)Ca,U
 Uo1)
+ (1 + a0A(
jm
Idko(s)l
+O
+ r,)]
From this we arrive at the following assertion.
4.1. Consider the automatic control system represented by the EXAMPLE structural scheme in Fig. 3.11. Let the nonlinear element satisfy the Lipschitz conditions IW(t
+ 0))  F(y(t + 0)) I
jom
Iornd,R(s,x, Y ) I r2,
I(x(t  s)  Y(t  s)ld,R(s, x , Y )
v x, Y E QH
137
$4. Construction of Degenerated Functionals for Concrete Systems
Fig. 3.11. Structural scheme of control system
The work of the system presented in Fig. 3.11 is described by the equation
T i ( t )+ K , K , D i ( t  T) + ~ ( t+) KOK,x(t  T) + K o F ( x ( t  t + 6)) = K,y(t  t) Designate by z(t) the perturbation of the output signal corresponding to some fixed input signal j ( t ) . The function z ( t ) satisfies the equation
Applying the statement established above, we obtain that the system, shown in Fig. 3.1 1 is globally asymptotically stable if
T < K,K,D,
T  K , K , D > ( T + K,K,D)Ko(K,
+ r)
Depending on the concrete form of the systems under consideration, the functional Z ( q ) in Theorem 4.1 can sometimes be reasonably chosen in a different form. Let US formulate some results using this approach. Let
138
3. Stability of Neutral Functional Differential Equations
conditions (4.4) be satisfied, and (4.9)
Then the trivial solution of Eq. (4.3) is globally asymptotically stable. Introduce the functional VX,) = z : ( x t )
+ (P + r d
+ (1 + cll0 + xOl)r;’ + (P + r1)
dk,(s)l
jomdRl(s,cp) ~,‘s*’(tl)
jrnldko(s)l 0
p ( t d dr,
jt
dtl
dtl 11x2(t2) d t 2
(4.10)
tS
Here Z,(cp) is determined by the formula
jo
Zl(cp1 = cp(0)  jorncp(s) dk1W  jorndko(s)
s
cp(tl> dtl
From (4.9) follows the f stability of the solution of the inequality IZ,(Cp)l I C , and the estimates Z t ( 4 5 W t ) 5 clllxtll; Also, it is easy to evaluate that for almost all t > 0
V I 2x2(t)[p(1
 clol 
ale)
+ r,(l + aO1+ cclO)]
From this follows the validity of our statement. Modifying functional (4. lo), one can obtain different conditions of stability of Eq. (4.3). Consider some of them. Use the fact that an arbitrary function ko(s) with bounded variation can be represented in the form of the difference of two limited functions which d o not decrease: ko(s) = k3(s)  k4(s). In this case it is sufficient to add to functional (4.10), in which ko(s) is replaced by k3(s), P by aO3and cll0 by ~(13,the expression (l
+ “01 + “13)
+ “04 + “04
j0rndk4(s)
jt:/2(tl)
dtl
Iomdk,(s) l:2(tl) dtl
JOrndk,(J)
Jtlsdtl
11x2(t2)
139
&Construction I. of Degenerated Functionals for Concrete Systems
With the help of this new functional one can establish the following statement. Let conditions (4.4)be fulfilled and
+ M 1 3 < 1, c103(1
 MO1
+ @'I4 + c(23 <
 c(13)
'
+ rl)(l +
+ @13)
(c(04
Then the trivial solution of Eq. (4.3)is globally asymptotically stable. By slightly changing functionals (4.Q(4.10)one can obtain the stability conditions for the trivial solutions of the equations rm
rm
k(t) =  J
~ ( t s)d,ko(t, s)
+J
0
i ( t  s)
d,k,(t, s)
+
0
Since in this case we have to deal with rather cumbersome shall restrict ourselves to the consideration of the simplest case, from which it is, however, easily seen what changes should be made in functionals (4.8), (4.10)for general Eqs. (4.11).Find the stability conditions of the trivial solution of the equation i(t)=
b(t)x(t

h) + ck(t  h),
t 20
(4.12)
The function b(t) 2 0 is assumed to be continuous. With the help of the functional V ( X , )= [ ~ ( t ) C X ( ~ h) 
b(s
+ h)x(s)dsI2
h
+
b(s
lcl
+ 2h)x2(s) ds +
11'
b(t,
+ 2h) dt,
h
j,: + b(t,
h)X2(t2)
dt2
it is easily found that the trivial solution of Eq. (4.12)is asymptotically stable under the assumptions sup{ f 2 O ICI sup{2b(t
+ j,t+hb(s)d s j < 1
+ h) + Jc/(b(t + h) + b(t + 2h))
f 2 O
+ b(t + h) j t l h ( b ( s + h) + b(s + 2h)ds
I40
3. Stability of Neutral Functional Differential Equations
4.3. Use of Degenerate Functionals for Stability Investigations of RFDEs
The degenerated Lipaunov functionals may also be used with success for RFDEs. Consider the equation ~ ( t S) dko(s)
i(t) = 
+ ~ ( tx,),,
t20
(4.13)
lorn
studied in Chapter 2, $3.6.We shall use the assumptions and notations from Chapter 2, $6. Let the kernel k,(s) not decrease monotonically (i.e.. a,, = Boo) and also BlO
< 1,
aoo(1  810)  (1 a00
+
a20
+ lorn
S
+BlOP =Y
’0
dR(s) < oc)
Then the trivial solution of Eq. (4.13)is globally asymptotically stable. Consider only the case r > 0 with the aid of the functional
Under our assumptions, we get
 (1
+ Plo)r’
~ornx2( t s) dR(s)
141
@ Construction I. of Degenerated Functionals for Concrete Systems
Using the inequalities 2x(t)a(t, x,) I rx2(t)
+ rl
x2(t  s) dR(s) lom
2 4 4 x,) Iorndko(s) l  s x ( t l ) dtl I
+ r Jomdko(s) [sxz(tl)
Iornx”(t  s) dR(s)
810rl
dt,
we obtain that 3 I  y x 2 ( t ) . From the assumption Pl0 < 1 and Lemma 3.1 follows that f stability of the trivial solution of the inequality /x(t)  Jorndkds) ]t;sx(tl)
dtli 5 Cl
Thus, from Theorem 3.1 follows the stability of the trivial solution of Eq. (4.13). Some supplementary considerations show that it is asymptotically stable [108(5)]. If the kernel ko(s) is not monotone then it may be represented as a difference of two monotonically nondecreasing bounded functions, k,(s) = kl(s)  k,(s). In this case it is sufficient to add the expression El2
Jorndk1(s)
j‘ ts
+ (1 + B l l )
d t , ],:x’(t,)
dt,
]omdk2(s) jt;sx’(td dtl
to functional (4.14) in which ko(s) is replaced everywhere by kl(s). In this case the following statement holds. Let P11
UOl(1  811) > (1 + 811)(r
< 1, goo
+ clZl + u l 2 + o]*
+ aoz)
s dR(s) < co
Then the trivial solution of (4.13) is globally asymptotically stable.
EXAMPLE 4.2. Consider the equations i ( t ) = ax(t>  bx(t  h)
f ( 0 ) = 0,
If(x1)
+ f(x(t)),
t20
 f(x2)l I C1lx,  ~ 2 1 ,
a
> 0, b > 0.
(4.15)
The trivial solution of (4.15) is asymptotically stable, if bh < 1, b(1  bh) > (1 + bh)(a Cl). Notice that for a > C, the trivial solution of the equation i ( t ) = ax(t) f ( x ( t ) )is unstable. Thus, the time lag may have the stabilizing effect on the system.
+ +
142
3. Stability of Neutral Functional Differential Equations
Modifying functional (4.14) one may establish some other stability conditions. Let El0
< 1% t120
(1  .1o)Poo  (1
+ ao0 +
jams
+ cr1,)r
>0
dR(s) < cc
Then the trivial solution of (4.13) is asymptotically stable. For the proof one uses functional (4.14) in which the second integral contains Idk,(s)l instead of dK0M Consider now one example of a nonautonomous linear equation from which it is seen how general equations may be treated. Consider the equation i(t) =

b(t)x(t  h),
h >0
(4.16)
Here the function b(t) is continuous, bounded and also b(t) > 0. If
PI = sup f t O
l+h
b(s) ds < 1,
infb(t) > 0 f t O
then the trivial solution of (4.16) is asymptotically stable. The proof is based on the functional V ( t,x,)
=
[
x(t) 
jff+h
b(s)x(s  h ) ds
1
+ l  h b ( t l + 2h) d t , l : b ( s + h)x2(s)ds Some applications of degenerated Liapunov functions for study of adaptive systems, governed by ordinary differential equations, are given by Kolmanovskii and Nosov [108(10)].
$5. INSTABILITY OF NEUTRALTY P E EQUATIONS
5.1. Statement of the Problem In this section various problems of instability of NFDEs are studied. The dependence of instability conditions on the set of allowed initial disturbances is marked. We formuiate general instability theorems with the use of degenerate functionals defined on the paths of disturbed motion. It is possible to extend the classical theorem of Chetaev [39] on the systems with aftereffect and to find the selfexcitation conditions for certain distributed selfoscilla
143
$5. Instability of NeutralType Equations
tory systems. For certain equations some instability conditions are obtained. As in Chapter 2, 91 we denote M a metric space of the continuous function rp(t), rp: ( a, 01 + R , with a metric p and Q His a sphere in the space M . Let F and G be two continuous mappings and also IG(t,
CP)I + / F ( t ,CP)I5 F o ,
~ E C O , a), ~ ( Q E Q ,
(5.1)
Consider the initialvalue problem ( d / d t ) [ x ( t ) G(t, x,)]
x,(e)
= F(t, xJ,
= rp(e),
t 20
(5.2)
a < e Io
Below it is assumed that a solution of problem (5.2) exists and that it is unique and continuously depends on the initial data. Let G(t, 0) = F(t, 0) = 0. Then Eq. (5.2) has a trivial solution.
Definition 5.1. The trivial solution of (5.2) is unstable if for any positive E and 6 there exist an initial function rp(0)E M and an instant t , = t,(6) > 0 such that P(CP,0 ) < 6 but Ix(ti, ~ p ) l 2 E5.2. Influence of the Choice of an Admissible Class of Disturbances on Stability
A correct formulation of the stability problem for FDEs must also include the determination of the allowed initial disturbances for the problem under consideration. However, such an analysis depends essentially on the form of the problem studied and, in general, it is hardly realizable. In the theory of RFDE stability, the class of allowed disturbances is usually taken in the form of continuous functions. In this connection we must bear in mind that a solution which is stable under such disturbances will be stable also in cases when actually only a narrower class of initial disturbances is possible. O n the other hand, a solution that is unstable under any continuous disturbances may prove to be stable under actually existing disturbances. This shows that the instability criteria are relatively inferior. We now present an appropriate example. Consider an RFDE which is stable under disturbances actually feasible in such a system and unstable in the case of arbitrary continuous disturbances. In Shimbell [206] the following model was used to describe the behavior of the central nervous system during learning i ( t ) = k [ x ( t )  x(t  I)]"
~(t)],
t>1
(5.3)
i ( t ) = kx(t)[N  ~ ( t ) ] 0I t I 1, ~ ( 0=) x0, 0 < x0 < N , k > 0
(5.4)
144
3. Stability of Neutral Functional Differential Equations
Examine the stability of a solution x(t) = N of Eq. (5.3) [108(8)].As the class of admissible disturbances of the initial function in the interval [O, 11 let us take a set of solutions of Eq. (5.4) which have the form x ( t ) = Nxo[xo
+ ( N  x o ) exp(

