APPROXIMATE CONTROLLABILITY OF DISCRETE SEMILINEAR SYSTEMS AND APPLICATIONS

. In this paper we study the approximate controllability of the following semilinear diﬀerence equation z n + 1) = A ( n ) z ( n ) B ) n ) ,u )) , n ∗ , z ( n ) ∈ Z , u ( n ) ∈ U , where Z , U are Hilbert spaces, A ∈ l ∞ ( N ,L ( Z )), B ∈ l ∞ ( N ,L ( U,Z )), u ∈ l 2 ( N ,U ) and the nonlinear term f : N × Z × U −→ Z is a suitable function. We prove that, under some conditions on the nonlinear term f , the approximate controllability of the linear equation is preserved. Finally, we apply this result to a discrete version of the semilinear wave equation.

1. Introduction. One of the main sources for applications of discrete control systems methods are continuous control systems; that is to say, those models described by differential equations instead of difference equations. The reason for this is that while physical systems are modeled by differential equations, control laws are implemented often in a digital computer, whose inputs and outputs are sequences. A common approach to design controls in this case is to obtain a difference equation model that approximates the continuous system that will be controlled.
Considering this observation and using some ideas presented in [1]- [6] we will give sufficient conditions for the approximate controllability of the following semilinear difference equation z(n + 1) = A(n)z(n) + B(n)u(n) + f (n, z(n), u(n)), n ∈ N * , z(0) = z 0 , where N * = N \ {0}, z(n) ∈ Z, u(n) ∈ U , Z and U are Hilbert spaces, A ∈ l ∞ (N, L(Z)), B ∈ l ∞ (N, L(U, Z)), u ∈ l 2 (N, U ), L(U, Z) denotes the space of all bounded linear operators from U to Z and L(Z, Z) = L(Z). The nonlinear term f : N × Z × U −→ Z satisfies the following conditions for n ∈ N, u ∈ l 2 (N, U ), z ∈ Z Φ(n, k)f (k−1, z, u(k−1)) ≤ M k , , 1 ≤ k ≤ n, and Φ ={Φ(n, m)} (n,m)∈∆ with ∆ ={(n, m) ∈ N × N : n ≥ m} is the evolution operator associated to A, i.e., where I is the identity operator in the space of bounded and linear operator L(Z). Then, for z 0 ∈ Z the equation (1) has a unique solution given by Corresponding to the nonlinear system (1) we consider also the linear system: From now on we shall use the following notation [m, n 0 ] N = [m, n 0 ] ∩ N with 0 ≤ m < n 0 .
Definition 1.1. (Approximate Controllability of system (5) ) The system (5) is said to be approximately controllable on [m, n 0 ] N if for every z 0 , z 1 ∈ Z and > 0 there exists u ∈ l 2 (N, U ) such that the corresponding solution of (5) satisfies z(n 0 ) − z 1 < . Definition 1.2. (Approximate Controllability on Free Time of (1) ) The system (1) is said to be approximately controllable on free time if for every z 0 , z 1 ∈ Z and > 0 there exists u ∈ l 2 (IN , U ) and n 0 ∈ IN such that the corresponding solution of (1) satisfies z(n 0 ) − z 1 < .
We will prove the following statement: If conditions (2)-(3) hold and the linear system (5) is approximately controllable on [m, n 0 ] N , for all 0 ≤ m < n 0 , then the semilinear system (1) is approximately controllable on free time. Moreover, we can find a sequence of controls steering the system from the initial state z 0 to an -neighborhood of the final state z 1 on some time n 0 .
There is a broad literature on the study of difference equations, but the study of controllability of such equations is still in effervescence, however there are good references in this respect. In [17], [18] the concept of approximate controllability for non-autonomous linear systems described by linear skew-products semiflows is characterized, and the authors showed that stabilizability implies approximate controllability for the case of systems associated to uniformly*-positive linear skewproducts semiflows. In [19] and [20], it was shown that the stability of the linear difference equation implies the controllability of the system. In [11], the authors considered the approximate controllability of abstract discrete-time systems similar to (1). They get results on approximate controllability of semilinear system (1) by perturbing the reachable set of linear system (5).
We have already obtained some results on exact and approximate controllability for linear and semilinear difference equations, which appear in [14], [15], [16].
2. Controllability of the linear equation. In this section we will present some characterization of the approximate controllability for the linear difference equation (5). To this end, we note that the solution of (5) is given by the discrete variation constant formula Definition 2.1. For the linear system (5) we define the following concepts: a) The controllability map (for 0 < m < n 0 ∈ N) is defined as follows B mn0 : The proof of the following Proposition can be seen in [14]. and The following Lemma will be used to stablish our next Theorem and it can be founded in( [1], [2], [7], [8] and [16])).
Lemma 2.1. If W , Z are Hilbert spaces and G : W → Z is a linear bounded operator, then the following statements are equivalent: So, lim α→0 Gu α = z and the error Ez of this approximation is given by the formula f ) Moreover, if we consider for each v ∈ L 2 (τ − δ, τ ; U )) the sequence of controls given by we get that: with the error E τ δ z of this approximation given by the formula Theorem 2.1. The linear system (5) is approximately controllable on [m, n 0 ] N if, and only if, one of the following statements holds: So, lim α→0 B mn0 u α = z and the error E α z of this approximation is given by Remark 2.1. The foregoing theorem implies that the family of operators Γ αmn0 : is an approximate right inverse of the operator B mn0 , i.e., lim α→0 + B mn0 Γ αmn0 = I.
Lemma 2.2. If linear system (5) is approximately controllable on [m, n 0 ] N , a sequence of controls steering the initial state z 0 to a ε-neighborhood of a final state z 1 in time n 0 is given by and corresponding solutions y(n) = y(n, m, y 0 , u α ) of the initial value problem satisfy lim α−→0 + y(n, m, y 0 , u α ) = z 1 . i.e., 3. Controllability of the nonlinear equation. In this section we shall study the approximate controllability on free time of the nonlinear difference equation (1).
To this end, we note that for each z 0 ∈ Z and a control u ∈ l 2 (IN , U ) the equation (1) has a unique solution given by for all n ∈ N * . Now, we are ready to present and prove the main result of this paper, which is the approximate controllability of semilinear equation (1) on free time. Proof. Given an initial state z 0 , a final state z 1 and ε > 0, we want to find a control u m α ∈ l 2 (N, U ) steering the system from z 0 to an ε-neighborhood of z 1 on time n 0 . Specifically, Consider any u ∈ l 2 (N, U ) and the corresponding solution z(n) = z(n, 0, z 0 , u) of initial value problem (1). For α ∈ (0, 1], we define the control u m α ∈ l 2 (N, U ) as follows: where u α = B mn0 * (αI + L B mn 0 ) −1 (z 1 − Φ(n 0 , m)z 0 ) ∈ l 2 (N, U ). (16) Now, assume that 0 < m < n 0 and n 0 > 1. Then the corresponding solution z m α (n) = z(n, 0, z 0 , u m α ) of initial value problem (1) at time n 0 can be written as follows: Therefore, the solution z m α (n) = z(n, 0, z 0 , u m α ) of initial value problem (1) at time n 0 can be written as follows: The corresponding solution y m α (n) = y(n, m, z(m), u α ) of initial value problem (10) at time n 0 is given by Hence, for m, n 0 ∈ IN big enough, with 0 < m < n 0 , we obtain that , for a big enough m.
On the other hand, from Lemma 2.2, there exists α > 0 such that Therefore, from the above two inequalities we get the following estimate which completes the proof of the theorem.

