Controllability of linear difference equations in Hilbert spaces and applications

In this paper, we present necessary and sufficient conditions for the exact and approximate controllability of the following linear difference equation: $$z(n + 1) = A(n)z(n) + B(n)u(n),\quad n\in \mathbb{N}^{*}, z(n) \in Z, u(n) \in U,$$where Z, U are Hilbert spaces, A(·) ∈ l^∞( $A(\cdot) \in l^{\infty}(\mathbb{N},$ L(Z)), B(·) ∈ l^∞( $\mathbb{N}$, L(U, Z)), u ∈ l²( $\mathbb{N}$, U) and $\mathbb{N}^{*}$ = $\mathbb{N}$ ∪ {0}. Moreover, in the case of exact controllability, the control u ∈ l²( $\mathbb{N}$, U) steering an initial state z0 to a final state z1 in time n0 is given by the formula $u=\mathcal{B}^{n0*}L_{\mathcal{B}^{n0}}^{-1}(z_{1}-\Phi(n_{0},0)z_{0})$ according to Lemma 2.1. As a particular case, we consider the discretization on flow of the following controlled evolution equation z′ = Az + Bu, z ∈ Z, u ∈ U, t > 0, where B ∈ L (U, Z), u ∈ L²(0, τ;U) and A is the infinitesimal generator of a strongly continuous semigroup {T(t)}t ≥ 0 in Z, given by $$T (t)z = \sum_{j=1}^{\infty}\mathrm{e}^{A_{j}}P_{j}z,\quad z\in Z,\ t\geqslant 0,$$according to Lemma 1.1. These results are applicable to a broad class of reaction–diffusion systems such as the heat equation, the wave equation, the equation modelling the damped flexible beam, the strongly damped wave equation, the thermoelastic plate equation, etc. In Section 4, these results are applied to a discrete version of the n-dimensional heat and n-dimensional wave equation.