kNt)]
(5.5)
From (5.5) it follows that any solution x(t) of Eq. (5.4) is increasing for t E [0, 1 ) . Any solution of Eq. (5.3) which corresponds to the initial function (5.5) in [0, 11 is not decreasing in [0, co),in addition remaining smaller or equal to N . Actually, the solution x(t) cannot cross the line x = N , since in this case i ( t , ) = 0 at the point t,, where x ( t l ) = N ; and for t 2 t , we should have x ( t ) = N . Further, by virtue of (5.3) we have i(1) > 0, since x(1) > x(0) and N > x(1). This means that the solution x ( t ) increases to the right of the point t = 1. Let t , be the first instant such that i ( t , ) = 0. Then either x ( t l )  x ( t ,  1) = 0 or N  x ( t l ) = 0. In the first case, because of Rolle’s theorem there exists a point t , < t , such that i ( t , ) = 0. But this contradicts the assumption that t , is the first moment at which i ( t ) = 0. In the second case, x ( t ) = N for t > t,. Hence the solution x(t) = N of Eq. (5.3) is stable under arbitrary initial disturbances (5.5). We can show, however, that the solution x(t) = N will be unstable if the allowed initial disturbances in [0,1] are taken in the form of space C[O, 13. Take a twice continuously differentiable function cp(8) such that llcp(8)  NII I 6, cp(1) < N , @(8)< 0, (p(8) < 0, 0 I 8 I 1. The derivatives i(1) and x(1) are negative on the basis of (5.3). Hence the solution x(t, 1, cp) decreases to the right of the point t = 1. Show that x(t, 1, cp) +  co for t + co. Assume the contrary. Then there must exist a first point t , > 1 such that x(t,) = 0. But x(t) < 0 for t E [ l , t,). Therefore N  x ( t l ) > 0, i(t, > 0, x(t)  x ( t ,  1 ) < 0, i ( t , )  i ( t l  1) < 0. Hence x(tl) = k [ i ( t , )  i ( t ,  1)][N  x(t,)] k[x(t,)  x(t,  l ) ] [  i ( t , ) ] < 0. The contradiction that results shows that x(t, 1, cp) < 0 for 1 It < 00. In addition, it is obvious that x(t, 1, cp) +  co for t + co.Thus the solution x(t) = N of (5.3) will be unstable if allowed initial disturbances coincide with the space C [  h, 01.
+
5.3. Instability Conditions Consider a continuous functional V ( t ,x,, Z(t, x,)), where Z(t, x,) = x(t) G(t, x,). The region V > 0 is a connected open region in the product [0, co) x QH which is bounded by the surface V = 0 and in which the functional V takes only positive values. Theorem 5.1. Let there exist a continuous functional V ( t )= V( t,x f ,Z(t, x,)) such that for t = 0 the region V > 0 has an open section whose boundary
145
$5. Instability of NeutralType Equations
contains the element cp(0) = 0. Further, in the region V > 0 let the following conditions take place for Eqs. (5.3), (5.4),( 5 . 3 , respectively,
Then the trivial solution of (5.2) is unstable. Proof: Let 6 > 0 be arbitrarily small. There exists an initial function cp(0) such that P(cp(@, 0)
6,
V(0) = V(0,cp, Z(0, cp))
=a
>0
(5.6)
Show that a solution x(t, cp) of (5.2) with such an initial function leaves the sphere QH for t + co. The solution x(t, cp) cannot leave the region V > 0 by passing through the part of the boundary on which V = 0. In fact, if t , > 0 is such that V ( t , ) = 0,
V ( t )> 0
for 0 I t < t ,
(5.7)
then by virtue of (5.4) it follows that P ( t ) 2 0, 0 5 t < t , . Hence V ( t ) 2 V(0)= a > 0, and by continuity V ( t , ) 2 V(0)= a > 0, which contradicts (5.7). Hence the solution x(t, cp) cannot leave the region V > 0 across the part of the boundary on which V = 0. But the solution x(t, cp) cannot always remain in the region V > 0. Assuming this, we would successively obtain V ( t ) 2 0,
V ( t ) 2 V(0)= a > 0,
P ( t ) 2 B(a) > 0
(5.8)
We find by integrating (5.8) that V ( t >xt, Z ( t , x,)) 2 V(0,cp, Z(0, cp))
+ @(a>
(5.9)
If t b ' ( a ) [ q ( H ) a ] , then inequality (5.9) contradicts condition (5.3). So the solution x(t, cp) must leave the region V > 0 through the part of the boundary on which p ( x t ,0) = H . This completes the proof of Theorem 5.1.
REMARK5.1. Suppose that we can select a subregion D which belongs to the region V > 0 and such that all the conditions of Theorem 3.1 are fulfilled in D. Suppose also that x(t, cp) E D for all t > 0. Then the assertion of Theorem 3.1 is valid. A region D which satisfies the preceding conditions is called a sector C1911.
146
3. Stability of Neutral Functional Differential Equations
EXAMPLE 5.1. Consider the equation i(t) ci(t
c 2 0,
a(t) 2 0,

z) = a(t)x(t)
b(t) 2 0,
a(t)
+ b(t) x
( t  z)
+ b(t) 2 A > 0,
z >0
(5.10) (5.1 1)
Introduce the functional V ( t ) = Z ( t , x , ) = x(t)  cx(t  z). Choose the initial function cp(8) such that cp(8) > 0, IIcp(0)ll I 6, Z ( 0 , cp) = cp(8)  ccp(z) = ct > 0. Show that x(t, cp) > 0 for t > 0. Assume the contrary and let t , > 0 be the first instant such that x(t,, cp) = 0. For 0 < t < t,, Z(t, x , = i ( t )  c i ( t  z)
= a(t)x(t)
+ b(t)x(t  z) > 0
Hence Z ( t , x , ) 2 Z ( 0 , cp) = ct > 0. By going to the limit for t + t ,  0, we obtain Z(t,, x , , ) 2 ct > 0. Therefore x ( t l ) = Z ( t , , x,,) c x ( t ,  z) 2 c1 > 0. The latter inequality contradicts the preceding assumption. Thus for any t > 0 we get v ( t )= a(t)x(t) b(t)x(t  z) 2 Act > 0. By virtue of Remark 5.1 the trivial solution of (5.10) will be unstable under conditions (5.11). In the same way it is possible to prove that the trivial solution of the equation i ( t )  c i ( t  z) = g(x,),g(0) = 0, c > 0 is unstable if the functional g(x,(8))> 0 for min, x,(8) > 0, z I 8 I 0.
+
+
5.4. Connection between Instability of NFDEs and Ordinary Differential Equations
As a consequence of Theorem 5.1, investigate the connection between the instability of an ordinary differential equation ~ ( t ,= ) x,,
i ( t ) = F ( t , x,),
to 2 0 , F(t, 0) = 0
(5.12)
and an NFDE i ( t ,X J
=~
( tz(t, , xt)~
xO(0) = ~p(0)
G(t, 0) = 0
Z(t, x , ) = ~ ( t) G(t, x,),
(5.13)
Theorem 5.2. I f the trivial solution of ordinary equation (5.12) is unstable then the trivial solution of N F D E (5.13) is also unstable. Prooj By the inversion theorem for Chetaev’s theorem [114(5)] there exists for (5.12) a function W ( t ,x ) which satisfies the conditions of Chetaev’s theorem. In particular, by virtue of (5.12)
aw
W(t,x ) = __ + at
1 aw axi n
~
i=l
i i
= @(t,x )
(5.14)
Now consider the functional W ( t ,Z ( t , x,)). Using (5.14) and (5.13) we find that W ( t ,Z ( t , x , ) ) = @ (t, xr)). The ‘validity of the other assumptions of
147
$5. Instability of NeutralType Equations
Theorem 5.1 about the functional w ( t , Z(t,x,)) is obvious. Hence the functional W(t,Z(t, x J ) satisfies all the conditions of Theorem 5.1. SO the trivial solution of (5.13) is unstable. EXAMPLE 5.2. Consider the equation (d"/dt")[x(t)+ cx(t  T)]
=f
(x(t)
+ cx(t  7))
(5.15)
Let c be a constant, n 2 3, and let y = 0 be an isolated root of the equation f ( y ) = 0. Then the trivial solution of the equation y"(t) = f ( y ) is unstable ([191], p. 148). According to Theorem 5.2, the trivial solution of (5.15) is also unstable.
5.5. Other Instability Conditions Denote q ( u ) as scalar nondecreasing functions of the argument u 2 0, such that q ( 0 ) = 0 and q ( u ) > 0 for u > 0. Theorem 5.3. Let all the conditions of Theorem 5.1 hold except conditions (5.3) and (5.9, which are replaced in the region V > 0 by v t , x,, Z(t, x,)) 2 %( IZ(4 x,) I
(5.16)
P(t, x,, a t , x,)) 2 %(lZ(t, x,)l
(5.17)
Then the trivial solution of (5.2) is unstable. Proof: Choose any initial function cp satisfying the conditions (5.6). By an argument similar to that made in the proof of Theorem 5.1, it is easy to verify that for this initial function the solution x(t, cp) cannot leave the region across that part of the boundary at which V = 0. However, the solution x(t, cp) cannot remain all the time in the region V > 0. Otherwise V ( t ) 2 0,
~ ( t2) ~ ( 0=) a > O
Take B > 0 such that w2(B) = a. Then 0 2 ( B ) = a = V(0)I V ( t ) I 0 2 ( I Z ( t ,x,)l). Hence /? I IZ(t, x J . Consequently, (5.18)
V ( t ) 2 w3(B) > 0
Integrating (5.18) we obtain a contradiction with the boundedness of the V ( t ,x,, Z(t, x,)) in the sphere Q H :V(r, x,, Z(t, x,)) I functional co2(]Z(t,x,)l) 5 u 2 ( H Fo). Theorem 5.3 is proved.
+
EXAMPLE 5.3. Given the equation (d/dt)[x(t) g(t, ~ (t h))] = U X ( ~ ) , g: R, x R,
+
R,
u > 0, h > 0
(5.19)
I48
3. Stability of Neutral Functional Differential Equations
Let V(t,x,, Z(t, x,)) = Z2(t,x,)  a
L
x2(s)ds (5.20)
Z(t, x,) = x(t)  g(t, x(t  h))
Functional (5.20) satisfies condition (5.16). Further if gz(t, x2(t  h), then
x ( t  h)) I
P = 2ax(t)Z(t,x,)  axZ(t) + uxZ(t  h) =d
+
( t )  2ax(t)g(t,~ (th)) + &(t, ~ (t h)) agz(t,x(t  h)) &(t  h) 2 aZ2(t,x,)
+
Hence by virtue of Theorem 3.3 the trivial solution of (5.19) is unstable. 5.6. Instability of Equations with Bounded Delay
Investigate Eqs. (5.2) with finite or infinite delay but such that the initial set coincides with the interval [  h, 01. So F and G are continuous functionals mapping from [0, 00) x C [  h, t ] into R,. Theorem 5.4. Let there exist a continuous functional V(t,x,, Z(t, x,)) and the region V > 0 satisfy the conditions of Theorem 5.1. Let in the region V > 0 V(4
IG(t
XI,
(5.21)
Z(4 x,)) I w4(11x,ll)
+ s, x,+,)  G(t, x,)l I w5(s) + v
max
IX(T
+ s)  x(z)l
(5.22)
hsrst
Then the trivial solution of (5.2) is unstable. Proofi The solution x(t, 9)cannot leave the region V > 0 through the part of the boundary V = 0. Show that the solution x(t, cp) cannot remain in the region V > 0. Really, let the initial function satisfy (5.6) and p > 0 such that w&) = a. Then w4(p) = CI = V(0)I V ( t ) I w4(llxtll). Consequently, there exists a sequence {ti} such that t i + 00 and Ix(ti)I = p. Show that the function x(t) = x(t, cp) is uniformly continuous in [  h, m). By using (5.1), (5.2) and (5.22) for any A > 0, we get Ix(t
+ A)

x(t)l = IG(t
+ A, x,+J

G(t, x,)
Denote p(q) the function p(q) = sup suplx(t r>O
Asv
+ A)

+
149
4.5. Instability of NeutralType Equations
From (5.24) and (5.22) it follows that (1  v)p(q) 5 max max Ix(0
+ A)  cp(0)l + qFo + w5(q)
ASq h5BSO
The righthand side of this inequality tends to zero for q + 0 due to the uniform continuity of x(t, cp), t E [  h, q]. Hence the uniform continuity of x(t, cp), t E [  h, 00) is proved. Therefore there exists an q such that Ix(s) 1 2 p/2 for ti  q I sI ti q and any i. Integrating (5.23) we obtain for t + co
+ V2u +
s:
o ~ ( ( x ( s ) ( )ds 2
O!
+ 2N(t)~,(p/2)
where N ( t ) is a number of points ti such that ti E [0, t ] . Theorem 5.4 is proved.
EXAMPLE 5.4. Obtain the instability conditions for a scalar NFDE
s : s:
i ( t ) = aOx(t)
k,(s)
=
+
A(t) dt
1:
~ ( t S ) dkO(s) +
+ 1 pn,
i ( t  S)
dk,(s) + ~ ( tx,) ,
(5.25)
h, 2 0
h,st
Here the continuous functional a(t, cp) is defined on [0, co) x CB[  co,01 and also 144 @ I 2
0 ) = 0,
johla(s)12W s )
Kernel R(s) is nondecreasing and bounded. Introduce a functional V,
Here Ph
r2
=
J
Ph
dR(s), 0
Z ( t , x,) = x ( t ) 
aoi =
c
J 0 Idk,(s)I
x ( t  s) dk,(s)
Relations (5.25), yield v(t)2 2x2(t)[ao(l  sol)  (1 + aO1)(r+ aoOl.By virtue of Theorem 5.4 the trivial solution of (5.25) is unstable under condition
150
3. Stability of Neutral Functional Differential Equations
+
+
a,(l  sol) > (1 aol)(r aoo). By using the methods of functionals I/ construction described earlier in this book, it is possible to obtain other conditions of instability of system (5.25).
5.7. Distributed SelfOscillatory Systems Consider the distributed lossless system from Chapter 1, $2 described by the equation
_1 _d [ ~ ( t) K x ( t  T)] C , dt
1 K 21 t =  (5.26) ~ ( t) ~ ( t t)  g(x(t)) + K g ( x ( t  T)), Z Z S Assume that the nonlinear element g( V ) has a negative differential resistance at certain points. Such an element can be the tunnel diode whose currentvoltage curve is plotted in Fig. 3.12 [147]. Find the conditions of selfexcitation of oscillations which are equivalent to the conditions of instability of the solution x ( t ) = uo. The value uo satisfies the equation 0 = CI Z  ' u ,  K Z  ' u ,  g(u,) Kg(u,). Denote y ( t ) = x ( t )  u,. Equation (5.26) in the neighborhood of uo can be written in the form = CI

~
+
= (S  Z  ' ) y ( t ) 
K(S
+ Z  ' ) + O(y(t))+ O(y(t

T)) (5.27)
where S is the steepness of the currentvoltage curve at the point u,; i.e., s = g(uo). Show that the condition of selfexcitation of the distributed
"0
Fig. 3.12. Currentvoltage curve.
V
96. Asymptotic Properties of NeutralType Equations
151
selfoscillatory system represented in Fig. 1.12 has the form S > K 1 .Study for this purpose the directly nonlinear equation (5.27). Take an E > 0 so small that S > (1
+ K + 2~)(1
K)'Z'
[(K(S
+ Z)')

(5.28)
and also in the sphere Q
2 ( S  2'

E)y(t)


