Functional Differential Equations with Causal Operators

ISSN 1749-3889 (print), 1749-3897 (online) International Journal of Nonlinear Science Vol.11(211) No.4,pp.499-55 Functional Differential Equations with Causal Operators Vasile Lupulescu Constantin Brancusi University, Republicii str., 21152 Targu Jiu, Romania (Received 21 August 21, accepted 29 March 211) Abstract: The aim of this paper is to established the existence of solutions and some properties of set solutions for a class of functional differential equation with causal operator under assumption that the equation satisfies the Carathéodory type condition. Keywords: functional differential equation; causal operator; existence results 1 Introduction The study of differential equations with causal operators has a rapid development in the last years and some results are assembled in a recent monograph [1]. The term of causal operators is adopted from engineering literature and the theory these operators has the powerful quality of unifying ordinary differential equations, integro - differential equations, differential equations with finite or infinite delay, Volterra integral equations, and neutral functional equations, to name a few. In the paper [6] is considered the class S of all infinite - dimensional nonlinear M- input u, M- output y systems (ρ, f, g, Q) given by the following controlled nonlinear functional equation { y (t) = f(p(t), ( ˆQy)(t)) + g(p(t), ( ˆQy)(t), u(t)) y [ σ,] = y C([ σ, ], R M, (1) ) where σ quantifies the memory of the system, p is a perturbation term, and ˆQ is a nonlinear causal operator. The aim of the control objective is the development of a adaptive servomechanism which ensures practical tracking, by the system output, of an arbitrary reference signal assumed to be in the class R of all locally absolutely continuous and bounded with essentially bounded derivative. More exactly, the control objective is to determine an (R, S) servomechanism, that is, to determine the continuous functions Φ : R M R M and ψ λ : R + R + (parametrized by λ > ) such that, for every system of class S and every reference signal r R, the control u(t) = k(t)φ(y(t) r(t)), k(t) = ψ λ ( y(t) r(t) ), k [ σ,] = k, (2) applied to (1) ensures convergence of controller gain,and tracking of r( ) with asymptotic accuracy quantified by λ >, in the sense that max{ y(t) r(t) λ, } as t. For more details (see [6, 1]). Using (2), we can write (1) as x (t) = F (t, (Qx)(t)), x [ σ,] = x C([ σ, ], R N ), (3) where N = M + 1, x(t) := (y(t), k(t)), x = (y, k ), and Q is an operator defined on C([ σ, ], R N ) by (Qx)(t) = (Q(y, k))(t) := (( ˆQy)(t), y(t), k(t)). The purpose of this article is to study the topological properties of the initial value problem (3). For this we will use ideas from papers [4], [5]. Also, we give an existence result for this problem, assuming only the continuity of the operator Q. In the paper [6] is also obtained an existence result assuming that the operator Q is a locally Lipschitz operator. Corresponding author. E-mail address: lupulescu v@yahoo.com Copyright c World Academic Press, World Academic Union IJNS.211.6.3/51

