K. BALACHANDRAN, J.P. DAUER, AND P. BALASUBRAMANIAM more details, we refer to the work of Anderson [1], Gyori [7, 8] and Gyori and Wu [9]. A concrete

Size: px

Start display at page:

Download "K. BALACHANDRAN, J.P. DAUER, AND P. BALASUBRAMANIAM more details, we refer to the work of Anderson [1], Gyori [7, 8] and Gyori and Wu [9]. A concrete"

Ashley Alaina Lawson
6 years ago
Views:

1 Journal of Mathematical Systems, Estimation, and Control Vol. 7, No. 2, 1997, pp. 1{14 c 1997 Birkhauser-Boston Asymptotic Null Controllability of Nonlinear Neutral Volterra Integrodierential Systems K. Balachandran y J.P. Dauer P. Balasubramaniam y Abstract In this paper, sucient conditions are obtained for asymptotic null controllability of nonlinear neutral Volterra integrodierential systems. The results are obtained by using the Leray{Schauder xed point theorem. Key words: controllability, Volterra integrodierential systems 1 Introduction Compartmental models are frequently used in, e.g., theoretical epidemiology, physiology, population dynamics, the analysis of ecosystems, and chemical reaction kinetics. These models are used to describe the evolution of systems which can be divided into separate compartments, marking the pathways of material ow between compartments and the possible out- ow to and inow from the environment of the system (see [1, 7{11] and the references particularly in [1, 7, 8]). For some such models the time required for the material ow between compartments cannot be neglected, i.e., is not instantaneous. A paradigm for such systems can be visualized as one in which compartments are connected by pipes through which material passes in denite time. Because of the time lags caused by the passage though the pipes, the model equations for such systems are dierential equations with deviating arguments; this is contrary to the classical case where model equations have transport time that can be considered negligible and thus can be modelled satisfactorily using ordinary dierential equations. For Received February 22, 1994; received in nal form October 14, appeared in Volume 7, Number 2, y This research was supported by CSIR, New Dehli,India. Summary 1

2 K. BALACHANDRAN, J.P. DAUER, AND P. BALASUBRAMANIAM more details, we refer to the work of Anderson [1], Gyori [7, 8] and Gyori and Wu [9]. A concrete example of a compartmental model is the radiocardiogram, where the two compartments correspond to the left and right ventricles of the heart and the pipes between these compartments represent the pulmonary and systematic circulation. Pipes coming out from and returning into the same compartment may represent shunts and the coronary circulation (see [7]). Other applications arise in tracer kinetics, in modeling the uptake of potassium by red blood cells, as well as in such environmentally oriented applications as modeling the kinetics of lead in a body (see [1]). A system for representing such models is obtained using nonlinear neutral Volterra integrodierential equations of the form (1.1) given below. The aim of this paper is to study the controllability problem for such systems and to signicantly extend the application of controllability to the above class of models (see [1, Section 21]). Consider the nonlinear neutral Volterra integrodierential systems of the form d dt x(t), C(t, s)x(s)ds, g(t) = Ax(t) G(t,s)x(s)ds B(t)u(t)f(t; x(t);u(t)); x() = x (1.1) where x 2 E n, u 2 E m, t 2 J = [; 1), C(t) and G(t) are continuous n n matrix valued functions, B(t) is a continuous n m matrix valued function, A is a constant n n matrix and f: J E n E m! E n and g: J! E n are, respectively, continuous and absolutely continuous vector valued functions. In this paper we develop conditions for the system (1.1) to be asymptotically null controllable. Let C k denote the space of bounded continuous k-vector valued functions dened on J with the usual sup norm. Denition 1.1 The system (1.1) is said to be asymptotically null controllable if for every x 2 E n there exists a control u dened on J such that x() = x and t!1 x(t) =. Balachandran and Dauer [2] introduced the concept of,-controllability by using an idea of Russell [14]. A similar concept was developed by Chukwu [4] to study controllability to ane manifolds, and Eke [5, 6] extended the concept to perturbed nonlinear systems. Recently Balachandran and Balasubramaniam [3] studied the null controllability of nonlinear systems. The work reported in [2{6] are based on the work of Kartsatos [12, 13] on generalized boundary conditions in ordinary dierential equations. 2