~ ] y (t T)
(5.29)
Consider the functionals I/=
y ( t )  Ky(t  t) [ K ( S
+ 2)' + E ]
LZ
Y(S)
ds
(5.30)
The region B = (I/ > 0 ) n ( y > 0) n Q, is a sector. The trajectories of system (5.26) cannot cross the boundary V = 0 because
V
=
[IS  2'
+ K ( S + 2  ' ) + 2~]y(t)2 0.
In ,a manner similar to Example 5.1, we conclude that y ( t ) > 0 for Otherwise, if y ( t J = 0, then V ( t ) 2 V(0)2
ff, y(t1) =
V(t1)
t
> 0.
+ K y ( t ,  t)
+ [ ( K ( S + Z)' +
E]
J;&y(s) ds.
Hence the region B is a sector in which i/ 2 0. By virtue of Remark 5.1, the solution x(t) = uo is unstable.
$6. ASYMPTOTIC PROPERTIES OF NEUTRALTYPE EQUATIONS
6.1. Some Introductory Remarks One of the important problems in the qualitative theory of NFDEs is the study of the asymptotic behaviour of its solutions. Asymptotic behavior for t + a3 of NFDE solutions may be different. They can tend to stationary or periodic solutions or their behavior can be more intricate. Neutral functional differential equations have properties that are similar to the properties of difference equations for some cases and properties that are similar to the
152
3. Stability of Neutral Functional Differential Equations
properties of differential equations for other cases. This duality of NFDEs is essential for a study of the asymptotic behavior of their solutions. In this section some results of both types are described. The method of investigation that we use is based essentially on degenerate Liapunov functionals. It is possible with the aid of degenerate functionals to separate the cases when the asymptotic behaviour of NFDEs is similar to that of difference equations from the cases in which this is not valid.
6.2. Basic Definitions Consider the NFDE [108(7)] Z ( t ) = F(t, Z),
Z ( t ) = x ( t )  G(t, X J ,
t20
(6.1)
Here F and G are given continuous mappings, F(t, x), F : [0, co) x R, + R,; G(t, xt), G : [0, co x C [  h, 01 + R,. The solution of (6.1) is determined by the initial data xo = cp,
cpEcCCh,01
(6.2)
Let lF(t, x)l I M , for 1x1 IH and IG(t, cp)  G(t, $)I I K,llq  $11 for llcpll I H , Il$l/ I H . Assume that there exists a unique solution of problem (6.I), (6.2). Scalar equations (6.1) named completely integrable are considered in Sharkovskii and Romanenko [200]. Let M c C [  h, 01 be a closed set. The distance p between the element cp E C[  h, 01 and the set M is equal to p(cp, M ) = min, 11 cp  Y 11, Y EM . A set M c C [  h, 01 is called an inoariant set for Eq. (6.1) if for arbitrary cp E M , the solution xt(cp)E M .
Definition 6.1. The closed invariant set M of Eq. (6.1) is called stable if for any E > 0 there exists 6 > 0 such that p(x,(cp), M ) E, t > 0,for cp: p(cp, M ) < 6. ZA in addition, lim p(x,(cp), M ) = 0, t + co,V cp ER, then the set M is called asymptotically stable and the set R is called the attraction domain of the set M . Stability of invariant sets is investigated with the aid of the functional inequality Iz(t, Yt)l
=
IY(t> G(t, Yt)l I f ( t > ,
fER1
cpEah,Ol Let M , c C [  h, 01 be a closed invariant set of equation Yo(cp> = cp?
a t ,YJ
= Y(t)  G(t, Y J = 0
(6.3) (6.4)
(6.5)
Definition 6.2. The closed invariant set M of functional Eq. (6.5) is called f stable if for any E > 0 there exists 6 > 0 such that for all solutions of inequality (6.3),(6.4) we have p(y,, MI) < E ifp(cp, M , ) 5 6 a n d f ( t ) 5 6. Thefstable set
153
56. Asymptotic Properties of NeutralType. Equations
M , is called asymptotically f stable if lim p(y,(cp), M , ) = 0, t + 03, for all cp E IR c C [  h, 01 and for all f ( t ) such that f ( t ) + 0, t + 00.
For the case in which the sets M and M , contain the only zero element, Definitions 6.1 and 6.2 coincide with Definitions 3.2.
6.3. Stability of Invariant Sets Theorem 6.1. Let the trivial solution of the ordinary diflerential equation (6.6)
i ( t ) = F(t, x(t))
be uniformly stable (uniformly asymptotically stable). Further assume that the bounded invariant set M of (6.5) is f stable (asymptotically f stable). Then M , will be a stable (asymptotically stable) invariant set of Eq. (6.1).
,
ProoJ By virtue of the inversion theorem [114(5)] there exists for Eq. (6.6) a Liapunov function V(t,x) such that ol(lxl) I v(t,
s 02(IxI),
p(6.6) I O,
cv6.6)
I 03(Ixo]
Here q ( u ) are continuous scalar nondecreasing functions such that oi(0) = 0, q ( u ) > 0 for u > 0 and e 6 . 6 ) is the derivative of the function V along the trajectories of (6.6). Now consider the functional V(t,Z(t, XJ). Clearly, o l ( I Z ( t ,x,)l) IV(t, Z(t, x,)) I m2(1Z(t,x,)l), p(6.1) I 0. Assume any E > 0. From the f stability of the set M , follows the existence E for p(cp, M , ) I 6 and f ( t ) I 6. Choose of a 6 > 0 such that p(y,(cp), M , ) I 6, > 0 such that ~ ~ ( 6K,6,) , I o,(S). Let p(cp, M , ) = 1140  $11 I 6,, llcpll I H and 11$11 I H. Since Z(t, $) = 0 for $ E M , ,
+
IZ(t9 cp)l = IZ(4 $ + (cp  $111 I IZ(t, $)I + IZ(t3 $ + (cp  $))  Z(t?$11 5 (KH
+ 1Pl.
Using the properties of functional V(t, Z(t, xt)), we have
w,a t , 4)IV(O,Z(O,cp))
w,(lZ(t, 41)I
+
I ~ 2 ( 6 , KH6,) I~ , ( 6 )
The monotonicity of the function o l ( u )implies that lZ(t, xt)1 I 6. But the set M , is f stable. Hence p ( X , , M , ) IE ; i.e., the set M , is stable. Similarly, asymptotic stability of the set M , may be established (see Theorem 3.3). Theorem 6.1 is proved.
EXAMPLE 6.1. Consider the scalar equation i ( t ) = f (Z(t)>,
z ( t ) = x ( t )  ax(t  1)[1  x(t  I)] 1 < a < 3, t 2 0
(6.7)
154
3. Stability of Neutral Functional Differential Equations
Assume that the trivial solution of the equation i ( t ) = f ( x ( t ) ) , f ( 0 ) = 0, is asymptotically stable. For Eq. (5.7), difference Eq. (6.5) has the form x(t) = ax(t

1)[1

x(t  l)]
(6.8)
Equation (6.8) was studied in Sharkovskii and Romanenko [200]. For 1 < a < 3 there exists two stationary solutions of (6.8): x , ( t ) = 0 and x 2 ( t ) = 1  a  l . Denote u(t) = x ( t )  x2(t). Then by using (6.8) we get
I u ( t )  (2  a)u(t  1) + auyt lu(t)l I 12  a1 lu(t  s)l

+ ad(t
1)1 I f ( t )

1)
+f ( t )
(6.9)
Let E > 0 be sufficiently small. Choose 6 ~ ( 0E ) , such that 12  a16 + + 6 I E. Remark that such a 6 always exists for 1 < a < 3 and sufficiently small E. For these E and 6 from (6.9) it follows that for cp, llcpll I 6 and J; llfll I 6, lu(t)l I 12  al lcp(t  1>1+ acp2(t  1) + f ( t ) I 12  a16 a62 6 I E, 0
+
+
(6.10)
By the step method similar to (6.10) we obtain lu(t)l I E, t > 0. Hence the set M , is f stable. The asymptotic stability of M , follows now from Kolmanovskii and Nosov [108(5)], p. 177, Lemma 2.2. The attraction domain of the set M , consists of functions q ~ C [  h 01 , such that p(cp, M , ) I C ’ ( 1  12  al). By virtue of Theorem 6.1 the set M , is asymptotically stable for NFDE (6.7).
6.4. Instability Investigate the conditions for which Eq. (6.5) does not determine the asymptotic behaviour of the solutions of NFDE (6.1) solutions. Let the initial functions 9 E C[  h, 01 be bounded, IIcp 11 I H. Assume that the corresponding solutions y,(cp) of (6.5) are defined for all t > 0. The set of such solutions is designated by N ( H ) .
Definition 6.3. The set N ( H ) is f unstable if for any E > 0 there exist cp, II 40 /I 5 H and = q ( ~such ) that I Z(q, x,(cp))  Z(q, Y,) I = I Z(q, x,(cp) I 2 E. If the set N ( H ) is f unstable, then in general it does not define the asymptotic properties of the solutions of Eqs. (6.1). Theorem 6.2. Let the trivial solution Eq. (6.6) be unstable. Then the set N ( H ) is f unstable.
155
96. Asymptotic Properties of NeutralType Equations
Prooj According to the inversion theorem [114(5)], there exists for Eq. (6.6) a function V ( t , x ) satisfying the conditions of the Chetaev theorem. Consider the functional V(t,Z(t, x,)) = V(t).There are elements cp with an arbitrarily small norm in the domain V > 0. In this domain the conditions $ 6 . 1 ) 2 0 and 2 p > 0 if V ( t ) 2 c1 > 0 are fulfilled. In addition, the solution x,(cp) leaves the set N ( H ) . In fact, if V(O,Z(O,cp)) = c1 > 0, then $6.1)(t) 2 0. Hence V ( t )2 V ( 0 )= c1 > 0 and $6.1)( t ) 2 b > 0, V ( t ) 2 P t + 00 for t + 00. But V(t,2)is continuous and bounded in Z for all t . Consequently, the functional 2 cannot be bounded for all t . So Z ( t , x , ) will be greater than any arbitrary E > 0 as soon as t is sufficiently large. Theorem 6.2 is proved.
e6.1)
EXAMPLE 6.2. Consider once more Eq. (6.7). Assume that Eq. (6.7) is unstable and 0 < a < 1. Then there exists the unique asymptotically stable solution y ( t ) = 0 of (6.8). But for NFDE (6.7) this solution will be unstable. So in the considered case, the behaviour of solutions of difference equations (6.8) and NFDE (6.7) are quite different. 6.5. Interconnection between Functional and Differential Equations Sometimes the asymptotic properties of solutions to NFDE (6.1) are determined both by the behaviour of solutions of functional equations (6.5) and by the behavior of differential Eq. (6.6). Consider Eq. (6.1) with Z ( t , x,) = x ( t )
+ q(t, x(t  l), x(t  2), ... , x(t  p ) )
x(t)ER,,
4 : R, x
... x R , + R ,
(6.11)
Here q is a continuous function and [email protected],x(t  11, x(t  21,. . . x(t  P))l I Ix(t  1)1 ... J x ( t p)J 7
+ +
(6.12)
Theorem 6.3. Let the trivial solution of E q . (6.6) be globally exponentially stable and condition (6.12) be fuEfilled. Then all solutions of E q . (6.1) with Z ( t , x,) given by (6.1 1) are bounded for t 2 0 and its trivial solution is stable. Pro05 Global exponential stability of the trivial solution of Eq. (6.6) means that there exist two constants C ( x o ) > 0 and c1 > 0 such that Ix(t, xo)l I C(x0)eFa',
V xo
(6.13)
So any solution of (6.1) by (6.13) satisfies the estimate
Iz(t,x,(cp>>lIC ( Z ( 0 ,cp>>e'' 5 Cl(llcpll)e"',
C,(llrPll) I C(P11cpIl)
(6.14)
156
3. Stability of Neutral Functional Differential Equations
From (6.14) and (6.1 1) it follows that
I4t) + d t , x ( t