4.
Application. In general, given a controlled evolution equation where z ∈ Z, u ∈ U , Z and U are Hilbert spaces, A is the infinitesimal generator of a C 0 -semigroup {T (t)} t≥0 , one can consider a discretization on its flow, the same that is used in [9] and [10] to study the exponential dichotomy of evolution equation. That is to say z(n + 1) = T (n)z(n) + Bu(n), n ∈ N * , where the control u = {u(n)} n≥1 belongs to l 2 (N, U ) and z(n) ∈ Z.
As an application of the main result of this paper we shall consider a discretization on flow of the controlled nonlinear wave equation.
where < ·, · > is the inner product in X and So, {E n } is an orthonormal and complete family of projections in X and x = ∞ n=1 E n x, x ∈ X. c) −A generates an analytical semigroup {e −At } given by d) The spaces of fractional powers X r are given by: with the norm Also, for r ≥ 0 we define Z r = X r × X, which is a Hilbert space with the norm Now, using the change of variables y = v, the second order equation (19) can be written as a first order system of ordinary differential equations in the Hilbert space Z = X 1/2 × X as where A is an unbounded linear operator with domain D(A) = D(A) × X, u ∈ L 2 (0, τ, U ) with U = X. The proof of the following Theorem follows directly from Lemma 2.1 in [13].
Theorem 4.1. The operator A given by (25), is the infinitesimal generator of a strongly continuous group {T (t)} t∈I R given by where {P j } j≥1 is a complete family of orthogonal projections in the Hilbert space Z given by and Note that and there exist M > 1 such that T (t) ≤ M . Now, the discretization of (24) on flow is given by where u ∈ l 2 (N, U ), B : U −→ Z, Bu = 0 u .
First, we will show that (29) is approximately controllable on [m, n 0 ] N . In this case, we have and The verification that P j BB * = BB * P j and T * (t) = T (−t) is trivial. Then T (Θ(n 0 , k))BB * T * (Θ(n 0 , k))z In consequence, by Theorem 2.1 part (c), the equation (27) is approximately controllable on [m, n 0 ] N .
Finally, if we consider a perturbation of the equation (29), say z(n + 1) = T (n)z(n) + B(n)u(n) + f (n, z(n), u(n)), z ∈ Z z(0) = z 0 where the nonlinear term f : N × Z × U −→ Z is a suitable function, then from the results obtained in section 3, we have that (32) is approximately controllable on free time.