3 ASYMPTOTIC NULL CONTROLLABILITY 2 Preinary Results Consider the linear system d x(t), dt = Ax(t) C(t, s)x(s)ds, g(t) The solution of (2.1) can be written as (see [15]) x(t) =Z(t)[x(), g()] g(t) G(t,s)x(s)ds B(t)u(t): (2.1) Z(t, s)b(s)u(s)ds where Z(t) isannncontinuously dierentiable matrix satisfying Z d t Z(t), C(t, s)z(s)ds = AZ(t) G(t,s)Z(s)ds (2.2) dt with Z() = I, the identity matrix. Assume the following its exist where Z(t) =Z6=; t!1 t!1 _Z(t) = Z; g(t) =k(constant), W (t) =W t!1 t!1 W (t) = ZB(s)(ZB(s)) ds; (2.3) the asterisk denotes the matrix transpose. Further, (see Wu [15]) the solution of the system (1.1) is given by x(t) = Z(t)[x(), g()] g(t) Z(t, s)[b(s)u(s)f(s; x(s);u(s))]ds: Theorem 2.1 The system (2:1) is asymptotically null controllable if and only if W is nonsingular. Proof: Assume that W is nonsingular, then for each x 2 E n, x 6=, dene the control function u on J as u(t) =,(Z(t)B(t)) W Z[x(),1, g()] k Zg(s)ds : 3

4 K. BALACHANDRAN, J.P. DAUER, AND P. BALASUBRAMANIAM Clearly x() = x and t!1 x(t) =, and so system (2.1) is asymptotically null controllable. Conversely, assume that W is singular. Then there exists a vector y 6= such that y Wy =. It follows that Therefore, y ZB(s)(y ZB(s)) ds =: y ZB(s)= for s 2 J: Since the solution is asymptotically null controllable, there exists a control u() such that x(t) = t!1 t!1 Z(t)[x(), g()] g(t) Z(t, s)b(s)u(s)ds =: Letting g =,wehave Zx ZB(s)u(s)ds =: So that y Zx y ZB(s)u(s)ds =; which implies that y Zx =. So y =, which is a contradiction to the fact that y 6=. Hence W must be nonsingular. 2 Example 2.1 Consider the linear system (2.1) with G(t, s) =C(t,s)=,exp(t, s); g(t) = exp(,t) A =,1; B(t) = 1 2 exp(,t): (2.4) Therefore Z(t) = 2 exp(,t), 1 satises (2.2), so that Z()=1; t!1 Z(t) =,1: R Hence from (2.3) we have W = exp(,2s)ds = 1 8 and jw j = 1 8 is nonsingular. Therefore the system is asymptotically null controllable. To approach the asymptotic null controllability for the system (1.1), we dene an operator S: C n C m! C n C m, S(x; u) = (z; v), for which 4

5 ASYMPTOTIC NULL CONTROLLABILITY any xed point (x; u) 2 C n C m will satisfy (1.1) with x() = x and t!1 x(t) =. Now, for each (x; u) 2 C n C m,we dene z(t) = Z(t)[x(), g()] g(t) Z(t, s)[b(s)u(s)f(s; x(s);u(s))]ds v(t) = (Z(t)B(t)) W,1,Z[x(), g()], k, Zg(s)ds, Zf(s; x(s);u(s))ds Note that if (x; u) 2 C n C m is a xed point ofs,wehave x(t) = Z(t)[x(), g()] g(t) Z(t, s)[b(s)u(s)f(x(s);u(s))]ds u(t) = (Z(t)B(t)) W,1,Z[x(), g()], k, Zg(s)ds, Zf(s; x(s);u(s))ds Thus x(t) is the solution of (1.1) corresponding to the control u(t) with x() = x and t!1 x(t) =. To nd such a xed point, we introduce a parameter 2 [; 1] into the problem (1.1) as follows, Z d t x(t), C(t, s)x(s)ds, g(t) = Ax(t) dt Consider the operator where z(t) = G(t, s)x(s)ds fb(t)u(t) f(t; x(t); u(t))g: S(x; u; ) =(z; v) Z(t)[x(), g()] g(t) Z(t, s)[b(s)u(s)f(s; x(s);u(s))]ds 5 : :