I), . . . , x ( t  p))l
s Cl(llcpll)ear
Hence I Ix(t

+
+
+
Ix(t  2)l .... Ix(t  p)I + Cl(llcpll)eat 211 + ... + Ix(t  p)l + Ix(t  p  1)l
Ix(t)l I lx(t  1)l
+ C,(llcpll)[e"' + e"('')] I Iq(t  N ) ( + ... + Iq(t  N  p ) ( + cl(llcpll)[ea'
+ e  Q ( t  1 ) + ... + e"('N+1)
1
N . Designate y Here the number N is such that N  1 < t I Then inequality (6.15) takes the form Ix(t)l I PIlcplI + cl(llcpll)c~Nl +YN2
expCa(t  N
+ 111 5 PIIcpII + C,(llcpll)(l
 y1l
(6.15)
= exp(  a )
+ ... + 11
< 1.
(6.16)
So all solution of (6.1) are bounded. But estimates (6.16) is uniform in cp with llcpll I H.Therefore from (6.16) follows the stability of the trivial solution of Eq. (6.11). The Theorem 6.3 statement is also valid if condition (6.12) is exchanged for the following one
Id t , x ( t
1)), x ( t  2), . . . x(t  P) I I K[Ix(t  1)1 + ... + Ix(t  p)[], 
9
K exp(a) < 1 (6.17)
For K > 1 the trivial solution of the difference equation x ( t ) + q(t, l), . . . , x(t  p ) ) = 0 is in general unstable in the Liapunov sense. Condition (6.17) coordinates the order of this instability with the speed of convergence to zero of all solutions of ordinary differential Eq. (6.6). In addition, all solutions of NFDE (6.1) are bounded and the trivial solution of (6.1) is stable. x(t 
57. LIAPUNOV FUNCTIONALS DEPENDING ON DERIVATIVES
7.1. General Stability Theorem Let us denote by W a normed space of absolutely continuous functions cp(f3) defined on [  h, 01 with square integrable derivatives. Define the norm in W
IIcp11~= cp
157
97. Liapunov Functionals Depending on Derivatives
Denote QH a sphere in W :QH = {cp(B)): IIcp(0)llw 5 H } . Consider the following NFDE i ( t ) = f (t, x,, if), x,,(e)
=
cp(e),
x(t)ER,,
t 2 to
(7.1)
h I e Io
i f ,= @(e),
Let the map f : R , x Q H+ R , be continuous and satisfy the Lipschitz condition in the second and third arguments, and let the Lipschitz constant in the third argument be less than 1. Then, as in Theorem 1.3.2, one can prove that for any cp(0)E W there exists a unique solution of NFDE (7.1) such that x,(e)E Wfor all t 2 to. Suppose that f(t,O,O)=O,
If(t>cp,+,)l2M,
IIcpllwI~,
tER
(7.2)
Definition 7.1. The trivial solution x(t) = 0 of Eq. (7.1) is said to be stable if for any E > 0, t o e R, there is a 6 = b ( ~to) , such that ( ( c p ( l w 2 6(e, to) implies x(t, to, q)l 5 E. The solution x(t) = 0 is said to be asymptotically stable if it is stable and q(t)ER(to)c Wimplies x(t, to, cp) + 0 as t + co.The domain Q(t,) is called an attraction one at moment to. Other definitions of stability can be introduced in a manner analogous to that presented in Chapter 2, $1. Let the functional V(t,cp, @)becontinuous in the sphere Q H and satisfy the , x(t) is a Lipschitz condition in cp and @. Denote by V ( t ) = V(t,x,, i f )where solution of Eq. (7.1). Suppose that the functional V ( t ) is absolutely continuous in t for any solution x(t) of Eq. (7.1). We denote P(t) the upper rightalong the solution of Eq. (7.1): hand derivative of V ( t ,x,, i f ) V(t)=
G
[V(t + At)  V(t)]/(At)
A,+ + O
Denote wi(u) continuous nondecreasing functions such that oi(0)= 0, wi(u) > 0 for u > 0.
Theorem 7.1. Let the continuous functional V ( t ,x,, i, exist ) and satisfy conditions ~ l ( I l X , l l W )2
V t , x,, if)I~ 2 ( I I X , I I w )
V ( t )I 0
(7.3) (7.4)
Then the trivial solution of Eq. (7.1) is stable. Zf P(t) I Wj( Ix(t)I) then the solution x(t) = 0 is asymptotically stable.
(7.5)
158
3. Stability of Neutral Functional Differential Equations
Pro05 Assume E > 0. Take h(e) such that wl(&) = w2(h).From (7.3), (7.4) it follows for llcp(0)llw I 6 ( ~ that )
~l(lIX,IlW)I V ( t )I V ( t 0 )
~2(IlcpIIw) 2 w 2 ( 4 = W,(E)
Ilx,ll$ 2 E ~i.e., ; Ix(t)l I E.This proves stability. To This implies that Ix(t)12 I prove asymptotic stability, choose H , such that w,(H) = w 2 ( H , ) .Then, as H previously demonstrated, \lcpllw I H , and inequality (7.2) implies ( I X , I ~ ~ I and l i ( t ) ( I M . Suppose that a solution x(t, to, cp), Ilcpllw I HI does not tend to zero as t + 03. Then for some E > 0 there exists a sequence ti co,i 03, such that Ix(t, to, cp)l 2 E. Therefore, on the intervals ti  A I t 2 t i A, A = &(2M)’ we have /x(t, to, cp)l 2 4 2 and k(t) I a3(&/2) = y < 0. Therefore, +
c j,LA k(s) d s
+
+
ti+A
V ( t )  V(t0)=
p(4ds 2
2 2y A N ( t )
t,
where N ( t ) is a number of points ti such that t i < t . Thus V ( t )  V(to)>  co as t + co. This contradicts inequality (7.3). The proof of Theorem 7.1 is complete.
7.2. Examples Consider the Eq. [l50] i(t) =
ax(t)
+ c(t)i(t

h),
t20
(7.6)
where c(t) is a continuous function. Define the functional [l50] V ( t , x,, i t )= x 2 ( t )
+
For p ( t ) we get V ( t ) = 2x(t)i(t)
+ a ‘[i’(t)
 i 2 ( t  h)]
Replacing i ( t ) according to (7.6) we obtain for Ic(t)l < 1 that P ( t ) = ux2(t)  a1[1  2(t)]i2(t

h) I ux2(t)
Therefore, the functional V ( t , x,, i t )satisfies all the conditions of Theorem 7.1. This implies the asymptotic stability of the solution x(t) = 0. For other examples see El’sgol’tz and Norkin [59] and Misnik [lSO]. Castelan and Infante [35], for a matrix autonomous NFDE i ( t ) = AX([) f
Bx(t  h) + C i ( t  h),
X(t)ER,
(7.7)
have constructed the functional V ( t ,x,, i , )satiisfying the conditions of Theorem 7.1 under the assumption that the trivial solution of (7.7) is
159
48. Stability and Boundedness of Linear Nonhomogeneous Equations
asymptotically stable. A Liapunov functional was obtained as the limit, in an appropriate sense, of a Liapunov function constructed by wellknown methods for a difference equation approximation of the original NFDE (7.7).
88. STABILITY AND BOUNDEDNESS OF LINEAR NONHOMOGENEOUS EQUATIONS 8.1. Connection between Stability and Boundedness Consider the NFDE
Here the operator L(t,x,, i t ) = A,(t, i t ) + A2(t,x,) is linear in the second and third arguments. Assume that for problem (8.1) conditions of existence, uniqueness and continuous dependence of solutions on initial data are met.
Theorem 8.1 [253(5)]. Zf any solution of problem (8.1) with f ( t ) = 0 is bounded for t 2 0, then its trivial solution is stable. Notice that, in general, the boundedness of all the solutions of a nonlinear equation does not imply their stability (see Example 2.1.1). Let the operators A,(t, i t ) and A,(t, x,) be bounded and o periodic in t. Consider functional equation z(t) = A , ( t , Z t )
+ g(t),
t > 0,
z(e>= 0, e I o
(8.2)
Problem (8.2) is said to satisfy the Perron condition if its solution defined by
Theorem 8.2 [162(13)]. Let operator L(t, x,, iJ be linear in x,, 2, and periodic in t. Assume that problem (8.2) satisjes the Perron condition and that Eq. (8.1) has a bounded solution for the functions f ( t )E L,[O, 00) and q(0) = 0. Then the trivial solution of Eq. (8.1) is asymptotically stable. 8.2. Boundedness of Derivatives Consider an ordinary differential equation with bounded coefficients
+
X(")(t) A,(t)x'"''(t)
+ . . . + A&)
x (t)=f ( t ) X(m) =
It is known the following Esclangon's theorem [36].
d"x/dt"
(8.3)
160
3. Stability of Neutral Functional Differential Equations
Let there exist a bounded solution x(t),  co < t < co of Eq. (8.3) for some bounded functionf(t). Then all derivatives x(t), . . . ,x('")(t), co < t < co,are bounded. This theorem, extended on differentialdifference equations in [1943, is not valid in general for NFDEs. Consider the scalar NFDE (2n It ) i ( t )
i(Jt22)= 2(t  J t 2 _ 2 r r ) x ( J m )
The coefficient of this equation is bounded for t 2 2n and f ( t ) = 0. But this equation has bounded solution x(t) = sin(?), t 2 2n, with unbounded derivative. Prove an Esclangontype theorem for the NFDE x'"t'(t) + A,(t,
X y )
+ A2(t,Xt,...,xi",))
x(e) = q e ) = ... = x ( m ) ( e ) = 0,
t >0
=f(t),
h Ie I o
(8.4)
Here continuous operators A , : R , x L,[h,O] + R,, A2:R, x C [  h, 01 x . . . x C [  h, 01 + R , are linear in all arguments beginning from the second. In addition, A ,(t, cp) E L,[O, T ] for any cp(s) E L,[  h, TI, T > 0, and A2(t, $f", . . . , E L,[O, T ] for any @")(s) E C [  h, T I . Assume also that vrai suplA,(t, $;I), . . . , $$'")I OStST
< a [ ~ ~ $ ( l ) ~ ~ C [  ! r+ , T ~" '
+ ll$(m)llC[h,T])
(8.5)
Condition (8.5) replaces the assumption about the boundedness of the Eq. (8.3) coefficients. Consider the functional equation z(t) + A,(t, Z t ) = q t ) ,
t > 0; R(t)EL,CO,
z(e)= 0,
h Ie Io
col
(8.6)
Theorem 8.3 [l08(5)]. Let operators A , and A , satisfy the formulated requirements and Perron condition be met for Eq. (8.6). If there exists a bounded solution x ( t ) of Eq. (8.4) for a bounded function f ( t )E L,[O, a),then vraisup,,,,,[li(t)] + .  + Ix(")(t>l]< co. $9. LINEAR PERIODIC EQUATIONS 9.1. Some Examples Linear periodic NFDEs are studied less than RFDEs because the monodromy operator for NFDEs is not completely continuous. We now illustrate by examples some features of periodic NFDEs.
161
59. Linear Periodic Equations
EXAMPLE 9.1. The Floquet multiplier of NFDE may have injinite multiplicity. For the equation i ( t )  i ( t  1) = (cos 2nt)[x(t)  x(t  l)], the Floquet multiplier p = 1 has infinite multiplicity since any continuously differentiable 1periodic function x ( t ) is its solution. EXAMPLE 9.2. Consider a scalar equation which has no Floquet solution i(t)  i ( t

2n) = (cos t)x(t  271)
(9.1)
If x ( t ) = exp(It)y(t) is a Floquet solution of (9.1), then y ( t ) must be a 2nperiodic solution of the equation j ( t ) [1  exp(  2nA)] = [(cos t ) exp(  2nA)  I( 1  exp(  25ry))]y(t) (9.2)
But for any complex number I Eq. (9.2) has no nontrivial periodic solutions. Let be I = mi, m = 0, f 1, + 2 , . . . . Then the unique solution of Eq. (9.2) having the form 0 = (cos t)y(t) is y ( t ) = 0. If I # mi, m = 0, f 1,. . . ,then the general solution of Eq. (9.2) is exp(  2nI) cos s  I ] d s } 1  exp(  2nI) This solution will be 2n periodic if and only if
But it is impossible in the domain I # mi, m = 0, f 1,. . . . Hence, there are no nontrivial periodic solutions of Eq. (9.2) and the Floquet solutions of Eq. (9.1). EXAMPLE 9.3. Consider an equation whose Floquet solutions do not give an asymptotic representation for other solutions: 2i(t)  i ( t