5 International Journal of Nonlinear Science, Vol.11(211), No.4, pp. 499-55 2 Preliminaries Let R n be the n-dimensional Euclidian space with norm. For x R n and r > let B r (x) := {y R n ; y x < r} be the open ball centered at x with radius r, and let B r [x] be its closure. If I = [, b) R, b (, ], then we denote by C(I, R n ) the Banach space of continuous functions from I into R n. If σ is a positive number then we put C σ := C([ σ, ], R n ) and let L loc (I, Rn ) denote the space of measurable locally essentially bounded functions x( ) : I R n. Definition 1 Let σ. An operator Q : C([ σ, b), R n ) L loc ([, b), Rn ) is a causal operator if, for each τ [, b) and for all x( ), y( ) C([ σ, b), R n ), with x(t) = y(t) for every t [ σ, τ], we have (Qx)(t) = (Qy)(t) for a.e. t [, τ]. Two significant examples of causal operators are: the Niemytzki operator and the Volterra integral operator (Qx)(t) = g(t) + (Qx)(t) = f(t, x(t)) k(t, s)f(s, x(s))ds. For i =, 1,..., p, we consider the functions g i : R R n R n, (t, x) g i (t, x), that are measurable in t and continuous in x. Set σ := maxσ i, where σ i, and let i=1,p (Qx)(t) = σ g (s, x(t + s))ds + p g i (t, x(t σ i )), t. Then, the operator Q, so defined, is a causal operator (for details, see [6], [1]). For another concrete examples which serve to illustrate that the class of causal operators is very large, we refer to the monograph [1]. We consider the initial-valued problem with causal operator under the following assumptions: (h 1 ) Q is continuous; i=1 x (t) = F (t, x(t), (Qx)(t)), x [ σ,] = φ C σ, (4) (h 2 ) for each r > and each τ (, b), there exists M > such that, for all x( ) C([ σ, b), R n ) with sup x(t) σtτ r, we have (Qx)(t) M for a.e. t [, τ]; (h 3 ) F : [ σ, b) R n R n R n is a Carathéodory function, that is: (a) for a.e. t [ σ, b), F (t,, ) is continuous, (b) for each fixed (x, v) R n R n, F (, x, v) is measurable, (c) for every bounded B R n R n, there exists μ( ) L 1 loc ([ σ, b), R +) such that F (t, x) μ(t) for a.e. t [ σ, b) and all (x, v) B. By a solution of (4) on [ σ, T ], we mean a function x( ) C([ σ, T ], R n ), with T (, b] and x σ,] = φ, such that x [,T ] is absolutely continuous and satisfies (4) for a.e. t [, T ]. We remark that, x( ) C([ σ, T ], R n ) is a solution for (4) on [ σ, T ], if and only if x σ,] = φ and x(t) = φ() + F (s, x(t), (Qx)(s))ds for t (, T ]. The existence of solutions for this kind of Cauchy problem has been studied in [6], when Q : C([ σ, b), R n ) L loc ([, b), Rn ) is a locally Lipschitz operator. The existence of solutions for this kind of Cauchy problem has been studied by [2], in the case in that Q : C([, b), E) C([, b), E) is a local Lipschitz operator. Also, for other results (see [3, 7 9]). IJNS email for contribution: editor@nonlinearscience.org.uk

V. Lupulescu: Functional Differential Equations with Causal Operators 51 3 Existence of solutions In the first half of this section, we present an existence result of the solutions for Cauchy problem (4), under conditions (h 1 ) (h 3 ). Theorem 1 Assume that the conditions (h 1 ) (h 3 ) hold. Then, for every φ C σ, there exists a solution x( ) : [ σ, T ] R n for Cauchy problem (4) on some interval [ σ, T ] with T (, b). Proof. Let δ > be any number and let r := φ σ + δ. If x ( ) C([ σ, b), R n ) denotes the function defined by { x φ(t), for t [ σ, ) (t) = φ(), for t [, b), then sup x (t) r. Therefore, by (h 2 ), we have (Qx )(t) M for a.e. t [, b). Since F is a Carathéodory t<b function, there exists μ( ) L 1 loc ([, b), R +) such that We choose T (, b) such that T F (t, x, v) μ(t) for a.e. t [, b) and (x, v) B r () B M (). μ(t)dt < δ and we consider the set B defined as follows B = {x C([ σ, T ], R n ); x [ σ,] = φ, sup x(t) x (t) δ}. tt Further on, we consider the integral operator P : B C([ σ, T ], R n ) given by { φ(t), for t [ σ, ) (P u)(t) = φ() + F (s, x(s), (Qx)(s))ds, for t [, T ], and we prove that this is a continuous operator from B into B. First, we observe that if x( ) B, then and so, by (h 2 ), (Qx)(t) M for a.e. t [, b). Hence, for each x( ) B, we have sup (P u)(t) x (t) = tt sup tt T T F (s, x(s), (Qx)(s))ds F (s, x(s), (Qx)(s)) ds μ(t)dt < δ sup x(t) < r tt and thus, P (B) B. Moreover, it follows that