6 K. BALACHANDRAN, J.P. DAUER, AND P. BALASUBRAMANIAM v(t) =, (Z(t)B(t)) W,1 [,Z[x(), g()], k Zg(s)ds, Zf(s; x(s);u(s))ds] We want to show that, in an appropriate Banach space D, there exists a function pair (x; u) 2 D with S(x; u; 1)=(x; u). For that we need the following Leray{Schauder xed point theorem [6]. Note that S(x; u; ) is completely continuous in (x; u) if, for each 2 [; 1], S(x; u; )iscontinuous in (x; u) and maps every bounded subset of D into a relatively compact set. Theorem 2.2 Let D be a Banach space. For the equation assume the following: S(x; u; ), (x; u) = (2.5) (i) S(x; u; ) is dened on D [; 1], with values in D and is completely continuous in (x; u). Moreover, if K is a bounded subset of D, S(x; u; ) is continuous in uniformly with respect to (x; u) 2 K. (ii) S(x; u; )=, for some 2 [; 1] and for every (x; u) 2 D. (iii) If there are any solutions of (2:5), then they belong to some closed ball B of D, independently of. Then there exists a continuum of solutions of (2:5), corresponding to all values of 2 [; 1], and all of these solutions lie in B. 3 Main Result We now state and prove sucient conditions for the system (1.1) to be asymptotically null controllable. For this result, we denote = kw,1 k and for any bounded interval ^J, we let C( ^J;E d )(d=nm) be the Banach space of all continuous E d -valued functions on ^J with the sup norm. Theorem 3.1 Assume the following to hold for system (1.1): (i) The fundamental matrix solution Z(t) is such that kz(t)k k 1 and kg(t)k k 2 ; for all t 2 J; where k 1 and k 2 are some positive constants. (ii) The n m continuous matrix B(t) is bounded on J, with : max kb(t)k ; t2j for some >: 6

7 ASYMPTOTIC NULL CONTROLLABILITY (iii) There exists constants H, P, R, N such that for all s 2 J we have sup t!1 sup t!1 sup t!1 (iv) The matrix W is nonsingular. _Z(t, s)g(s)ds P; Z(t, s)b(s)ds H; Z(t, s)f(s; x(s);u(s))ds R(kxk kuk)n: If we can choose the constants satisfying [(H R)k 1 R] < 1, then the system (1:1) is asymptotically null controllable. Proof: Let J 1 =[;1], d = n m and let the sup norm of C(J 1 ;E d ) be kk 1. Assume that w =(x; u) 2 C(J 1 ;E d ) and consider the function w() dened by w(t) = w(t); t 2 J 1 w(t) = w(1); t 2 [1; 1): Clearly, the set of all such w is a Banach space, which we designate by C 1 with the norm k wk 2 = kwk 1. Note that C 1 = D 1 H 1, where D 1 is dened with elements x 2 C(J 1 ;E n ) and H 1 is dened with elements u 2 C(J 1 ;E m ). So k wk 2 = kxk D1 kuk H1 ; where w =(x; u) 2 C 1 = D 1 H 1 : Consider the operator S: C 1! C 1 dened by where z(t) = v(t) =, S(x; u;)=(z; v) Z(t)[x(), g()] g(t) Z(t, s)[b(s)u(s)f(s; x(s); u(s))]ds (Z(t)B(t)) W,1 [,Z[x(), g()], k Zg(s)ds, Zf(s; x(s); u(s))ds 7 :

