Constrained Controllability of Nonlinear Systems

Transcription

1 Ž. JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 01, ARTICLE NO. 060 Constrained Controllability of Nonlinear Systems Jerzy Klamka* Institute of Automation, Technical Uni ersity, ul. Akademicka 16, Gliwice, Poland Submitted by Firdaus E. Udwadia Received December 19, 1994 In this paper infinite-dimensional dynamical systems described by nonlinear abstract differential equations are considered. Using the generalized open mapping theorem sufficient conditions for constrained exact local controllability are formulated and proved. It is generally assumed that the values of controls are in a convex and closed cone with vertex at zero. As an illustrative example a constrained exact local controllability problem for a nonlinear delayed dynamical system is solved in detail. Some remarks and comments on the existing results for controllability of nonlinear dynamical systems are also presented Academic Press, Inc. 1. INTRODUCTION Controllability problems for finite-dimensional nonlinear dynamical systems have been investigated in many publications Žsee 5 for an extensive review of the literature.. However, there exist only a few papers on controllability problems for infinite-dimensional nonlinear dynamical systems defined in Banach or Hilbert spaces 1,, 6, 13. Moreover, it should be pointed out that only paper 6 contains results on constrained controllability. In this paper we shall consider constrained controllability problems for infinite-dimensional nonlinear dynamical systems defined in Banach spaces. More precisely, we shall formulate and prove sufficient conditions for constrained exact local controllability in a prescribed time interval for nonlinear dynamical systems which possess Frechet derivatives with respect to all arguments. It is generally assumed that the control values are in a convex and closed cone with vertex at zero, or in a cone with * address: jklamka@ia.polsl.gliwice.pl X 96 $18.00 Copyright 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

2 366 JERZY KLAMKA nonempty interior. The proof of the main result is based on the so-called generalized open mapping theorem presented in 7. Roughly speaking, under suitable assumptions constrained exact global controllability of a linear approximation implies constrained exact local controllability of the original nonlinear dynamical system. This is of course a generalization to the infinite-dimensional case of some previous results concerning finite-dimensional nonlinear dynamical systems 4, 1, where some special case of the open mapping theorem has been used 3. Controllability conditions for infinite-dimensional dynamical systems can be found for example, in the papers PRELIMINARIES In this section we shall present some important facts from the theory of nonlinear operators in Banach spaces. Let U and X be real Banach spaces and let Gu:U Xbe Ž. a mapping that is continuously Frechet differentiable near the origin u 0. Let us suppose for convenience that GŽ It is well known from the implicitfunction theory Žsee, e.g., 3 or. 7 that if the Frechet differential DG0 Ž. u transforms U onto the whole space X, then the nonlinear map Fu Ž. transforms a neighborhood of zero in the space U onto some neighborhood of zero in the space X. Now, let us consider the special case when the domain of nonlinear operator G is not the whole space U, but G is defined only on a piece of a cone near its vertex at zero. We shall denote by L the closed convex cone in the space U, and by an open set in the space U containing 0. In the following, we shall use for controllability investigations some property of the nonlinear operator G which is a consequence of a generalized open mapping theorem 7, Lemma 1. This result seems to be known, but for the sake of completeness we shall present it without proof and in a slightly less general form which is sufficient for our purpose. LEMMA.1 7, Lemma 1. Let U, X, L, and be as described abo e. Let G: X and suppose that on, G has Frechet deri ati e DG,which u is continuous in u at 0, where 0 L, and suppose that G Ž Assume that DuG carries L onto the whole space X. Then there exist neighborhoods Nx about 0 and Mu about 0, such that the equation x GŽ u. has for each x Nx at least one solution u Mu L.

