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Exact controllability for semilinear difference equation and application

Abstract:

In this paper, we study the exact controllability of the following semilinear difference equation z(n + 1) = A(n)z(n) + B(n)u(n) + f(z(n), u(n)), n ∈ IN*, z(n) ∈ Z, u(n) ∈ U, where Z, U are Hilbert spaces, IN* = IN ∪ {0}, A ∈ l∞(IN, L(Z)), B ∈ l∞(IN, L(U, Z)), u ∈ l2(IN, U) and the nonlinear term f:Z × U → Z satisfies: ∥f(z2, u 2) - f(z1, u1)∥ ≤ L{∥z2 - z1∥ + ∥u2 - u1∥}. We prove the following statement: If the linear equation is exactly controllable and L ≪ 1, then the nonlinear equation is also exactly controllable. That it to say, the controllability of the linear equation is preserved under nonlinear perturbation f(z,u). Finally, we apply this result to a discrete version of the semilinear heat equation. © 2008 Taylor & Francis.