Balachandran, K.

Somasundaram, D.

English

Institute of Mathematics AS CR, v. v. i.

KybernetikaKrishnan Balachandran; D. SomasundaramControllability of nonlinear delay systems with delay depending on state variableKybernetika, Vol. 22 (1986), No. 5, 439--444Persistent URL: http://dml.cz/dmlcz/124576Terms of use:© Institute of Information Theory and Automation AS CR, 1986Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitizeddocuments strictly for personal use. Each copy of any part of this document must contain theseTerms of use.This paper has been digitized, optimized for electronic delivery and stamped withdigital signature within the project DML-CZ: The Czech Digital Mathematics Libraryhttp://project.dml.czK Y B E R N E T I K A - V O L U M E 22 (1986), N U M B E R 5 CONTROLLABILITY OF NONLINEAR DELAY SYSTEMS WITH DELAY DEPENDING ON STATE VARIABLE K. BALACHANDRAN, D. SOMASUNDARAM A theorem is proved on the existence of solution of a certain type of nonlinear delay systems with implicit derivative by Darbo's fixed point theorem. Sufficient conditions are established for global controllability of such systems. 1. INTRODUCTION Controllability of nonlinear systems with various types of delays in control variables has been studied by several authors [2, 9, 10, 11, 14] by means of Schauder's fixed point theorem. Using the same principle Mirza and Womack [12] and Dauer and Gahl [7] have discussed the controllability of nonlinear delay systems. In [8] Gahl investigated this problem for nonlinear systems of neutral type. Dacka [4] introduced a new method of analysis to study the controllability of nonlinear systems with implicit derivative with aid of Darbo's fixed point theorem. This method is extended by Dacka [5, 6] to nonlinear delay systems having implicit derivative and Bala-chandran and Somasundaram [1, 3, 13] to systems with different types of delays in control variables. In this paper we shall study the existence of solution and controlla-bility of nonlinear delay systems in which the delay depends on the state variable. This paper is similar in spirit to the authors' paper [1, 3] and hence the mathemati-cal preliminaries relating to the measure of noncompactness of a set and Darbo's fixed point theorem are omitted. For details the reader can refer to [3, 4]. 2. BASIC ASSUMPTIONS AND DEFINITIONS Consider the following nonlinear delay system with implicit derivative as represent-ed by the differential equation (1) x(i) = f(u(t), x(t - h(x(t), t)), x(t), t) + B(t) u(t), t = t0 x(t) = <j)(t) for t = f0 439 where x e R", u e Rm,f is an n-vector function, Bf) is n x m matrix and h(x{t), t) = 0. Let the initial function cj)(t) and the delay h(x(t), t) be continuous. Set a(f) = f — — h(x(f), f), a = inf a f̂) and — oo < a < t0 therefore x(f) = </>(f) on [a, f 0]. Assume that the function j(u, x, y, t) is continuous and satisfies the following condition (2) |/(u, x, y, f)| < M for ueRm , x, y e R", te [f0, f j and for each y, y e R" and u e Rm, xe R",te [f0, ft] (3) \f(u,x,y,t)-f(u,x,y,t)\^k\y- y\ where M and k are positive constants such that 0 < k < 1. Moreover assume that |B(f)| = N, where N is positive constant. Define W(t0,tl)= Гв(s)B'(s)ds J ío where the prime indicates the matrix transpose. The norm of a continuous matrix valued function is taken as in [2, 3]. Let c/)(t) be a continuous Revalued function defined on [a, f0] and let «(f) be a fixed continuous function on [f0, f .] . Definition 1. The solution of (1) is the function x(f) such that: (i) x(f) is defined and continuous on the interval [a, f.] and of class C1 on [f0, f.] such that at the point f0 the right side derivative only is taken into account, (ii) Equation (1) is satisfied by the function x(f) in the interval [f0, ft] where on the interval [a, f0] the function x{f) = </>(f). In this paper we shall use the following definition of controllability due to Mirza and Womack [12]. Definition 2. For system (1), an initial function <f> e C„[a, f0] at time f0 is said to be controllable to the origin if, for some ft > f0, there exists a control