3 CONSTRAINED CONTROLLABILITY 367 As a matter of notation, X* denotes the dual space of X and Ž x*, x. is the value of the functional x* X* at the point x X. If M X,we define the so-called polar cone by 0 M x* X*: Ž x*, x. 0, for x M SYSTEMS DESCRIPTION AND BASIC DEFINITIONS Let us consider a nonlinear infinite-dimensional control system described by the abstract ordinary differential equation xž t. fž xž t.,už t.., t 0,T Ž 3.1. with zero initial condition xž. 0 0, where f: X U X is a nonlinear mapping such that fž 0, X and U are real Banach spaces. It is assumed that the mapping f is continuously Frechet differentiable with respect to both arguments. Let U0 U be a closed convex cone with nonempty interior. The set of admissible controls for the system Ž 3.1. is given by U L Ž 0,T,U.. ad 0 In the following we shall use the notation V L Ž 0, T, U.. Hence Uad V. For dynamical system Ž 3.1. it is possible to define many different concepts of controllability. In the following we shall focus our attention on the so-called constrained exact controllability in time interval 0, T. In order to do that, first of all let us introduce the notion of an attainable set at time T 0 from zero initial state xž. 0 0, denoted by K Ž U. T 0 and defined as K Ž U. x X: x xž T,0,u.,uŽ t. U, a.e. in 0, T 4, Ž 3.. T 0 0 where xt,0,u Ž., t 0, is the unique solution of the equation Ž 3.1. with zero initial condition and control u. Under the assumptions stated on the nonlinear mapping such a solution always exists 1. DEFINITION 3.1. The dynamical system Ž 3.1. is said to be U0-exactly locally controllable in 0, T if the attainable set K Ž U. T 0 contains the neighborhood of the origin in the space X. For the finite-dimensional case, i.e., when X R n, we may omit the word exact in Definition 3.1 since in this case exact controllability is equivalent to approximate controllability Žsee 5, Chap. 3 for details.. DEFINITION 3.. The dynamical system Ž 3.1. is said to be U0-exactly globally controllable in 0, T if K Ž U. X. T 0

4 368 JERZY KLAMKA Similarly to the previous case, for X R n we may omit the word exact from Definition 3.. In this paper, we study constrained controllability for the dynamical system Ž 3.1. and for the associated linear dynamical system of the form Ž z t. AzŽ t. B Ž t. t 0,T Ž 3.3. Ž. with zero initial condition x 0 0, where Ž. A D f xž t,0,u.,u x Ž. B D f xž t,0,u.,u. u Here Dx f and Duf are the Frechet derivatives of the mapping f with respect to x and u, respectively. In order to compare constrained controllability results for nonlinear and associated linear dynamical system we need the following lemmas, which are proved in 1, Lemma 3.1 and Lemma 3., respectively. LEMMA 3.1 1, Lemma 3.1. Under the assumptions stated abo e weha e DuxŽ t,0,u. zž t,0,., Ž 3.4. Ž. Ž. where z t,0, is the solution of the linear dynamical system 3.3 LEMMA 3. 1, Lemma 3.. Consider the map F: L Ž 0, T, U. X gi en by FŽ u. xž t,0,u., Ž 3.5. Ž. Ž. where x t,0, u is the solution of dynamical system 3.1. Then the Frechet deri ati e of F with respect to u, DF:L u Ž 0,T,U. X, Ž 3.6. is a surjecti e linear mapping if and only if the linear dynamical system Ž 3.3. is U-exactly globally controllable in 0, T. Lemmas 3.1 and 3. allow us to connect the constrained exact controllability problems for nonlinear and associated linear dynamical systems. More precisely, we shall show in the next section that under some rather weak assumptions global constrained exact controllability of a linear associated dynamical system implies local constrained exact controllability of the original nonlinear dynamical system.

5 CONSTRAINED CONTROLLABILITY CONTROLLABILITY CONDITIONS In this section we shall formulate and prove some sufficient conditions for constrained exact local controllability of nonlinear dynamical system Ž The main result is given in the following Theorem 4.1. THEOREM 4.1. Suppose that: Ž. i fž 0, 0. 0, Ž ii. U0 U is a closed con ex cone with ertex at zero, Ž iii. The linear dynamical system Ž 3.3. is U0-exactly globally controllable in 0, T. Then the nonlinear dynamical system Ž 3.1. is U0-exactly locally controllable in 0, T. Proof. Let us define for the nonlinear dynamical system Ž 3.1. a nonlinear map G: L Ž 0, T, U. X by 0 GŽ u. xž T,0,u.. Ž 4.1. Similarly, for the associated linear dynamical system Ž 3.3. we define a linear map H: L Ž 0, T, U. X by 0 HŽ. zž T,0,.. Ž 4.. Since by assumption Ž iii. the linear dynamical system Ž 3.3. is U0-exactly globally controllable in 0, T, then by Definition 3. the linear operator H is surjective, i.e., it transforms the cone Uad on the whole space X. Furthermore, by Lemma 3. we have that Ž. Ž. Ž. Ž. DG u H. 4.3 u u 0 Since U is a closed convex cone, then L Ž 0, T, U. 0 0 is also a closed convex cone. Therefore, the nonlinear operator G satisfies all assumptions of the generalized open mapping theorem stated in Lemma.1. Hence, the operator G transforms a conical neighborhood of zero in the space of admissible controls onto some neighborhood of zero in the space X, which by Definition 3.1 is equivalent to the U0-exact local controllability of the nonlinear dynamical system Ž Hence, our theorem follows. In practical applications of Theorem 4.1 the most difficult problem is to verify the assumption Ž iii. about the constrained global controllability of the linear dynamical system Ž In order to avoid this disadvantage we may use the following corollary.

