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In this paper we study the approximate controllability of the following semilinear difference equation \[ z(n+1)=A(n)z(n)+B(n)u(n)+f(n,z(n),u(n)), \quad n\in \mathbb{N}^*, \] $z(n)\in Z$, $u(n)\in U$, where $Z$, $U$ are Hilbert spaces, $A\in l^{\infty}(\mathbb{N},L(Z))$, $B\in l^{\infty}(\mathbb{N},L(U,Z))$, $u\in l^2(\mathbb{N},U)$ and the nonlinear term $f:\mathbb{N} \times Z\times U\longrightarrow Z$ 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.

In this paper, we study the exact controllability of the following semilinear difference equation z(n) ∈ Z, u(n) ∈ U, where Z, U are Hilbert spaces, , , , and the nonlinear term f : Z × U → Z satisfies: We prove the following statement: If the linear equation is exactly controllable and L

In this paper we present necessary a sufficient conditions for the exact and approximate controllability of Block-Diagonal Control Systems representing a broad class of linear evolution equations in Hilbert spaces. We apply these results to the controlled nD wave equation, heat equation and thermoelastic plate equation.