function u: [t0, fj] -» Rm such that the solution x(f) of (1) exists and satisfies x(ft) = 0. If this is true for all </>eC„[a, f0] then the system is globally controllable to the origin. We shall prove that the solution (1) exists under certain conditions. If u(f) is considered as fixed for the instant then we may consider the equation (4) x ( f )= / (x ( f -h(x ( f ) , f ) ) ,x ( f ) , f ) , t0gt<tt x(t) = <j>(t), a <;t=t0 where the initial function (j>(t) is continuous. Here the whole dependence on a is concealed in the dependence on t. Therefore it is enough to give sufficient conditions for the existence of a solution of (4). 440 3. EXISTENCE THEOREM Theorem 1. If the function f(x, y, t) satisfies the conditions (2) and (3) with h(x(t), t) — 0 then (4) has at least one solution for any initial function <j> e C,,\a, «0]. Proof. Consider the Banach space Cx„\t0, t^\ and the subset H = [x : x e Cl [t0, tx] , x(t0) = (fi(t0)] . For any function x e H, we shall mean by x(a(t)), the function defined in such a way that if a(t) < t0 for t e [t0,11] then x(a(t)) = 4>(«(t)). Define the mapping Tby T(x) (t) = ct>(t0) + P f(x(s - h(x(s), s)), x(s), s) ds. Moreover, consider the bounded closed set B in H by B = [XEH : | x | ^ r, \\x\\ = M] where M and r are positive constants such that r = \ct>(t0)\ + M(tl - t0). Since/is continuous, Tis continuous and maps B into itself. Using similar argument as in [1, 3] we obtain w(DTx) = k w(Dx, h) + /?(/?). Therefore, we have H(TE) = k n(E) for any bounded s e t J J c B c / / . Hence by Darbo's fixed point theorem, the mapping Thas a fixed point x e C^[t0, ti] such that x(t)=T(x)(t). Clearly the extension of this function to the interval [a, tg] by means of the function cj) is a solution of equation (4) of the following form x(t) = cj)(t0) + f f(x(s - h(x(s), sj), x(s), s)ds t = t0 x(t) = (j)(t), a• = t g < 0 . Remark 1. It should be observed that if we assume that the function / and h satisfy also Lipschitz condition with respect to the state variables, then the uni-queness of the solution of (4) can be established by standard techniques used in proving uniqueness theorems. 441 4. MAIN RESULT Theorem 2. Given the system (1) with conditions (2) and (3) and the matrix W(t0, t{) is nonsingular for some fx > f0. Then the system (l) is globally controllable to the origin. Proof. Let <f> e C„[a, f0] be an arbitrary initial function. Consider the Banach space e = Cm[to» ti] x c«[to>ti]- Define the following nonlinear transformation of the space Q by Tl[u,x])(f) = [T1([U,x])(f),V[u,x])(f)] where Tt and T2 are defined as follows (5) T1([u,.x])(f) = = -B'(t) Wt-'Co.'OUCo) + P/(<*) ,x(-) , i» ,s)ds L , J to and (6) ra([M,x])(f) = = </>(f0) + P B(s) Tt([u, x]) (s) ds + [' / (T.([H, X]) (S), X(-), x(s), S) ds J (0 J <o where if a(f) < f0, then we assume that x(«(f)) = <Ka(f)Y By the definition of T, the operator T is continuous and maps the space Q into Q. Let us consider the closed convex subset of Q H = {[«, x] e Q: H < K.; ||x|| < K2, j|Dxfl < K3} where the positive constants Ku K2 and K3 are as follows ---!--iVj JT-^o, MB [K*o)| + (*i - .o)M] K2 = |0((o)| + (tx - f0) iVKj + (f. - f0) M X3 = NK, + M . It is easily seen that T transforms H into H and for each pair [u, x] e H, we have w(Tt([u, xj), h) ^ w(B', h) q where q = sup IIW-'Oo.'OlUW + f l/[«(s),x(-),x(s),s)dsl. [« . ' * - • L J to J Since the function B does not depend on the choice of the points in H, all the functions T!([u, x]) (f) have a uniformly bounded modulus of continuity, therefore, they are equicontinuous. Further the function / is bounded, all the functions ^([w, x]) (f) 442 are equicontinuous and uniformly bounded, hence they form a compact subset in the space Cm[t0, tt~\. Note that all the functions T2([u, xj) (t) are equicontinuous and by similar argument we can find the modulus of continuity of DT2([u, xj) (t) for t, s e [t0, 11] as \DT2([u, x])(f) - DT2([u, x])(s)\ S k\x(t) - x(s)\ + P(\t - s\) hence w[DT2([u, x]), h) < k w(Dx, h) + 0(h). Thus we have for any set E <= H w0(T1E) = 0 and vv0(DT2E) <, k w0(DE2) where