6 370 JERZY KLAMKA COROLLARY 4.1. Suppose that assumptions Ž. i and Ž ii. of Theorem 4.1 are satisfied, and moreo er U0 that has nonempty interior in the space U. Then the nonlinear dynamical system Ž 3.1. is U -exactly locally controllable in 0, T 0, if the associated linear dynamical system Ž 3.1. is U-exactly globally controllable Ž i.e., without any constraints. and Ker si A* BU 04 for e ery s R Ž. Ž 0. Ž. Proof. In the proof of Corollary 4.1 we shall use the condition for constrained exact global controllability of the linear dynamical system Ž If the cone U0 has nonempty interior in the space U and the condition Ž 4.4. is satisfied, then the U-exact global controllability in 0, T of the linear dynamical system Ž 3.3. implies its U0-exact global controllability in 0, T 8, Theorem.1. Therefore all the assumptions of Theorem 4.1 are satisfied and our corollary follows. The conditions for exact global controllability of linear dynamical systems which are needed in Theorem 4.1 and Corollary 4.1 are known to be quite a strong requirement Exact controllability for linear dynamical systems does not hold, for example, if the corresponding semigroup of solution linear operators is compact or if the operator B is compact Žsee, e.g., 9 for details.. This situation includes for example partial differential equations defined in a bounded domain or infinite-dimensional dynamical systems with finite-dimensional controls. However, it should be mentioned that exact controllability may occur in the case of dynamical systems described by partial hyperbolic equations or in retarded functional differential dynamical systems with a suitably chosen state space. This last case will be carefully considered in the next section of the paper. 5. APPLICATIONS As an application of the results presented in Section 4, we shall consider the constrained exact controllability problem for a nonlinear autonomous retarded functional differential equation of the form xž t. fž x,už t.. t 0,T, Ž 5.1. t Ž. n where x t R, x Ž s. xž t s., for s h,0, t and xt Ž1. W Ž h,0,r n. is the Sobolev space of absolutely continuous functions with square integrable first derivative on h, 0. Let us assume that the Ž1. Ž n. n n nonlinear mapping f: W h,0,r R R is continuously differentiable with respect to both variables and moreover that fž 0, 0. 0.

7 CONSTRAINED CONTROLLABILITY 371 Assume that the linear approximation associated with the nonlinear dy- Ž. namical system 5.1 is given by the equation 1, Example 3.1. zž t. D1fŽ 0,0. zt D fž 0,0., Ž 5.. Ž. where D f 0, 0 B is the n m constant matrix k N Ý D fž 0,0. z A zž t h., A are n n constant matrices, 1 t k k k k 0 0 h h h h h are constant delays. 0 1 k N Ž m. m The control space is L 0, T, R, and U0 U R is the closed convex cone with nonempty interior. The characteristic equation for the linear retarded functional differential equation Ž 5.. is given by 5, Chap. 4 det Ž s. 0 Ž 5.3. Ž. k N where s si Ý A expž sh. k 0 k k. For each root s of the characteristic equation the condition Ž 4.4. is equivalent to the relation Žsee 8, Theorem 4.3 for details. Ker s BU 04 for every s R, 5.4 Tr 0 Ž. Ž 0. Ž. where the term Tr means the matrix transposition. Hence, we are in a position to formulate the sufficient condition for constrained exact controllability of the nonlinear retarded functional differential equation Ž PROPOSITION 5.1. Suppose that Ž. i fž 0, 0. 0, Ž ii. m U0 R is a closed con ex cone with nonempty interior, Ž iii. The condition Ž 5.4. holds, Ž iv. rank B n, i.e., n m and B is a nonsingular matrix. Then the nonlinear retarded functional differential equation Ž 5.1. is U0- exactly locally controllable in 0, T for e ery T 0, in the state space Ž1. W Ž h,0,r n.. Proof. The condition Ž iv. is the necessary and sufficient condition for global exact controllability of the linear retarded dynamical system Ž 5.. in Ž1. the Sobolev space W Ž h,0,r n. and in every time interval 0, T, T 0 5, Chap. 4. Therefore all the assumptions of Corollary 4.1 are satisfied and hence our proposition follows.

