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Exact controllability for semilinear difference equation and application. (English) Zbl 1146.93012

Summary: 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 * ,

z(n)Z, u(n)U, where Z, U are Hilbert spaces, * ={0}, Al (,L(Z)), Bl (,L*U,Z)), ul 2 (,U) and the nonlinear term f:Z×UZ satisfies:

f(z 2 ,u 2 )-f(z 1 ,u 1 )L{z 2 -z 1 +u 2 -u 1 }·

We prove the following statement: If the linear equation is exactly controllable and L1, 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.

93C55Discrete-time control systems
93C10Nonlinear control systems
35K05Heat equation