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) ∈ 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 < < 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.

Keywords:

• difference equations
• exact controllability
• heat equation
• Lipschitz

Keywords:

• primary: 93C55
• secondary: 93B05