8 37 JERZY KLAMKA Finally, let us consider a simple numerical example which illustrates the theoretical considerations. Let us consider a nonlinear retarded functional differential equation of the form Ž 5.1. given by the following equalities: 1 x Ž t. sin x Ž t. x Ž t 1. x Ž t 1. u Ž t. u Ž t. Ž 5.5. x Ž t. x Ž t. x Ž t. x 3 Ž t 1. x Ž t 1. 4u Ž t. u Ž t Hence the linear approximation is given by 1 z Ž t. z 1 Ž t. z 1 Ž t 1. z Ž t 1. 1 Ž t. Ž 5.6. z Ž t. z Ž t. z Ž t Ž t. Ž t.. Therefore n m, h 1, N 1, and the constant matrices A, A, 0 1 and B have the following forms: b1 0 A0, A1, B. Ž b b 3 4 Let us assume that the control functions are nonnegative, i.e., the closed convex cone The characteristic equation ½ 5 u 1 0 u 1 U u R : u 0, u 0. Ž. det Ž s. det si A A expž s. s 3s e s e s has 0 as the only real root 8, Sect. 4. Moreover, we have 0 1 Tr 0 0 Ž 0. and Ž Ž Ž. Tr. 3 u w Ž u u w Therefore Ker 0 d, where d R is an arbitrary real number. Since BU, u 0, u 0, then BU R : w 0, w 04. Therefore the condition Ž is satisfied. It is easy to verify that f Ž 0, 0. 0 and that rank B n. Hence by Proposition 5.1 the nonlinear retarded functional differential equation Ž 5.5. is U0-exactly Ž1. locally controllable in 0, T in the state space W Ž h,0,r..

9 CONSTRAINED CONTROLLABILITY CONCLUDING REMARKS In this paper sufficient conditions for constrained exact local controllability of nonlinear abstract dynamical systems have been formulated and proved using the generalized open mapping theorem. These conditions extend to the case of constrained controllability the results published in 1, 13. On the other hand, the presented theorems are generalizations to infinite-dimensional cases of the finite-dimensional constrained controllability results given in the papers 4, 1. The method presented in this paper is general and covers wide classes of nonlinear dynamical systems defined in Banach spaces. However, it is especially useful for retarded nonlinear dynamical systems, for which it is rather easy to verify conditions for constrained global exact controllability of the linear approximation. Moreover, it should be stressed that a similar method can be used to consider the so-called distributed parameter dynamical systems described by partial differential equations of parabolic and hyperbolic types with mixed boundary conditions. Finally, it should be pointed out that the generalized open mapping theorem can be used to consider the constrained exact controllability of so-called semilinear dynamical systems defined in infinite-dimensional Banach spaces. In such dynamical systems there exist linear and pure nonlinear parts of the state equation, 13. Constrained controllability conditions for such dynamical systems will be the topic of further consideration. REFERENCES 1. E. N. Chukwu and S. M. Lenhart, Controllability questions for nonlinear systems in abstract spaces, J. Optim. Theory and Appl. 68, No. 3 Ž 1991., H. O. Fattorini, Local controllability of a nonlinear wave equation, Math. Systems Theory, 9, No. 1 Ž 1975., L. M. Graves, Some mapping theorems, Duke Math. J., 17 Ž 1950., D. Idczak and S. Walczak, On the controllability of nonlinear Goursat systems, Optimization, 3, No. 1 Ž 199., J. Klamka, Controllability of Dynamical Systems, Kluwer, Dordrecht, G. Peichl and W. Schappacher, Constrained controllability in Banach spaces, SIAM J. Control and Optim. 4, No. 6 Ž 1986., S. M. Robinson, Stability theory for systems of inequalities. II. Differentiable nonlinear systems, SIAM J. Numer. Anal. 13, No. 4 Ž 1976., N. K. Son, A unified approach to constrained approximate controllability for the heat equations and the retarded equations, J. Math. Anal. Appl. 150, No. 1 Ž 1990., R. Triggiani, On the lack of exact controllability for mild solutions in Banach spaces, J. Math. Anal. Appl. 50, No. Ž 1975.,

10 374 JERZY KLAMKA 10. R. Triggiani, Controllability and observability in Banach space with bounded operators, SIAM J. Control, 13, No. Ž 1975., R. Triggiani, Extensions of rank conditions for controllability and observability to Banach spaces and unbounded operators, SIAM J. Control Optim. 14, No. Ž 1976., S. Walczak, A note on the controllability of nonlinear systems, Math. Systems Theory, 17 Ž 1984., H. X. Zhou, Controllability properties of linear and semilinear abstract control systems, SIAM J. Control Optim., No. 3 Ž 1984.,