271) = (cos t)x(t  27~)
(9.3)
Research of Floquet solutions of Eq. (9.3) is equivalent to the determination of a I such that the equation j(t)[2  exp(  2n;l)l = [(cos t ) exp( 2nI)  4 2  exp(  2nI))]y(t) (9.4) has a nontrivial period solution. If [2  exp(  2nI)l = 0, then Eq. (9.4) has no nontrivial periodic solutions. If [2  exp(  2nI)] # 0, then the periodic solution of Eq. (9.4) exists only for I = 0. Then this periodic solution y ( t ) is simultaneously a Floquet solution x,(t) of Eq. (9.3) x,(t) = y ( t ) = exp(sin t). Now find solution xl(t) of Eq. (9.3) with initial data x l ( t ) = sin t , i l ( t ) = cos t, 0 I t I 271. By the step method we have m+ 1
x,(t) =
C &,(sin tIk,
k= 1
2mn I t I 2n(m + I), m = 0, I , . . . (9.5)
162
3. Stability of Neutral Functional Differential Equations
Here Ak, are defined by the formulae Ak, = Ak, J2
+ Ak,ll1/2,
A f , = 1, A:
= A'",+' = 0
(9.6)
From (9.6) it follows that Ak, > 0, m = 1, 2, ...; 1 I k Im + 1 and m+ 1
max
1 Ak,,
Ix,(t)J= xl(tm)=
k=l
Znrn
1 "
1 A;1
~
t,
71
=
2
+ 27rm (9.7)
m+ 1
I
k=l
" 1 A;< 3 1 AL1
k= 1
4,=
so 1 x,(trn 1) 5 xl(trn> 2
~
3 qxl(tm 1 )
(9.8)
By virtue of (9.7) and (9.8) we get Ix,(t)l I ($)(r'2n)1 . H ence lim x l ( t ) = 0, t + co. But inf, x,(t) > 0. Consequently, the function Cx,(t) is the best approximation of solution x l ( t ) if and only if C = 0. The obtained Floquet representation is not asymptotic for the solution x l ( t ) . From (9.8) it follows that x l ( t m )2 2". Therefore the difference between the solution x l ( t ) and its Floquet representation Cx,(t) = 0 decreases less quickly than does exp[In 2(t/271 So the Floquet solution x,(t) does not give an asymptotic representation for x l ( t ) since from Theorem 2.12.2 must be equal or greater than (In 2/2n). Examples 9.19.3 show that a full generalization of Floquet theory for NFDEs is impossible. In particular, Theorem 2.12.2 is not valid for NFDEs. Notice, however, that a system of Floquet solutions is complete in the space C [  qw, 01, q = max(m, n) for the scalar equation
a)].
i ( t ) = a i ( t  mw)
+ q(t)x(t  no),
la1 < 1
Here the function q ( t ) 2 0 does not equal zero in whole intervals [116].
9.2. Stability Theorem Consider the equation ( d / d t ) [ x ( t ) G(t, xJl
= L(t, X J
(9.9)
Here operators G: R , x C [ h, 01 + R , and L: R , x C [ h, 01 + R, are continuous in both arguments, linear in the second argument and w periodic in t. The operator G(t, cp) is called a stable one if the trivial solution of the homogeneous functional equation x ( t )  G(t, x,) = 0, t 2 0, is uniformly asymptotically stable.
$9. Linear Periodic Equations
163
Theorem 9.1 [81(4), 108(5)]. Let the previously formulated assumption be met and let the operator G be stable. Then there exists a number a E (0,l) such that Eq. (9.9) has a Jinite number of Floquet multipliers p j for which lpjl 2 a. The trivial solution of Eq. (9.9) is stable i f and only i f all multipliers p j satisfy the condition lpjl 1 and those with lpkl = 1 have simple elementary Jordan divisors. Further, the trivial solution of Eq. (9.9) is asymptotically stable if and only i f lpjl < 1.
Chapter 4
Stability of Stochastic Functional Differential Equations $1. SOME PREREQUISITES FROM THE THEORY OF STOCHASTIC RETARDED FUNCTIONAL DIFFERENTIAL EQUATIONS
1.1. Statement of the InitialValue Problem
This chapter is devoted to the study of the theory of SRFDEs dx(t) = a(t, x,) dt x, = x(t
+ b(t, x,) d t ( t ) ,
+ e),
t20
co < e 5 0,
x(t)E~,
(1.1)
Here S(t)ER1 is a standard Wiener process, defined on the probability space (Q, 3,P). Recall that the elements w are called elementary events, and the sets in 3 are called random events (see [64,84,108(9)]). The probability measure P defined on the r~ algebra 3 is a nonnegative countably additive set function on 3 such that P(Q) = 1. A random variable [(w) E R , is a function defined on Q such that { w : [(a) < x } E 3 for any x E R,. The family of random variables [(t, ~ ) E R , t, 2 0 is called a stochastic process. For any fixed weR the function i ( t , w ) considered as a function of t is called a trajectory or sample function of the stochastic process. The stochastic process [ determines the minimal cr algebra whose related random variables [(s, w ) are measurable for s It. Usually the argument w of the stochastic process [(t, w ) is omitted. The standard Wiener process < ( t )has independent stationary gaussian increments and
t(0) = 0,
ECS(t)  S(S)l = 0
ECt(t)  t(s)lCt(t)  tWl‘ =
m 4
where E denotes expectation, I is a unit matrix and “prime” is a transportation sign. Let B,,,,(dt) c 3 be a minimal cr algebra whose related random variables t(s)  t ( t ) are measurable for any t , 5 t 5 s t , . The sample functions of 164
165
$1. Some Prerequisites from the Theory of SRFDEs
the process t(t)are continuous, nowhere differentiable, have infinite variations on any finite time interval and satisfy Holder's condition with an index less than f. The upper limit of Wiener process samples equals +co with probability 1 for t + co and the lower limit equals  03. The continuous functionals a(t, cp) and b(t, cp) in Eq. (1.1) are determined on [0, co) x CB[  co,O]. Given the stochastic process cp(8), 8 I0, the initial condition for (1.1) is x(e) = q(e), eI o (1.2) Assume that (i algebra B  ,o(cp) is independent from Bo,(d t).The stochastic process x ( t ) is called a solution of initialvalue problem (l.l), (1.2) if B,,(dt) is independent from B  ,,(x) w Bor(d5) for any t 2 0 and with probability 1 we will have
+
~ ( t=) ~ ( 0 )
1:
a(s, x,) ds
+
sd
b(s, x,) d t ( s )
(1.3)
where x(8) = q(8) for 8 I0. The last integral in (1.3) is It83 stochastic integral having the following properties [64,84]:
jo T
lo T
CC,fl(t) + [email protected])1 &(t)
= c1
E joTfl(t) &(t)
= 0,
fl(t) [email protected]) + c2 I O T f 2 ( tdt(t) )
lo T
lOTh(t)W t ) {oTf2(s)a
s )=
Efl(t)fi(t>dt
Here the arbitrary processesf,(t), f2(t), t~ [0, T] are measurable relative to (i algebra B  mo(cp) u Bor(dt)and
~oTElfl(t)12 dt < 00 1.2. Existence Theorem
Assume that for any x(8), y(8) E CB[  m,O] and t 2 0
M4
144 x)  44 Y)I2 I
 Y(S)12
dR,(s)
JOm
IW, x)  b(t, Y)12
I Ix(s>

Y(s>12 dR,(s)
( 1.4)
166
4. Stability of Stochastic Functional Differential Equations
where R , and R , are nondecreasing functions with bounded variations r:
=
i = 1, 2
jomdRi(s)< co,
) continuous samples and It is always assumed that the process ( ~ ( 8has
supEIcp(6)I4 < a
eso
Theorem 1.1 [91]. Let Eq. (1.1) and the initial function satisfy the conditions formulated above. Then there exists a unique solution of problem (l.l), (1.2) with bounded fourth moment on any finite time interval. This solution is measurable relative to the processes cp(6), 8 I 0 and <(s), 0 I s I t, i.e., B  m,(x)c B,,(cp) u Bot(d(). The process x, is Markooian.
1.3. It6’s Formula It6’s formula will be useful in the investigation of the stability of system (1.1). Given a scalar continuous function u(t, x), t 2 0, x E R , with continuous partial derivatives ut(t, x), u,(t, x ) = (du/dx,), u,,(t, x ) = (d2u/dxidxj), i, j = 1,. .. , n. Then the stochastic differential dr](t) of the process v](t)= u[t, x ( t ) ] [where x ( t ) is a solution of Eq. (1.1)] equals
+ u:CG x(t)Ib(t, X r ) Here Tr bb‘ is a trace of the matrix bb‘.
1.4. Equations for Moments Consider the linear SRFDE dx(t) =
[d,A(t, s)]x(t  S) dt lom
+
[d,B(t, s ) ] x ( ~ S) d ( ( ~ ) (1.7) jom
with initial condition (1.2). Here A and B are ( n x n ) matrix functions. From It6’s formula we obtain the deterministic equations for the moments of solution to SRFDE (1.7). For m(t) = M x ( t ) we get [d,A(t, s)]rn(t  s), m(0) = Ecp(0)
62. Formulation of Stability Problems for SRFDEs
167
and for k(t, s) = Ex(t)x’(s), we get dk(t, t> ___ = dt
Iom {k(t, t  S ) d,A’(t, S )
Let Eq. (1.7) be of the form d x ( t ) = a(t)x(t  h) dt Then J
= E x ’ ( T ) H x ( T )(where
+ b(t) dt;(t),
t 2 0, h 2 0
T > 0 and a matrix H are given) is equal to
Here P
= B;(O)HB,(O),
Q
= B;(O)HB,(s)
Ns, P ) = B;(S)HB;(P) c1
=
loTTrB;(t)HB,(t)b(t)b’(t)dt
The matrix B , is easily calculated by the step method from the equation B,(t) = B,(t
+ h ) ~ ( +t h),
B,(T) = I, B,(s) = 0, s > T
92. FORMULATION OF STABILITY PROBLEMS FOR SRFDEs. LIAPUNOV DIRECT METHOD
2.1. Basic Definitions of Stability In this paragraph various definitions and theorems relating to stochastic stability are formulated for vector SRFDEs (1.1). It is always assumed that Eq. (1.1) satisfy requirements from $1 and also that 4 t , 0) = 0,
b(t, 0 ) = 0
(2.1)
168
4. Stability of Stochastic Functional Differential Equations
From condition (2.1) it follows that Eq. (1.1) has a trivial solution x ( t ) = 0 defined by a zero initial condition. The stability of the trivial solution of system (1.1) is investigated relative to the disturbances of the initial function cp(0). Denote x(t, cp) the solution of problem (l.l), (1.2).
Definition 2.1. The trivial solution of SRFDE (1.1) is called: ( a )p stable i f for each E > 0 there exists S(E) > 0 such that for any initial process cp(0) with continuous trajectories and independent from Bo,(d(), the inequalities
I
SUPE cp(e)ip 850
<
w,
sup^ I q(e) I 2 p < co
(2.2)
810
imply that Elx(t, cp)Jp < E, t 2 0 ; ( b ) asymptotically p stable i f it is p stable and for any arbitrary initial function q ( 0 ) E Q lim E ) x ( t ,q)ip= 0
(2.3)
l+ m
The domain Q is called an attraction domain of the trivial solution.
If p
=2
then the definition 2.1 is a definition of meansquare stability.
Definition 2.2. The trivial solution of SRFDE (1.1) is called: ( a ) stable with respect to probability $ f o r each E , > 0 and E~ > 0 there exists a 6 > 0 such that the solution x(t, cp) of problem (l.l), (1.2) satisfies the inequality
Lo
P supIx(t, q)l I E ,
I
2 1  E2
provided that SuP8coIq(0)l I 6 with probability 1; ( b ) asymptotically stable with respect to probability if it is stable with respect to probability and for any positive number L1 and A2 and arbitrary initial function lp(8)ESH with probability 1, there exists T(H,A,, A 2 ) > 0 such that
Definition 2.3. The trivial solution of SRFDE (1.1) is called exponentially p stable ( p > 0) i f for any positive constants c , and c2 Elx(t, cp)lp Ic,supEIq(0)lP exp(c2t),
t20
850
2.2. Asymptotic p Stability General theorems of the Liapunov direct method for SRFDEs may be obtained by formal generalization of the appropriate theorems for
169
$2. Formulation of Stability Problems for SRFDEs
deterministic equations from Chapter 2, 92. Let us formulate, therefore, the only theorem used in this section.
Theorem 2.1. Let Eq. ( 1 . 1 ) satisfy requirernentsfrorntjl and (2.1). Assume that on [0, 00) x CB[  00,01 there exists the continuous functional V ( t , q) such that V ( t ,q) 2 c*lq(0)lP,
t 2 0,p 2 2
(2.4)
Taking into account (2.4), (2.5) we get
Hence, p stability is proved. Justify (2.3). By (2.7) and It6's formula we conclude that Elx(t, q ) l P as a function of t satisfies the Lipschitz condition and
Therefore relation (2.3) holds true.
REMARK2.1. Theorem 2.1 is valid if, instead of inequality (2.6), the following inequality holds: d W , x,) I c,Ix(t)lP dt
+ U l ( t )dt;(t),
t >0
Here the continuous process a , ( t ) is measurable relative to B  ,o(q) u B0,(d5) and has a bounded second moment on any finite time interval.
170
4. Stability of Stochastic Functional Differential Equations
2.3. Exponential p Stability
Formulate the necessary and sufficient conditions [230(2)] of exponential p stability for a stationary SRFDE with finite delay
+ b(x,) d((t), xo(e)= Cp(e),
t 20
dx(t) = a(x,) dt x(t
+ 0) = xr,
h I
eI o
(2.8)
Theorem 2.2. The necessary and sufJicient condition of exponential p stability of the trivial solution of Eq. (2.8) is the existence of functional V ( q ) with the properties C,IICpIIP
I
UCp) I
 1
lim
r+
+o
;{EV[xr(~p)l

czll’PIIP
~ ( c PI )}C~II(PII~
REMARK2.2. Theorems 2.1 and 2.2 are valid when the stochastic disturbance t ( t ) is a general process with independent increments
2.4. SRFDEs with Random Delays [99, 1291 Consider the system
Here A and B are constant matrices and q(t) E [0, h] is a homogeneous scalar Markovian process with a finite number of states ~ ( 0=) qo. Here q0 is any point from the interval [0, h].
Theorem 2.3. Let the continuous functional V ( 9 ,q ) exist such that, unijormly in ‘l, Wl(llCpII)
lim
r++o
1 
t
5
VCp,‘l)5 ~2(1lCpII),
[EV[x,(cp), d t ) l