P (B) is uniformly bounded. Further, let x m x in B. We have sup (P u m )(t) (P u)(t) tt = sup [F (s, x(s), (Qx m)(s)) F (s, x(s), (Qx)(s))]ds tt sup F (s, x(s), (Qx m)(s)) F (s, x(s), (Qx)(s)) ds tt T F (s, x(s), (Qx m)(s)) F (s, x(s), (Qx)(s)) ds and so, by (h 3 ), it follows that sup (P u m )(t) (P u)(t) as m. Since x m [ σ,] = φ for every m N, we tt deduce that P : B B is a continuous operator. Further, we show that P (B) is uniformly equicontinuous on [ σ, T ]. Let ε >. On the closed set [, T ], the function t μ(s)ds is uniformly continuous, and so there exists η > such that t μ(τ)dτ ε/2, for every t, s [, T ] with t s < η. s IJNS homepage: http://www.nonlinearscience.org.uk/

52 International Journal of Nonlinear Science, Vol.11(211), No.4, pp. 499-55 On the other hand, since φ C σ is a continuous function on [ σ, ], then there exists η > such that φ(t) φ(s) ε/2, for every t, s [, T ] with t s < η. Let t, s [ σ, T ] are such that t s η, where η = min{η, η }. If σ s t then, for each x( ) B, we have (P u)(t) (P u)(s) =. Next, if σ s t T then, for each x( ) B, we have (P u)(t) (P u)(s) = φ() + F (τ, x(τ), (Qx)(τ))dτ φ(s) Finally, if s t T then, for each x( ) B, we have φ() φ(s) + F (τ, x(τ), (Qx)(τ)) dτ φ() φ(s) + t μ(τ)dτ ε 2 + ε 2 ε. (P u)(t) (P u)(s) = F (τ, x(τ), (Qx)(τ))dτ s F (τ, x(τ), (Qx)(τ))dτ t s F (τ, x(τ), (Qx)(τ)) dτ t s μ(τ)dτ ε. Therefore, we conclude that P (B) is uniformly equicontinuous on [ σ, T ]. Therefore, by Arzela-Ascoli theorem we deduce that P (B) is a compact operator. Since B is nonempty, convex closed subset of C([ σ, T ], R n ), then the Schauder fixed point theorem implies that P has a fixed point in B. Theorem 2 Assume that the conditions (h 1 ) (h 3 ) hold. Then, there exist T (, b) and a sequence of absolutely continuous functions x m ( ) : [ σ, T ] R n such that, extracting a subsequence if necessary, x m (t) x(t) uniformly on [ σ, T ] and x( ) : [ σ, T ] R n is a solution for the Cauchy problem (4). Proof. Let δ > and T (, b) are that in the proof of the Theorem 1. For each m 1 we consider the sequence (x m ( )) m 1 given by { x (t), for σ t T/m x m (t) = φ() + T/m F (s, x m (s), (Qx m )(s))ds, for T/m t T Then, for all m 1 we have x m ( ) B. Moreover, for t T/m, we have. and for T/m t T, we have (P u m )(t) x m (t) T/m F (s, x m (s), (Qx m )(s)) ds (P u m )(t) x m (t) = (P u m )(t) (P u m )(t T/m) = = t T/m t T/m F (s, x m (s), (Qx m )(s))ds F (s, x m (s), (Qx m )(s))ds T/m F (s, x m (s), (Qx m )(s)) ds T/m μ(s)ds F (s, x m (s), (Qx m )(s))ds t T/m μ(s)ds. Therefore, it follows that sup (P u m )(t) x m (t) as m. On the other hand, the set A = {x m ; m 1} is tt relatively compact in C([, T ], R n ). Hence, extracting a subsequence if necessary, we can assume that x m ( ) x( ) in C([, T ], R n ). Therefore, since sup (P u)(t) x(t) sup (P u)(t) (P u m )(t) + sup (P u m )(t) x m (t) + sup x m (t) x(t) tt tt tt tt IJNS email for contribution: editor@nonlinearscience.org.uk

V. Lupulescu: Functional Differential Equations with Causal Operators 53 then, by the fact that P is a continuous operator, we obtain that sup (P u)(t) x(t) =. It follows that x(t) = tt (P u)(t) = φ() + F (s, x(s), (Qx)(s)ds for every t [, T ]. Now, if we define x( ) : [ σ, T ] Rn by { φ(t), for t [ σ, ) x(t) = φ() + F (s, x(s), (Qx)(s)ds, for t [, b), then x( ) solve the Cauchy problem (4). 