8 K. BALACHANDRAN, J.P. DAUER, AND P. BALASUBRAMANIAM Our aim is to show that S(x; u; 1) has a xed point. First, we shall prove that S(x; u; ) is continuous in. To see this, let 1 ; 2 2[; 1] and (x; u) 2 C 1. Then we have Because js(x; u; 1 )(t), S(x; u; 2 )(t)j j 1, 2 j jz(t)[x, g()]j jg(t)j _Z(t, s)g(s)ds Z(t, s)[b(s)u(s)f(x(s); u(s))]ds j 1, 2 j (Z(t)B(t)) W,1 [,Z[x(), g()], k, Zg(s)ds, Zf(s; x(s); u(s))ds] j 1, 2 j[k 1 jx, g()j k 2 P Hkuk R(kuk kxk)n] j 1, 2 j[k 1 fk 1 jx, g()j k P R(kuk kxk)ng]: ks(x; u; 1 ), S(x; u; 2 )k 2 = sup t2j 1 js(x; u; 1 )(t), S(x; u; 2 )(t)j it follows that the operator S(x; u;)iscontinuous in, uniformly on any bounded subset of C 1. Let w =(x; u); w n =(x n ;u n ); w; w n 2 C 1 ; n =1;2;:::; Suppose that y n =(z n ;v n )=S(x n ; u n ;); y =(z; v) =S(x; u;): that is Then we have Now, n!1 k w n, wk 2 = n!1 kw n, wk 1 =; n!1 kx n, xk = and n!1 ku n, uk =: ky n,yk= sup t2j 1 jy n (t), y(t)j = sup t2j 1 [jz n (t), z(t)j jv n (t), v(t)j]: sup t2j 1 jz n (t), z(t)j f(s; w n (s)), f(s; w(s))]kds 8 kz(s)[b(s)(u n (s), u(s))

9 ASYMPTOTIC NULL CONTROLLABILITY and the integrand tends to zero as n! 1. Thus, it follows from the Lebesgue dominated convergence theorem that n!1 kz n, zk =. Similarly, n!1 kv n, vk =: Therefore, n!1 ky n, yk =: This proves the continuity of S(w; ) with respect to w. Let K be a bounded subset of C 1 with bound b k. We now show that the family of functions y = S(w; ), w 2 K, 2 [; 1], are equicontinuous. Let t 1 ;t 2 2[; 1]. Then, jy(t 1 ), y(t 2 )j = jz(t 1 ), z(t 2 )j jv(t 1 ), v(t 2 )j: Now, and jz(t 1 ), z(t 2 )jkz(t 1 ),Z(t 2 )kjx,g()j jg(t 1 ), g(t 2 )j _Z(t 1, s)g(s)ds,2 _Z(t 2, s)g(s)ds 1 2 1, 1 2 t Z 1 t1 2 Z(t 1, s)[b(s)u(s)f(s; x(s); u(s))]ds Z(t 2, s)[b(s)u(s)f(s; x(s); u(s))]ds kz(t 1 ), Z(t 2 )kjx,g()j kg(t 1 ), g(t 2 )k, t 1 k _Z(t 1, s), _Z(t 2, s)kkg(s)kds k _Z(t 2, s)kkg(s)kds kz(t 1, s), Z(t 2, s)kkb(s)u(s)f(s; x(s); u(s))]kds kz(t 2, s)[b(s)u(s)f(s; x(s); u(s))]kds jv(t 1 ), v(t 2 )j k(z(t 1 )B(t 1 )), (Z(t 2 )B(t 2 )) k[k 1 jx, g()j k P Rk wk N]: These estimates show that the given family of functions is equicontinuous. That the family is uniformly bounded follows from the following argument kyk 2 = kzk D1 kvk H1 ; 9