‘lE[O,
VCp,‘lo11 I ~AII(PII>,
hl
d o ) = ‘lo
Then the trivial solution of Eq. (2.9) is stable with respect to probability.
Theorem 2.3 permits us to relate the stability of SRFDE (2.9) to the stability of the corresponding deterministic equations. Let, for example, the deterministic system (2.9) with r](t)= r] be exponentially stable uniformly in for arbitrary r] E [0, h] and also the derivative dEl r](t) ‘loI/dt
(2.10)
171
$3. Stability of Scalar Stochastic Equations
be sufficiently small. Then the trivial solution of (2.9) is stable with respect to probability. Notice that in general this statement is not valid if the derivative (2.10) is not small (see Example 2.1.4).
$3. STABILITY OF SCALAR STOCHASTIC EQUATIONS 3.1. Preliminaries Derive some sufficient conditions for meansquare stability of solutions of the scalar SRFDE r rm 1 dx(t) = J x(t  s) dKo(s) a(t, x,) dt
1
+
1
0
+ b(t, x,) dat),
x(e)= cp(e),
t >0
co <
e I0,
(3.1) x(t)ER1
Here the kernel Ko(s) has a bounded variation on [O, co). Continuous functionals a(t, cp) and b(t, cp) are determined on the space [O, m) x CB[  00, 01 and satisfy conditions (1.4), (2.1). The initial process cp(0) satisfies the assumptions from $1.
3.2. Case of Autonomous Linear Part Full
proofs
of
the
theorems
in
this
section
are
available
[105(5), 105(11), lOS(5)l. Here we limit our presentation only by the form of
the Liapunov functionals and stochastic differentials used in the proofs. Denote
J
s dR,(s) 0
+J
s d ~ , ( s )<
a3
0
Theorem 3.1. Let Eq. (3.1) satisfy the requirements from $1 and kernel Ko(s) have a jump at zero of value a , > 0 and
Then the trivial solution
Of
Eq. (3.1) is asymptotically meansquare stable.
172
4. Stability of Stochastic Functional Differential Equations
Introduce the functional V ( t , x,)
= x2(t)
+
d K 0 ( s )+ r;' d R , ( s ) lom[
(3.2)
The stochastic differential of functional (3.2), calculated with the aid of ItG's formula, is equal to
+
[
x 2 ( t  ~)ldKO(s)l 2 4 t )  [+mo
+ a(t, X J ] }
dt
~ ( t S) dKO(s) lom
+ ~ t ) b ( txr) , di~t)
+ jomx2(t  s)[r;'
dRl(s)
+ dR,(s)]
By virtue of (1.4) and (2.1) 2x(t)
jam
~ ( t S ) dKo(s)
=  2 ~ o x ( t ) 2x(t)
l+mo s +I ~ ( t S) dKO(s)
m
m
+O
+O
+ x 2 ( t ) IdKo(s)I 2x(t)a(t, x,) Ir l x 2 ( t ) + r; la2(t, x t ) x 2 ( t  s ) dRl(s) Ir , x 2 ( t ) + r;' I22a0x(t)
x2(t  s)IdKo(s)I
jom
Hence dV(t, x,) I2x2(t)B1 dt
+ 2x(t)b(t, x,) d t ( t )
(3.3)
The validity of Theorem 3.1 now follows from (3.2), (3.3), Theorem 2.1 and the obvious estimations V ( t ,x,) 2 x 2 ( t )
+
E V ( ~X,) , I s u p ~ x ~ ( te) eso
s dR,(s)
+ r;'
I o m s dR1(s)]
$3. Stability of Scalar Stochastic Equations
173
For the scalar equation dx(t) = ax(t) dt
+ bx(t  h) d{(t),
t 2 0, h 2 0
Theorem 3.1 gives that the trivial solution is asymptotically meansquare stable if 2a > b2.
Theorem 3.2. Let conditions (1.4), (1.5) are (2.1) be met, let the kernel Ko(s) be nondecreasing and let
Then the trivial solution of Eq. (3.1) is asymptotically meansquare stable.
Consider the functional V ( 4 X,)
where
=
Vo(4 x,)
”
vo = [X(t)  jorndKO(S)
ts
+ [omdR2(s)[sx’(tl)
x(t1) dt,]i
+ (a00 + rl)
dtl
(3.4)
dKo(s)
. 6 ’ , d t 1 11x2(t2)dt2 + (1 + a1o)r;l [omdRl(s) [;i(r,) dt, For the stochastic differential of functional (3.4) the following relation is valid: dV(t, x,) S  f12X2(t)dt
+2
[
x(t)
 jorndKO(S) l  s x ( t l )dtl]b(t, x,) a t )
(3.5)
Theorem 3.2 follows from (3.4), (3.5) and Theorem 2.1.
Theorem 3.3. Let conditions (1.4), (lS), (2.1) befulJilled and let
Then the trivial solution of Eq. (3.1) i s asymptotically meansquare stable.
174
4. Stability of Stochastic Functional Differential Equations
The proof of this theorem is based on consideration of the functional V ( t ,XI)
=
"((t)

jorndK0(s)jt;sx(tl) dt1l2
+ (Boo + r1)
+
j'
jorndK0W
1s
jOrnC(l+ @1o)r;l dR,(s)
dtl 61x2 d t 2
+ dR2(S)PI
jt;sx2(tl) dtl (3.6)
For dx(t)
= [ u x ( ~) bx(t

h)] dt
+ C X ( ~d t)( t ) ,
t20
(3.7)
from Theorem 3.2 it follows that the trivial solution of Eq. (3.7) is asymptotically meansquare stable if
0 < bh < 1,
2b(l

bh) > c2
+ 2(1 + bh)lal
Note that the trivial solution of Eq. (3.7) with b = 0, a > 0 is meansquare unstable. Consequently it is possible to stabilize a system by introducing time lag.
3.3. Case of Nonautonomous Linear Part Some modifications of functionals (3.2), (3.4), (3.6) permit us to investigate Eq. (3.1) where kernel Ko(s) is exchanged on Ko(t, s). Consider, for example, the equation dx(t)
=
[
~ ( txt) ,
+
s:
~ ( t s)f(t,
+ b(t, X') d t ( t ) ,
S)
1
ds  @l(t)x(t) dt
t20
(3.8)
The functionals a(t, xr)and b(t, x,) in (3.8) are the same as in Eq. (3.1), the the function f ( t , s) is function crl(t) is continuous and bounded for t E [0, a), continuous for all t, s and
pxt,
s)l ds 5 462)  d s d ,
s2
2
s1
where q(s) is a bounded nondecreasing function. The meansquare asymptotic stability of Eq. (3.8) takes place under the conditions 2a1(t)  r:  2r1 
If(t Jom
+ s, s)l ds 
jorn
I f ( t , s)l
1
ds > 0
(3.9)
w. Stability of SecondOrder Equations
175
These stability conditions are proved with the aid of the functional V ( t ,X t ) = X y t )
+
+
s,”
ds p ( t 1
+ s, s)lx2(t1)d t ,
jom
Cr;’ d R , ( s ) + dR,(s)l
lt;sx2(tJ
dt,
For the equation d x ( t ) = a(t)x(t)
t > 0,
+ bx(t  h) @ ( t )
h 2 0,
b a const
where the function a(t) is continuous and bounded, the stability condition is 2a(t) > b2, t 2 0.
54. STABILITY OF SECONDORDER EQUATIONS
4.1. General Assumptions Obtain some conditions for the meansquare stability of the secondorder SRFDE i ( t ) = y(t),
dy(t) =
[

t >0
j o m y ( t  s) dKo(s) 
+ b(t, y,) d&t),
y,
=~
joa
+
1
x ( t  s) d K , ( s )
( t 0),
GO
dt
< 8 I0
(4.1)
The coefficients of Eq. (4.1) will always be assumed to satisfy the conditions from $1. The kernels K,, K O have bounded variations on [0, 00) and the integrals in (4.1) are RiemannStieltjes ones [l05(l l)]. Let, for the arbitrary functions cp,, cp2 E CS[  co,01,
and let
b(t, 0)
=0
(4.3)
where K z ( s ) is a nondecreasing function of bounded variation on [0, 00). Under these assumptions there exists a unique solution [ x ( t ) , y(t)] of system (4.1) satisfying the initial conditions X(O)
= cp(e),
Y(e) =
wo, o I 0
(4.4)
176
4. Stability of Stochastic Functional Differential Equations
Here q ( B ) is a given process with continuously differentiable realization such that sw[v4(e) + ~ ~ ( e ) i (4.5) eso
It is assumed that o algebra B_,,(cp, Q) is independent from the o algebra Bo,(d(). In this case the inclusion B  ,*(x, y ) c [ B  mo((p, 4) u B0,(d5)] is valid. From (4.1)(4.5)it follows that for all t 2 0 the solution of problem (4.1)satisfies the inequality
+
E [ x ~ ( ~ y4(t)] ) I c1supE[v4(8) eso
+ Q4(e)]ec2',
ci 2 0
(4.6)
4.2. Stability Conditions The conditions formulated in this section are expressed in terms of moments of the kernels Kj(s).Denote
r(t) = ECX2(t) + Y2(t)l>
Y(v)= supECv2(@)+ 42(e>l eso
Theorem 4.1. Let the coefficients of system (4.1)satisfy requirements (4.1), (4.2), and let the kernel Ko(s) have a jump at zero of value a > 0, with Do1
> 0, a >
j+mo
IdKOWl
+ E l l + a029
a10
+ a21 + a12 = 00
Then the trioial solution of system (4.1)is asymptotically meansquare stable.
Proof: Introduce the functional w t ,
YJ
= 2B0lX2(t)
+ Y2(t)+ J+rOnIdKo(s)I
[scY2(h)
l;'(tl)
+ BOlX2(tl)ldtl + 2 jorndK2(s)
where
dt,
177
$4.Stability of SecondOrder Equations
On the basis of (4.1) and ItB's formula we have dy2(t) = b2(t,YJ dt
+ 2y(t)b(t, Y,)dt(t)
But by (4.2), (4.3) with probability 1 (4.10) Moreover
(4.11)
[
dy2(t) I lornxz(t s) d K 2 ( s )
+ j+wo y2(t  s)IdKo(s)I
j
m
2x(t)y(t)B01
+ y2(t)(2a + Ull +
IdKo(s)I)
(4.13)
+O
+ [oaldK'(s)l
[Jd,,]mdt + 2Y(t)b(t, Y J dt(t)
It should be noted that inequality (4.13) means that for arbitrary t 2 2 t , 2 0 the integral in the left side of (4.13) from t , to t , is, with probability 1, less than or equal to the integral in the right side of (4.13) between the same limits. From the second equation of Eqs. (4.1) and It8's formula we obtain dK%x,,Y,)
= 2V,(X,,
y,)[ x(t)Bo1 dt
+ b2(t,Y,)dt
+
Y,) d W 1 (4.14)
178
4. Stability of Stochastic Functional Differential Equations
Estimating as in (4.11) the terms on the right side of (4.14) we obtain according to (4.8)
+ j O m x 2 (t
S)
(4.15)
dK,(s) d t
From (4.Q (4.13) and (4.15) follows the formula
(4.16)
Let us integrate inequality (4.16) from zero to t 2 0 and take the expectation on both sides of the result. By (4.9, (4.6), (4.10) and the hypotheses of Theorem 4.1, the expectation of the stochastic integral in the right side of (4.16) is equal to zero. Hence for all t 2 0 W x t , Y,)  W c p , 4) IHere Yl = 2[n 
sd
sm
CYlEY2(s)+ YzEX2(s)lds
IdKo(s)l  all
 a02
+O
Y2
[
= 280, a 
r
o
IdKo(s)l  a l l
I
1
(4.17)
(4.18)
But by the hypotheses of Theorem 4.1 together with (4.8)
w c p ,4)
C,Y(cp)
w x , , Y,) 5 b o , ~ x Z ( t+) EY2(t)
(4.19) (4.20)
From these relations and (4.17) it follows that r(t) I c2y(cp) for arbitrary t 2 0. So the meansquare stability is established. For the proof of asymptotic
v. Stability of SecondOrder Equations
179
meansquare stability it suffices to show that the function r(t) is uniformly continuous on [0, co) and jomr(t)dt < cc
(4.21)
But inequality (4.21) follows immediately from (4.17)(4.20), since they imply that for all t 2 0
j ( ) Y l t Y 2 0 ) + Y z ~ X 2 ( s )ds 1
C,Y(Cp)
Let us prove now that the function r(t) is uniformly continuous on [0, a). For this we apply ItB’s formula to the function x 2 ( t )+ y2(t). From (4.10), (4.11) we obtain that for arbitrary nonnegative t , and t , M t l )  &)I
2C,Y(Cp)(l
+ a00 + a01 + “02)ltl  t z l
Theorem 4.1 is proved For the equation ( t 2 0)
W ) = YW,
dy(t) = [  a l v ( t )
+ a,x(t
 h d l dt
+ a&

h2) d t ( t )
Theorem 4.1 gives that the trivial solution is asymptotically meansquare stable if a, > 0 and a , > a,h, + a:.
Theorem 4.2. Let the conditions (4.2), (4.3) be satisfied and let 41 =
4 = Boo
a10
 P11
+ +a21 < 1,
BOl
>0
> maxC41, (Bo141 + ao,)/(l  4111 f aZl
f a20
+ @3l
<*
Then the trivial solution of system (4.1) is asymptotically meansquare stable.
The proof of this theorem is based on the functional V X t ,
YI) =
G<X‘,
YJ
+ G ( Y J + 2X2(t)B0l
+ 2 jorndK2(s)l;’(t,)
dt,
I80
4. Stability of Stochastic Functional Differential Equations
where V, is given by (4.8) and
V,(Y,)
= YO)

Ly(t1) dtl
joWdK0(s)
+ JomdK,(s)
s’
1S
dtl jt:y(t2)dt2
By It6’s formula we obtain
d W , , Y1)
 2X2(t)BOl(4  41) dt
+w
, Yt)[Vl(YJ 2Y2(t)[Bo1q1
+ VO(Xt?Y J l a t ) + q(1  41)  a021 dt
(4.23)
The validity of Theorem 4.2 follows from relations (4.22), (4.23) (for details see Kolmanovskii [105(7), 105(1l), and Kolmanovskii and Nosov 108(5)].) REMARK 4.1. The results of Chapter 2, $6 and Chapter 3,$4 show that it is not difficult to alter functionals (4.7), (4.22) in such a way that one can obtain the conditions of asymptotic meansquare stability of the nonautonomous Eq. (4.1).
REMARK4.2. Applying to functionals (4.7), (4.22) a slight modification of the arguments in Has’minskii [84] one finds, under the hypotheses of Theorems 4.1 and 4.2, not only meansquare stability, but also stability with respect to probability in the sense of Definition 2.2. 4.3. Stability Domains of Linear Autonomous Systems Obtained by the Liapunov Direct Method
Some sufficient stability conditions for second order SRFDEs are derived by use of quadratic Liapunov functionals by Shaichet [199( 1), 199(2)]. Consider the equation
dx(t) = [ay(t)

