APPROXIMATE CONTROLLABILITY OF DISCRETE SEMILINEAR SYSTEMS AND APPLICATIONS

摘要

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)), n ∈ 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 / :NxZxU → Z is a suitable function. We prove that, under some conditions on the nonlinear term /, the approximate controllability of the linear equation is preserved. Finally, we apply this result to a discrete version of the semilinear wave equation.