10 K. BALACHANDRAN, J.P. DAUER, AND P. BALASUBRAMANIAM where kzk D1 k 1 jx, g()j k 2 P H(kuk H1 kxk D1 )N kvk H1 k 1 [k 1 jx, g()j k P R(kuk H1 kxk D1 )N]: Thus, the given family of functions is uniformly bounded and equicontinuous. Hence S(k; ) is relatively compact in C 1 for each 2 [; 1]. Now, assume that the equation S(w; ), w = (3.1) has a solution in C 1 and that w is such a solution which corresponds to xed 2 [; 1]. Then where j w(t)j = jx(t)j ju(t)j k 1 jx,g()j k 2 P Hkuk Rk wk which implies N k 1 [k 1 jx, g()j k P Rk wk N] 1 2 k wk; t 2 [; 1] 1 = (1 k 1 )[k 1 jx, g()j P N]k 2 k 1 k 2 = (H R)k 1 R k wk C1 = kwk 1 (1, 2 ),1 : (3.2) By assumption [(H R) k 1 R] < 1. Thus the solutions of (3.1) are bounded uniformly with respect to 2 [; 1]. It is clear that all the assumptions of Theorem 2.2 are satised, so that S(w; 1) has a xed point y = S(w; 1) = w 2 C 1. We now inductively repeat the process as follows. Let J m =[;m] and let C m be the Banach space of all functions w which are obtained from the functions w 2 C([;m], E d ) as follows: with the sup norm w(t) = w(t); t 2 [;m]; w(t) = w(m); t2[m; 1); k wk Cm = kwk m = sup t2jm jw(t)j: Then, there is a sequence fx m g, m =1;2;:::and a corresponding sequence fu m g such that x m is a solution of Z d t x m (t), C(t, s)x m (s)ds, g(t) = Ax m (t) dt G(t, s)x m (s)ds B(t)u m (t)f(t; x m (t);u m (t)) 1

11 with the property that ASYMPTOTIC NULL CONTROLLABILITY y m =(x m ;u m )2C m ; ky m k m =ky m k; where is identied in (3.2), x m (t) = Z(t)[x(), g()] g(t) Z(t, s)[b(s)u m (s)f(s; x m (s);u m (s))]ds u m (t) = (Z(t)B(t)) W,1 [,Z(x(), g()), k, Zg(s)ds, Zf(s; x m (s);u m (s))ds]: Since, obviously, the sequence fy m (t)g = f(x m (t), u m (t))g, m =1;2;:::,is uniformly bounded and equicontinuous on [; 1], there exists a subsequence fy 1 m(t)g, such that m!1 y1 m(t) =y(t)=(x(t);u(t)) 2 C([; 1];E d ): In the same way, there exists a subsequence fy 2 m(t)g of fy 1 m(t)g which is uniformly convergent to a function p(t), t 2 [; 2], such that p(t) =y(t); t2[; 1]: Using the diagonal process, we see that there exists a subsequence fy km (t)g of the original sequence fy m (t)g such that m!1 jy km(t), y(t)j = (3.3) uniformly on every nite subinterval of J, where y 2 C(J;E d ) and kyk. Clearly, given an arbitrary nite interval I =[;c] J, there exists some m 1 such that k m1 c for m m 1. Hence, fy m (t)g is dened for m m 1, and the it of (3.3) is well dened. It is evident that m!1 jy km(t), y(t)j = uniformly on any nite subinterval of J. Now, x c > and let m(t) = (x(t);u(t)), where x(t) = Z(t)[x(), g()] g(t) Z(t, s)[b(s)u(s)f(s; y)]ds 11