dy(t) = [  a x ( t )
+ c,x(t) d < , ( t )  by(t  h)] dt + czy(t) d < 2 ( t ) ,
bx(t  h)] dt
t >0
(4.24)
Sufficient asymptotic meansquare stability conditions for Eq. (4.24) are la/ < [h2(1
 sb’)’
 b2]1’2,
s = c2/2,
c = rnax(Ic,I, Ic21).
Stability domains of (4.24) for different values of parameter sh are represented in Fig. 4.1. The case in which h = 0 is represented in Fig. 4.2. For the equations
+ c,x(t) d
d x ( t ) = [ay(t)  bx(t  h)] dt
t >0
(4.25)
181
g4. Stability of SecondOrder Equations
4
*h
sh = 0.245
sh=0.075 sh=O
ah
Fig. 4.1. Stability domains for system (4.24).
b 0
U
Fig.4.2. Degenerated case of h = 0.
asymptotic meansquare stability takes place if ( a (< [b(l  bh)  s] .(I + bh)’. Corresponding stability domains are shown in Figs. 4.3 and 4.4. Note that for a = 0 the boundary points of stability domains for Eqs. (4.24) and (4.25) are b l , z = (2h)’[1 +_ (1  4 ~ h ) ” ~ If ] . sh 2 0.25 then the stability domains degenerate. In Figs. 4.1 and 4.3 stability domains are situated inside corresponding curves. Consider the equation X(t)
+ [a2 + ~ & t ) ] ~ ( +t ) 2bi(t  h) + b2x(t  2h) = 0
(4.26)
Sufficientasymptotic meansquare stability conditions of Eq. (4.26) are c2/2 = s < a2b[1  h(u2
+ b2)”2],
b >0
(4.27)
182
4. Stability of Stochastic Functional Differential Equations
.f
bh
Fig.4.3. Stability domains for system (4.25).
I
10
Fig.4.4. Degenerated case of h = 0.
* a
$4.Stability of SecondOrder Equations
183
1
Fig. 4.5. Stability domains for system (4.26).
Fig.4.6. Degenerated case of h = 0,s > 0.
oh
184
4. Stability of Stochastic Functional Differential Equations
Fig. 4.7. Degenerated case of s = 0, h > 0.
Inequality (4.27) is impossible if s1 = sh3 > 3J3/128 z 0.040595. The stability domain is shown in Fig. 4.5. Degenerated cases in which h = 0, s = 0 or s = h = 0 are shown in Fig. 4.64.8. For the equations % ( t )  [aZ
+ c&t)]x(t) + 2b%(t h) + b2X(t  2h) = 0
(4.28)
the stability domains represented in Fig. 4.9 are given by the inequality b > 0. This inequality is impossible c2/2 = s < a2b[1  h(la1 + lbl)] for sh3 > (219,/%  1871)/128 z 0.001069. Degenerated cases are shown in Fig. 4.104.12.
Fig.4.8. Case of h
= s = 0.
185
$4. Stability of SecondOrder Equations
t’
,sl = 00001 ,s,= 00005 ,sl = 00008
Fig. 4.9. Stability domains for Eq. (4.28)
\ I
I
2
1
\
\ \
/ )
/
/
/
/
I
I
1
2
Fig. 4.10. Degenerated case ( h = 0, s > 0).
L
0
98 I
187
$5. Stationary and Periodic Solutions
g5. STATIONARY AND PERIODIC SOLUTIONS
OF STOCHASTIC RETARDED EQUATIONS This section is devoted to the stationary and periodic solutions of SRFDEs. Some results about existence of stationary solutions are derived on the basis of the Liapunov direct method. 5.1. Existence of Stationary Solutions Let [ ( t )E R, be a continuous stationary (in the strict sense) process, i.e., for every sequence (tl,. .. ,t N )the joint probability distribution of the variables [(tl + T ) ,. . . , [(tN T ) is independent of 7: Consider initialvalue problem
+
i ( t ) = ~ [ x ,[(t)], ,
x(e)= cp(e),
t > 0,
h I eI 0,
X E
(5.1)
R,
cp(e)E
qh,
01
(5.2)
Here continuous functional a(cp, z ) E R, is such that la(cp,,z>
 a(cp2, 41 5 rllcp1  V2IL
cp1,
cPzEC[hh,Ol
Definition 5.1. A stationary process x ( t ) is a stationary solution of Eq. (5.1) if x ( t ) is stationary connected with i ( t ) and satisjes Eq. (5.1). Theorem 5.1. Let there exist such a solution X ( t ) of ( 5 4 , (5.2) such that (1) uniformly in t > 0
lim; s:p(i(s) cmt
( 2 ) for any positive
E ~ E, ~ C ,
> C } ds = 0
and t l , t2 E [ h,
00)
Elx(t1)  x(t2)IE1I C l t ,  t211+E2
(5.3)
Then there exists a stationary solution of Eq. (5.1).
For the proof of Theorem 5.1 see Kolmanovskii [105(1) and Kolmanovskii and Nosov 108(5)]. 5.2. Relation between Stability and Stationarity of Solutions Let a(cp, 5)
= al(cp)
+ [(t), a l ( 0 ) = 0 in Eq. (5.1), k(t) = a,(x,)
+ [(t),
i.e.,
t >0
(5.4)
188
4. Stability of Stochastic Functional Differential Equations
The existence of a stationary solution of Eq. (5.4) depends on the stability “degree” of the deterministic system
Theorem 5.2. Assume that the trivial solution of Eq. (5.5) is unqormly exponentially stable and also that El{(t)(’ < co. Then there exists a stationary solution of Eq. (5.4). Proof. Consider the solution x ( t ) of (5.4) with zero initial condition (5.2). Show that Elx(t)12 I c < co.From the conditions of Theorem 5.2 follow the existence of the continuous functional V ( q ) (Chapter 2, $5) such that
Let the function z ( t ) be determined by
Then for any s 2 0
Hence, applying Lemma 1.4.2 we get
From (5.7) and (5.6) it follows that rs
189
55. Stationary and Periodic Solutions
so
From Chebyshev’s inequality [64] we obtain P { Ix(s)~
> C} ds I c7c’
+
0
for
(5.9)
c +co
By virtue of (5.4), (5.8) Elx(t
+ A)  ~(t)1’ +
I 2A L + A E [ ~ : ( ~ ,[’(s)] ) ds
jt
t+A
5 26
E[r211xsl12
+ (’(s)]
ds 5 c,A2
(5.10)
Theorem 5.2 now follows from Theorem 5.1 and relations (5.9), (5.10).
5.3. Periodic Solutions A stochastic process B(t) E R, is said to be periodic with period T if for every sequence (tl,. . .,t N ) the joint probability distribution of the stochastic variables B(tl + mT),. . . ,B(tN + mT) (where m = f 1k 2 , . . .) is independent of m. Consider the equation i ( t ) = b[t, ~
( t )B(t)], ,
t 2 0, X E R , , P E R ,
(5.11)
Here the continuous functional b(t, cp, z ) for any fixed cp and z is periodic in t with period T and also
INt, cp, 4  b(4 q,z)l I cllv  $11,
v 40,
$EC[hh, 01
Definition 5.2. A periodic process x(t) is called a periodic solution of Eq. (5.11) i f x ( t ) is periodically connected with B(t) and satisjies to Eq. (5.11).
190
4. Stability of Stochastic Functional Differential Equations
Theorem 5.3. Let conditions (5.3) and (5.12) hold and there exist at least one solution x ( t ) of Eq. (5.1 1 ) such that uniformly in k k
= 0,
=
1 , 2,. . .
Then there exists a periodic solution of Eq. (5.1 1).
The proof can be found in Kolmanovskii [l05(1)] and Kolmanovskii and Nosov [108(5)].
5.4. Ergodic Properties of Stationary Solutions We present here some ergodic properties of stationary solutions of the SRFDE. dx(t) = al(x,) dt
+ b[x(t)] d((t), x(t)ER,,
t20 xt = x(t
+ O),
h I 8I 0
(5. 3)
Ergodic properties of Eq. (5.1 3 ) solutions are investigated by means of stationary solutions of a stochastic equation without delay dY(t) = azCY(t)l dt
+ bCY(t)l d m ) ,
t20
(5. 4 )
Here ( ( t ) is a standard Wiener process, the functional al(cp), cp E C [  h, 01 and functions a,(x), b ( x ) are continuous and satisfy the Lipschitz conditions. Any stationary solution x,(cp) of Eq. (5.13) with initial condition (5.2) induces the invariant measure p on (T algebra 3, of Bore1 sets from C [  h, 01.
Definition 5.3. Stationary solution x,(cp) of Eq. (5.13) is called ergodic i f for an arbitrary 3, measurable functional A(cp), cp E C [  h, 01, lim t+m
1 ~
t
' ~ , A [ X s ( c p ) 1 ds =
1
A(cp)P(dcp)
C [  h , 01
almost everywhere with respect to invariant measure p.
Theorem 5.4. Let there exist the unique stationary ergodic solution of Eq. (5.14). I f there exists stationary solution of Eq. (5.13) then it is unique and ergodic. The proof of Theorem 5.4, which is available in Kolmanovskii [105(4)],is based on absolute continuity of measures induced by solutions of Eqs. (5.13) and (5.14). In Kolmanovskii [105(4)], and Kolmanovskii and Nosov [108(5)] the widesense mixing property of stationary solutions of SDFE (5.13) are also obtained.
191
$6. Stability with Respect to the First Approximation
56. STABILITY WITH RESPECT TO THE FIRST
APPROXIMATION 6.1. General Theorem
Conditions of stability theorems with respect to the first approximation depend on the form of the first approximation equations. Establish a theorem for the case in which the equation of the first approximation is a deterministic one
with initial conditions (1.2). Continuous functionals a(t, cp) and b(t, cp) satisfy the conditions
l ~ ( ~ , c p ~   ~ ~ , c p l ~ l I ~ l l I c p  c p , I l , a(t,O)=b(t,O)=O
ucp  cplllt
lb(G cp)  b(4 cpdl I
cp,
cp1 ECLhh,
01
(6.2)
Examine the stability of the trivial solution of the equation
qt>= 44 x,) + w,x,)5(Q
(6.3)
where [ ( t )is a stochastic process with continuous samples. The definitions of stability for Eq. (6.3) are similar to Definitions 2.12.3. Let the trivial solution of Eq. (6.1) be uniformly exponentially stable. Then there exists the functional V(t ,cp) satisfying (5.6). Define the function z(t) by conditions z ( t ) = x(t) for t 5 s and i(t) = a(t, zt) for t > s. Then for any s 2 0 v ( s , x,) =
i 6 (I/A)[V(s
+ A, Z,+A)

v(s,X,)]
A+ + O
+
KG (l/A)[V(s
+ A, x,+J  V ( S+ A,
Z,+A>]
A+ + O
5 (c4/c2)V(s?
xs>
+ c3 I b(s,xs>5(s)l
Hence according to Lemma 1.4.2 from Chapter 1 and (5.5) IIXt
II I (Cz/Cl)IIVII exP[ (C4/C&I
+ (CjL/Cl) j;exPcc4(t Applying Lemma 1.4.1 to the function y ( t )
 ~ ~ / ~ 2 l l l ~d ss l l l ~ ~ ~ ~ l =
(Ix,I(exp[(c4/c2)t] we get
192
4. Stability of Stochastic Functional Differential Equations
Further
Require that  c4t/c,
+ c,L/c,
s:
1
p 2 0 (6.4)
lC(s)lds Iexp(pt),
Then EllxtllZ I (cz/c,)ZEllip112exp(Pt). Thus the following theorem holds.
Theorem 6.1. Let condition (6.4) be fulfilled and the trivial solution of Eq. (6.1) is uniformly exponentially stable. Then the trivial solution of Eq. (6.2) is asymptotically meansquare stable. 6.2. Scalar Equations
Formulate some stability conditions for scalar equations of the form (6.3) f a
i ( t ) =  J x(t  S) dKo(s) 0
+ B[t, x(t)][(t),
t >O
The kernel ko(s) satisfies the inequality [s~idko(s)l2 C , [exp(Cso) exp(Cs)],
s 2 so 2 0
Also assume that one of the following conditions is valid: (1)
Boo
> 0, a10 < 1,
(2) the kernel ko(s) has a jump at zero of value a > 0 and
(3) ko(s) is nondecreasing and ko(s) = ko(h) for s 2 h 2 0,
Denote z(t) the solution of Eq. (6.5) for fl = 0 with initial conditions z(0) = 1, < 0. By virtue of (6.6) and one of conditions (6.7)(6.9) there exist constants y,, y z such that J z ( t ) JI y 1 exp(y,t), where 0 < y z < C , y , > 0. z(s) = 0, s
193
$7, NeutralType Stochastic Functional Differential Equations
Theorem 6.2. Let the above formulated assumptions about Eq. (6.5) be valid. Then: ( a ) the trivial solution of Eq. (6.5) is asymptotically stable with respect to probability if sup,,, El[(t)l < Y ~ ( Y ~ L )  ~
P{:
ccI(.)