E2 is the natural projection of the set E on C^[t0, f«J. Hence, it follows that fi(TE)<kfi(E). By the Darbo fixed point theorem the mapping Thas at least one fixed point; therefore, there exist functions u e Cm[t0, f.] and x e Cl[t0, t j such that (7) u(t)=Tl([u,x])(t) (8) x(t)=T2([u,x\)(t). Extending the function (8) by the function (p(t) to a J§ t <. t0 and hence we have (9) u(t) = -B'(t) W-\t0, h) U(t0) + r j « s ) , x(-), x(s), s) dsl and (10) x(t) = 4>(t0) + f f(u(s), x(-), x(s), s) ds + f B(s)u(s)ds, te[t0, t . ] J(o J to x(t) = 4>(t) for / e [a, t0~\ . Substituting (9) into (10) we see that x(f,) = 0. Hence there exists a control u(t) which satisfies the condition of Definition 2. Since <j> was arbitrary, the system (1) is globally controllable. Q Remark 2. Using the method parallel to that of Theorem 2, we can establish the sufficient conditions for local relative controllability of the system (1). Example. Consider the system . / \ Xi(t) + x2(t — (1 — sin x,(f)Y) , . . , . x.(») = l l > 2K y r^f— + cos t«.(. + s.n t u2(t) 1 + xf(t) + x\(t - (1 - sin Xl(t))) x (t) xz(i) = -—--tr- + i s i n (*i(0 + *i(0) - sin ("i(0 + cos f "2(0 1 + x2(f) then W(0, rj) = ^7, where I is the identity matrix, is nonsingular if fj > 0. The 443 function f is continuous and bounded and satisfies the Lipschitz condition with respect to x with constant k — §. Hence, by Theorem 2, the system is globally controllable. (Received January 15, 1985.) R E F E R E N C E S  [1] K. Balachandran and D. Somasundaram: Controllability of ,a class of nonlinear systems with distributed delays in control. Kybernetika 19 (1983), 6, 475—482. [2] K. Balachandran and D. Somasundaram: Controllability of nonlinear systems consisting of a bilinear mode with time varying delays in control. Automatica 20 (1984), 2, 257—258. [3] K. Balachandran and D. Somasundaram: Relative controllability of nonlinear systems with time varying delays in control. Kybernetika 21 (1985), 1, 65—72. [4] C. Dacka: On the controllability of a class of nonlinear systems. IEEE Trans. Automat. Control AC-25 (1980), 2, 263-266. [5] C. Dacka: On the controllability of nonlinear systems with time variable delays. IEEE Trans. Automat. Control AC-26 (1981), 4, 956-959. [6] C. Dacka: Relative controllability of perturbed nonlinear systems with delay in control. IEEE Trans. Automat. Control AC-27 (1982), 1, 268-270. [7] J. P. Dauer and R. D. Gahl: Controllability of nonlinear delay systems. J. Optim. Theory Appl. 2/(1977), 1,59-70. [8] R. D. Gahl: Controllability of nonlinear systems of neutral type. J. Math. Anal. Appl. 63 (1978), 1, 33 -42 . [9] J. Klamka: Controllability of nonlinear systems with delay in control. IEEE Trans. Automat. Control AC-20 (1975), 5, 702-704. [10] J. Klamka: Relative controllability of nonlinear systems with delays in control. Automatica 12 (1976), 6, 633-634. [11] J. Klamka: Controllability of nonlinear systems with distributed delays in control. Internat. J. Control 31 (1980), 5,811-819. [12] K. B. Mirza and B. F. Womack: On the controllability of nonlinear time delay systems. IEEE Trans. Automat. Control ^C-77 (1972), 6, 812-814: [13] D. Somasundaram and K. Balachandran: Relative controllability of a class of nonlinear systems with delay in control. Indian J. Pure Appl. Math. 14 (1983), 11, 1327—1334. [14] D. Somasundaram and K. Balachandran: Controllability of nonlinear systems consisting of a bilinear mode with distributed delays in control. IEEE Trans. Automat. Control AC-29 (1984), 6, 573-575. Dr. K. Balachandran, Department of Mathematics, Bharathiar University, Coimbatore 641 046, Tamil Nadu. India. Dr. D. Somasundaram, Department of Mathematics, Madras University PG Centre, Salem 636 011, Tamil Nadu. India. 444 

Controllability of nonlinear delay systems with delay depending on state variable

http://dml.cz/bitstream/handle/10338.dmlcz/124576/Kybernetika_22-1986-5_7.pdf

Controllability of nonlinear delay systems with delay depending on state variable

Abstract

Similar works

Full text

Available Versions

Institute of Mathematics AS CR, v. v. i.