12 K. BALACHANDRAN, J.P. DAUER, AND P. BALASUBRAMANIAM u(t) = (Z(t)B(t)) W,Z(x,1, g()), k, Zg(s)ds 1 Zf(s; y(s))ds : Then, by the Lebesgue dominated convergence theorem, m!1 jy km(t), m(t)j =; t2[;c]: Since c was chosen arbitrarily, m(t) =y(t) for t 2 [; 1). Thus x(t) = Z(t)[x(), g()] g(t) Z(t, s)[b(s)u(s)f(s; x(s);u(s))]ds u(t) = (Z(t)B(t)) W,Z(x(),1, g()), k, Zg(s)ds, Zf(s; x(s);u(s))ds : Hence the proof is completed. 2 Example 3.1 Consider the system (1.1) with G, C, g, A, B are as in (2.4) and f(t; x; u) ="exp(,t)[log(1 jxj)u] (sin x)=(1 t 2 ): Now, since Z(t) = 2 exp(,t), 1, we have kz(t)k 3, kz (t)k 3, kg(t)k 1 for all t 2 J and max t kb(t)k 1. Also, we have sup t!1 sup t!1 sup t!1 _Z(t, s)g(s)ds P =1; Z(t, s)b(s)ds H = 1 2 ; Z(t, s)f(s; x(s);u(s))ds R(kxk kuk)n; where R = ", N = and W is nonsingular. It follows from Theorem 3.1 that, for suciently small ">, there is at least one solution to the problem (1.1) and the system is asymptotically null controllable. 12

13 References ASYMPTOTIC NULL CONTROLLABILITY [1] D.H. Anderson. Compartmental Modeling and Tracer Kinetics, in Lecture Notes in Biomathematics. New York: Springer-Verlag, [2] K. Balachandran and J.P. Dauer. Controllability of nonlinear systems to ane manifolds, Journal of Optimization Theory and Applications, 64 (199), 15{27. [3] K. Balachandran and P. Balasubramaniam. Null controllability of nonlinear perturbations of linear systems, Dynamic Systems and Applications, 2 (1993), 47{6. [4] E.N. Chukwu. Total controllability to ane manifolds of control systems, Journal of Optimization Theory and Applications, 42 (1984), 181{199. [5] A.N. Eke. Total controllability for linear control systems, Institute of Mathematical and Computer Science, 3 (199), 149{154. [6] A.N. Eke. Total controllability for nonlinear perturbed systems, Institute of Mathematical and Computer Science, 3 (199), 335{34. [7] I. Gyori. Delay dierential and integrodierential equations in biological compartmental models, Systems Science, (Wroclaw) Poland, 8 (1982), 167{187. [8] I. Gyori. Connections between compartmental systems with pipes and integrodierential equations, Mathematical Modelling, 7 (1986), 1215{1238. [9] I. Gyori and J. Wu. A neutral equation arising from compartmental systems with pipes, Journal of Dynamics and Dierential Equations, 3 (1991), 289{311. [1] J.A. Jacquez. Compartmental Analysis in Biology and Medicine. New York: Elsevier, [11] J.A. Jacquez. Compartmental models of biological systems: linear and nonlinear, in Applied Nonlinear Analysis, (V. Lakshmikantham, ed.). New York, 1979, pp. 185{25. [12] A.G. Kartsatos. The Leray{Schauder theorem and the existence of solutions to boundary-value problem on an innite interval, Indiana University Mathematics Journal, 23 (1974), 121{

14 K. BALACHANDRAN, J.P. DAUER, AND P. BALASUBRAMANIAM [13] A.G. Kartsatos. A boundary-value problem on an innite interval, Proceedings of the Edinburg Mathematical Society, 19 (1974{75), 245{ 252. [14] D.L. Russell. Mathematics of Finite-Dimensional Control Systems. New York: Marcel Dekker, [15] J. Wu. Globally stable periodic solution of linear neutral Volterra integrodierential equations, Journal of Mathematical Analysis and Applications, 13 (1988), 474{483. Department of Mathematics, Bharathiar University, Coimbatore , Tamil Nadu, India. Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, TN Department of Mathematics, Bharathiar University, Coimbatore , Tamil Nadu, India. Communicated by Helene Frankowska 14

ON THE ASYMPTOTIC STABILITY IN TERMS OF TWO MEASURES FOR FUNCTIONAL DIFFERENTIAL EQUATIONS. G. Makay

ON THE ASYMPTOTIC STABILITY IN TERMS OF TWO MEASURES FOR FUNCTIONAL DIFFERENTIAL EQUATIONS G. Makay Student in Mathematics, University of Szeged, Szeged, H-6726, Hungary Key words and phrases: Lyapunov