E[(s)] ds + 0
I
=
1,
t +
00
(b) the trivial solution of Eq. (6.5) is asymptotically pstable for p I B1(2y,L)', ifthere exists a positive P1, B2 such that
Other stability conditions for equations disturbed by the process [ ( t ) can be found in Kolmanovskii and Nosov [108(5)], and Shaichet [199(2)].
$7. NEUTRALTYPE STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS 7.1. Definition of SNFDEs An SNFDE may be obtained from deterministic Eq. (3.3.2) by adding random perturbations to their right side. This class of equations was introduced in Kolmanivskii [108(5)]. We now present some results for the following equations d[x(t)  G(t, x,] = a(t, x,) dt
+ b(t, xt) d((t),
t20
(7.1)
Here x E R, and the continuous functional G defined on
[O,
00)
x CB[  00,0]
satisfies the conditions IG(4 CP)  G(t, $)I
1444
 $(s)l
dK,(s)
(7.2)
jom
=
l o m d K l ( s ) 1,
supIG(t, 0)l < 00 rto
where K , ( s ) is a nondecreasing function. The process ( ( t ) is a standard Wiener one. The functionals a(t, x,) and b(t, x,) are just the same as in Eq. (1.1). The initial conditions for Eq. (7.1) has the form (1.2), (1.5). The solution of problem (7.1), (1.2) is understood in the sense of corresponding integral identity.
194
4. Stability of Stochastic Functional Differential Equations
7.2. Existence Theorem
Equation (7.1) differs from Eq. (1.1) only by the term G(t, xt). But it is easy to generalize the theorems about the existence, uniqueness and increase of the solutions of problem (7.1), (1.2) and also the theorems about the existence and properties of stationary solutions known for G = 0. We state here without proof the following theorems.
Theorem 7.1. Let Eq. (7.1) and the initial function satisfy the formulated requirements. Then there exists the unique solution of problem (7.1), (1.2). This solution has a bounded fourth moment on any Jinite time interval and is measurable relative to the processes (p(8), 8 I 0 and &s), 0 I s I t. The process x , is Markovian. The proof of Theorem 7.1 is founded on the method of successive approximations [9l, 108(5)].
Theorem 7.2. Let the requirements of Theorem 7.1 be fuljlled and also the functionals a, b, G be independent o f t . Assume that there exists a solution of Eq. (7.1) with bounded fourth moments. Then there exists a stationary solution of Eq. (7.1). 7.3. Stability Conditions Assume that G(t, 0) = a(t, 0 ) = b(t, 0) = 0. Then there exists a trivial solution of Eq. (7.1). The stability of this solution can be understood in different senses, for example, in the meansquare sense, with probability 1 etc. Here we consider the meansquare stability (see Definition 2.1). Analysis of the stability SNFDE (7.1) as well as the deterministic NFDEs is reasonable to reduce to the considerations to two auxiliary problems. The first one is a construction of a nonnegative functional with a nonpositive derivative. The second one is investigation of the stability of the trivial solution of some auxiliary functional inequality. The form of this inequality depends essentially on the sense in which the stability of Eq. (7.1) is investigated. So for study of meansquare stability it is necessary to consider the inequality
The solution of inequality (7.3) for t 2 0 is determined by initial condition (1.2). The meansquare stability of inequality (7.3) means that for any E > 0 there exists a S(E) > 0 such that Elx(t)I2 < E if C , < d ( ~ )Elq(8)12 , < S(E) S O .
Theorem 7.3. Suppose that Eq. (7.1) satisjies the formulated requirements and the trivial solution of inequality (7.3) is meansquare stable. Further, let there
195
$7. NeutralType Stochastic Functional Differential Equations
exist on [0, 00) x CB[  co,01 a continuous functional V ( t ,cp) of theform V ( t ,cp)
=
W t , cp)
+ Icp(0)  G(t, CP) l2
(7.4)
such that
EV(4 cp) ICsupEIcp(W eso
(7.5)
EV(t,, xtJ  EV(t1, X t , ) 5 0
The functional W(t,cp) in (7.4) is deJined on [0, 00) x C B [  O O , ~ ] , continuous, nonnegative and satisfies estimate (7.5). Then the trivial solution of Eq. (7.1) is meansquare stable.
EXAMPLE 7.1. Consider the neutraltype stochastic equation arising in the theory of aeroelasticity (Chapter 3, $2)
=
[

1 Iomx(t  s) dK,(s)
t 2 0,
1
+ a(t, x r ) dt + b(t, x,) d&t)
X(t)ER,
Assume that K,(s) has a jump at zero of the value b , > 0, the bounded function A(s) is continuously differentiable and
jam 1&)1
A(s) ds < 1,
a,, = !om
ds < co
In addition, conditions (1.5) are fulfilled and
Then trivial solution of Eq. (7.1) is asymptotically meansquare stable. For the proof consider the functional V(t,xt) = b t ) 
lom
x(t  s)A(s) ds]
2
+
lom+ [(a,,
r;
I)
I A(s) I ds
(for details see Kolmanovskii [I 05(5)], Kolmanovskii and Nosov [108(5)]).
196
4. Stability of Stochastic Functional Differential Equations
58. STABILITY OF LINEAR AUTONOMOUS EQUATIONS
8.1. Systems of Linear Equations Derive the asymptotic meansquare stability conditions for systems C199( 1)I. N
d x ( t ) = Ax(t) dt
+ 1 Brx(t  h) d&(t),
h 2 0, t 2 0
(8.1)
r=l
where x E Rnand t r ( t )are scalar independent Wiener processes. The elements aij and bij of matrices A and B, are constant. Notice that necessary and sufficient conditions of asymptotic stability of the deterministic equation 1 = A x are [l55] i = 1, ..., n
Ai>O,
(8.2)
Here Ai are principal diagonal minors of matrix S,
S=
s, s, s, ... 1 s, s, ... 0 s, s, ... 0
0
0 0
0 ... (1)”s”
0
Denote N
O(X)
=
C X’B;B,X = C r= 1
n
dij(x)
=
dij(x)
i,j=l
N
c c
bikbSsxsxk
k,s=l r = l
Lemma 8.1. The trivial solution of Eq. (8.1) is asymptotically meansquare “;:xixj = stable ifthere exists a positivedefinite quadratic form V ( x ) = X’VX such that x’(A V V A ) x =  ~ ( x and ) also the quadratic form ~ ( x=) I j: = d,(x)(l ,  K j ) is positivedejinite.
cZj=
+
Proof. Let L be the generating operator of Eq. (8.1). Consider the functional W(t,x,) = V [ x ( t ) ]
+
N
(8.3)
197
$8. Stability of Linear Autonomous Equations
Then from ItB's formula LW(t,Xt)
(8.4)
= .(X)
Hence the validity of Lemma 8.1 follows from (8.3), (8.4) and Theorem 2.1. REMARK.
Let dij = 0 for i # j . Then a ( x ) is positivedefinite if all
Ki < 1.
Find and a(x). Let A1,,,be the cofactor of the first row and m th column of the determinant A,. Determine the numbers qij by the equation
c oksDik(A)Djs(A)
n1
n
(1)"
=
k,s= 1
qijA2(nr1) r=O
Here oks are coefficients of quadratic form o ( x ) and Dik(A)are cofactors of the determinant D(A) a,,

A ...
a1n
D(A) =
...
a,,
ann A
Then
K j = Aij/2A,, ..
c n
a(x) =
=
aksXkXsr
1
..
bikbSs(l
 I/j)
i, j = l r = 1
k,s= 1
Denote by di principal diagonal minors of the matrix
Theorem 8.1. Let conditions (8.2) be fulfilled and di > 0, i = 1 , . . . , n. Then the trivial solution of system (8.1) is asymptotically meansquare stable. The proof follows from Lemma 8.1 and relation (8.5).
EXAMPLE 8.1. Given two equations 2
aikxk(t)dt
dxi(t) = k= 1
+ bixi(t  h) d(,(t),
i = 1, 2
Sufficient conditions for asymptotic meansquare stability of these equations are
s, = a,, + a,, max[b:(S,
< 0,
+ a:,),
s, = alla2,  a12a21 > 0 b:(S,
+ a;,)]
< 2.S1s2
198
4. Stability of Stochastic Functional Differential Equations
8.2. Corollary for nthOrder Equations Consider a linear SRFDE
i=l
I
N
C fiijx("i)(t h ) t j ( t ) = 0
a,x(""(t) +
j= 1
(8.6)
This equation is equivalent to system of the form (8.1)
dxl(t) = x2(t)d t , . . . , dxn l(t)
c a i x n  i + l ( tdt)
n
n
dx,
= 
= x,(t) d t
N
1 C Pijxni+l(t h) d t j ( t )

(8.7)
i= 1 j = 1
i= 1
For system (8.7) matrices A and S are

...
0
1
Functions o ( w ) and dij(x)are dij(x)= 0 i f i + j < 2n, o ( x ) = dnn(x) n
dnn(x)
=
N
1
(8.9)
fikjPsjXn+lsXn+lk
k,s= 1 j = 1
Lemma 8.2. Trivial solution of Eq. (8.6) is asymptotically meansquare stable if there exists a positive definite quadratic form V ( x ) = x'Vx = j= Kjxixj such that x'(A'V+ V A ) x = dnn(x), v,, < 1
C:,
(0)
qnn
A=
q;'n'
1
a2
0
0
.
. . . 4nn ( n  1' ... 0 ...
an
199
$8. Stability of Linear Autonomous Equations
8.3. Necessary and Sufficient Stability Conditions of Scalar Equations
Consider the scalar equation
(8.10)
+J
~
x(t
+ s>t i ~ ( s ~)
t ) , t 2o
h
with initial condition (1.2). Here K ( s ) and R(s) are functions with bounded variations on the interval 0 < s h. The necessary and sufficient conditions
Fig.4.23. Stability domains for Eq. (8.13) for SE[O,31.
200
4. Stability of Stochastic Functional Differential Equations
for asymptotic meansquare stability of the trivial solution of the system (8.10) are [230(2)]: (1) all roots of the characteristic equation
z
I:,
(8.11)
exp(zs) d K ( s ) = 0
lie in the left half plane; (2)
5 {omI lo
exp (is 0) dR(B)
h
Remark that from condition (1) it follows that deterministic system (8.10) with R = 0 is asymptotically stable. Let, for example, Eq. (8.10) have the form dx(t) = [  u x ( ~ ) bx(t  t)] d t
+ C X ( ~ Z) d t ( t )
rb i
2015
10
t 07
Fig. 4.14. Stability domains for Eq. (8.13) for s
< 10.
(8.13)
20 1
$8. Stability of Linear Autonomous Equations
From (8.11), (8.12) we obtain that system (8.13) is stable in the case Ibl 2 la1 if c2 < 2(1 + bk' sin kz)'(a
+ b cos kj),
k
=
Jbza2
(8.14)
and in the case Ibl 5 la1 if c2 < 2(1 + bk'shkz)'(a
+ bchkz),
k
=
Jazbz
(8.15)
The stability domains of system (8.13) given by inequalities (8.14), (8.15) are represented in Figs. 4.14 and 4.13 for different values of the parameter s = c22/2. The domain of stability is to the right of the boundary.
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Index
A
Aeroautoelasticity, 120 Antenna, 67
random, 170 small, 27 unbounded, 19,21,74 Diagram, Vyshnegradskii, 55, 58, 117, 118 Dynamics, relativistic, 13
B Boundedness, 159 Buss, 124
E
C Condition Perron, 159 selfexcitation, 150 splicing, 22 Controller adaptive, 43, 96 P, 35, 37, 38, 81 PD, 37, 118, 132 PID, 37, 118 Reswick, 4 1 Smith, 42 Criterion frequency, 60 integral, 63, 70 Matrosov, 77 Michailov, 60, 1 19, 123 Nyquist, 6 1, 62, 1 15 Popov  Halanay, 107 Tzypkin, 65
Equation differential difference, 19, 22, 34, 53 for moments, 166 linear, 25 autonomous, 3 1,48, 1 13, 196 periodic, 108, 160 scalar, 83, 134, 159 secondorder, 85, 180 Estimate, Q qriori, 24, 26
F FDE, 36 Flutter, 125 Formula, It6, 166, 169, 172, 177, 180 Function almostperiodic, 129 characteristic, 31, 33, 50, 69, 114 transfer, 1, 35, 37, 69 unsteady unit step, 12 1 Functional degenerated, 132, 140, 152 Liapunov, 126, 156, 155 Krasovskii, 72, 80, 89
D Delay adjustment, 100 bounded, 19,72, 130, 148
H Hodograph, Michailov, 6 1, 62, 1 19, 120
215
Index 1
Ideal predictor, 36 Inequality Chebyshev, 189 functional, 127, 129, 152, 194 Instability, 42, 45, 142, 154
L Lag informational, 4 technological, 1 transportation, 3 Laplace transform, 49, 114
M Method D subdivision, 55, 1 17 Liapunov direct, 7 I , 103, 126, 167, 180, 187 steps, 19,48, 111, 154, 161, 167 Model cutting, 8 immune response, 15 infeed grinding, 8 linear stochastic aeroautoelastic, 123 population dynamics, 13 predatorprey, 15 virus disease, 90 Multiplier, Floquet, 108, 161
N NFDE, 10, 19, 21 linear, 27 autonomous, 33, 1 13 with bounded delay, 130, 148
0 Oscillator. 9
P Problem, initial value, 21, 23, 164 Process Markov, 166, 194
transient, 38 Wiener, 164
Q Quasipolynomial, 32, 53
R Reactor, 1, 8 I , 132 nuclear, 8, 87 Response, frequency, 61 RFDE, 19 linear, 25, 48 with bounded delay, 19, 72 with unbounded delay, 19, 2 1, 74
S
Sector, 145, 151 SFDE, 123, 164 SNFDE, 193 Solution ergodic, 190 fbounded, 128, 130 Floquet, 110, 161 periodic, 189 representation, 29, 33 stationary, 187 Space C[h,O], 19 C[m,O], I8,2 1 CB[m, 01, 19, 21, 165 L[h,OJ, 22 L,[~,O],24 SRFDE, 6, 164 Stability, 44, 113, 164 absolute, 106 domain, 117, 124, 180, 199 exponential, 79, 170, 188, 192 global, 14, 128, 132, 136, 140 in the first approximation, 75, 191 invariant set, 152 Liapunov, 44 linear autonomous NF'DE, 1 13 linear autonomous RFDE, 48 under steady acting disturbances, 75, 79, 99, 101
Index Stabilization, ship, 4 Stable, 44, 45, 73, 127, 157 asymptotically, 46, 5 1, 73, 115, 128 uniformly, 47, 73, 153 f; 127, 129, 130, 152 asymptotically, 127, 129, 130, 152 L,, 49, 50, 114 mean square, 168 asymptotically, 168, 171, 173, 176, 194, 196 P, 168 asymptotically, 168, 169, 193 with respect to probability, 168, 170, 193 System central nervous, 17, 143 distributed selfoscillatory, 150 feedback, 35 singleloop, 35, 38 twoloop, 39 manmachine, 5 with lossless transmission line, 10
217 T Theorem Barbashin Krasovskii, 76 Chebotarev, 55, 117 Chetaev, 142, 146, 155 Esclangon, 159 Krasovskii, 72 Pontriagin, 54, 117 Razumichin, 72 Zubov, 33 Time lag, 2, 3 Turbojet, 6 V
Viscoelasticity, 103
U Uniqueness, 21, 23, 166, 194