4 Continuous dependence In the following, for a fixed φ C σ and a bounded set K R n, by S T (φ, K) we denote the set of solutions x( ) of Cauchy problem (4) on [ σ, T ] with T (, b] and such that x(t) K for all t [ σ, T ]. By A T (φ, K) we denote the attainable set; that is, A T (φ, K) = {x(t ); x( ) S T (φ, K)}. Proposition 3 Assume that the conditions (h 1 ) (h 3 ) hold. Then, for every φ C σ, S T (φ, K) is compact set in C([ σ, T ], R n ). Proof. We consider a sequence {x m ( )} m 1 in S T (φ, K) and we shall show that this sequence contains a subsequence which converges, uniformly on [ σ, T ], to a solution x( ) S T (φ, K). Since K is a bounded set, then there exists r > such that K B r (). By (h 2 ), there exists M > such that (Qx)(t) M for every x( ) C([ σ, T ], R n ) with sup σtt x(t) < r. Since F is a Carathéodory function, there exists μ( ) L 1 ([, T ], R + ) such that F (t, x, v) μ(t) for a.e. t [, T ] and (x, v) B r () B M (). Since x m [ σ,] = φ, we have that x m ( ) φ( ) uniformly on [ σ, ].On the other hand, since x n (t) = φ() + F (s, x n(s), (s)(qx n )(s))ds for all t [, T ], we have that and hence x n (t) x n (s) = s F (τ, x n (τ), (Qx n )(τ))dτ for s, t (, T ] x n (t) x n (s) s F (τ, x n(τ), (Qx n )(τ)) dτ t s μ(τ)dτ r t s for s, t (, T ]. Therefore, {x n ( )} n 1 is equicontinuous on (, T ]. Since, for every n 1, x n ( ) satisfies the same initial condition, x n () = φ(), then we deduce that {x n ( )} n 1 is equibounded on (, T ]. Further, by the Ascoli-Arzela theorem and extracting a subsequence if necessary, we may assume that the sequence {x n ( )} n 1 converges uniformly on (, T ] to a continuous function x( ). If we extend x( ) to [ σ, T ] such that x [ σ,] = φ then is clearly x n ( ) x( ) uniformly on [ σ, T ]. Now, by (h 1 ), we have that lim Qx n = Qx in L ([, T ], R n ). Therefore, lim (Qx n)(t) = (Qx)(t) for a.e. n n t [, T ] and so, by the continuity of the function F (t, ), we have lim F (t, x n(t), (Qx n )(t)) = F (t, x(t), (Qx)(t)) for a.e. t [, T ]. n Since {x n ( )} n 1 is equibounded on [ σ, T ], then there exists r > such that sup σtt x n (t) < r for all n 1. It follows that there exists M > such that (Qx n )(t) < M for a.e. t [, T ] and for all n 1, and so, we have that F (t, x n (t), (Qx n )(t)) μ(t) for a.e. t [, T ] and for all n 1. Therefore, by the Lebesgue dominated convergence theorem, we have It follows that lim n for all t [, T ] and so x( ) S T (φ, K). F (s, x n (t), (Qx n )(s))ds = x(t) = lim n x n(t) = φ() + F (s, x(t), (Qx)(s))ds, for all t [, T ]. F (s, x(t), (Qx)(s))ds, IJNS homepage: http://www.nonlinearscience.org.uk/

54 International Journal of Nonlinear Science, Vol.11(211), No.4, pp. 499-55 Proposition 4 Assume that the conditions (h 1 ) (h 3 ) hold. Then, the multifunction S T : C σ C([ σ, T ], R n ) is upper semicontinuous. Proof. Let K be a closed set in C([ σ, T ], R n ) and B = {φ C σ ; S T (φ, K) K = }. We must show that B is closed in C σ. For this, let {φ n } n 1 a sequence in B such that φ n φ on [ σ, ]. Further, for any n 1, let x n (.) S T (φ n, K) K. Then, x n = φ n on [ σ, ] for all n 1, and x n (t) = φ n () + F (s, x n (s), (Qx n )(s))ds, for all t (, T ]. As in proof of Proposition 3 we can show that {x n ( )} n 1 is equicontinuous and equibounded on [, T ]. Therefore, by the Ascoli-Arzela theorem and extracting a subsequence if necessary, we may assume that the sequence {x n (.)} n 1 converges uniformly on [ σ, T ] to a continuous function x( ) K. Since x(t) = lim n x n(t) = φ() + F (s, x(s), (Qx)(s))ds for all t [, T ], we deduce that x( ) S T (φ, K) K. This prove that B is closed and so φ S T (φ) is upper semicontinuous. Corollary 5 Assume that the conditions (h 1 ) (h 3 ) hold. Then, for any φ C σ and any t [, T ] the attainable set A t (φ, K) is compact in C([ σ, t], R n ) and the multifunction (t, φ) A t (φ, K) is jointly upper semicontinuous. In the following, we consider the following control problem: x (t) = F (t, x(t), (Qx)(t)) for a.e. t [, T ] x [ σ,] = φ minimize g(x(t )), (5) where g : R n R is a given function. Theorem 6 Let K be a compact set in C σ and let g : R n R be a lower semicontinuous function. If the conditions (h 1 ) (h 3 ) hold, then the control problem (5) has an optimal solution; that is, there exists φ K and x ( ) S T (φ, K) such that g(x (T )) = inf{g(x(t )); x( ) S T (φ, K), φ K }. Proof. From Corollary 5 we deduce that the attainable set A T (φ) is upper semicontinuous. Then the set A T (K ) = {x(t ); x( ) S T (φ, K), φ K } = φ K A T (φ, K) is compact in R n and so, since g is lower semicontinuous, there exists φ K such that g(x (T )) = inf{g(x(t )); x( ) S T (φ, K), φ K }. 5 An example We consider the following integro - differential equation with delay with the initial condition Also, we assume that the following assumptions hold: (h 1) f : [ σ, b] R n R n is a Carathéodory function; that is, (1) f(, x) is a measurable function for each fixed x R n, (2) f(t, ) is continuous for each fixed t [ σ, b], x (t) = F (t, x(t), g(t) + k(t, s)f(s, x(s))ds), (6) t σ x [ σ,] = φ C σ. (7) IJNS email for contribution: editor@nonlinearscience.org.uk

V. Lupulescu: Functional Differential Equations with Causal Operators 55 (3) for each r > there exist μ r ( ) L 1 ([ σ, b], R + ) such that f(t, x) μ r (t) a.e. on [ σ, b] for every x R n with x r. (h 2) k : [, b) [ σ, b) R + is a continuous bounded function with (h 3) g( ) L ([, b], R n ). a = sup{k(t, s); t [, b], s [ σ, b]}; If we define the operator Q : C([ σ, b], R n ) L ([, b], R n ) by (Qx)(t) = g(t) + k(t, s)f(s, x(s))ds, t, t σ then Q is a causal operator. In the following, we prove that the operator Q satisfies the conditions (h 1 ) and (h 2 ). Let x m x in C([ σ, b), R n ). Then, for t [, b], we have (Qx n )(t) (Qx)(t) = k(t, s)f(s, x n (s))ds a f(s, x n (s)) f(s, x(s)) ds k(t, s)f(s, x(s))ds and, by (h 1), it follows that (Qx n )(t) (Qx)(t) as n. Hence, Q is a continuous operator. Next, let r be any positive number. If sup σtb x(t) r, then (Qx)(t) M, for a.e. t [, b], where Thus, (h 2 ) holds. Therefore, we obtain the following result. M := sup g(t) + a tb b μ r (s)ds. Theorem 7 Assume that the assumptions (h 1) (h 3) hold. If F : [ σ, b] R n R n R n is a Carathéodory function, then the problem (6)-(7) has a solution on some interval [, T ], with T (, b]. References [1] C. Corduneanu. Functional Equations with Causal Operators, Stability and Control: Theory, Methods and Applications. Taylor & Francis, London. (22). [2] Z. Drici, F.A. McRae and J. Vasundhara Devi. Differential equations with causal operators in a Banach space. Nonlinear Analysis, 62(25):31-313. [3] Z. Drici, F.A. McRae and J. Vasundhara Devi. Monotone iterative technique for periodic value problems with causal operators. Nonlinear Analysis, 64(26):1271-1277. [4] A. F. Filippov. On certain questions in the theory of optimal control. Siam J. Control. 1(1962):76-84. [5] H. Hermes. The Generalized Differential Equation. x R(t, x). Advances in Mathematics, 4(197):49-169. [6] A. Ilchmann, E.P. Ryan and C.J. Sangwin. Systems of Controlled Functional Differential Equations and Adaptive Tracking. Siam J. Control Optim., 4(22):1746-1764. [7] T. Jankowski. Boundary value problems with causal operators. Nonlinear Analysis, 68(28):3625-3632. [8] V. Lupulescu. Causal functional differential equations in Banach spaces. Nonlinear Analysis, 69(28):4787 4795. [9] D. O Regan. A note on the topological structure of the solutions set of abstract Volterra equations. Proc. Roy. Irish Acad. Sect. A, 99(1999):67-74. [1] E.P. Ryan and C.J. Sangwin. Controlled functional differential equations and adaptive tracking. Systems & Control Letters, 47(22):365 374. IJNS homepage: http://www.nonlinearscience.org.uk/