Title: Exact controllability of semilinear evolution equation and applications

Authors: Hugo Leiva

Addresses: Departamento de Matematicas, Universidad de Los Andes, Merida, 5101, Venezuela

Abstract: In this paper we characterise the exact controllability for the following semilinear evolution equation z′ = Az + Bu(t) + F(t, z, u(t)), t>0, z∈ Z, u∈ U, where Z, U are Hilbert spaces, A : D(A) ⊂ Z → Z is the infinitesimal generator of strongly continuous semigroup {T(t)}_t≥0 in Z, B &insin; L(U,Z), the control function u belongs to L²(0, τ; U) and F : [0, τ] × Z × U → Z is a suitable function. First, we give a necessary and sufficient condition for the exact controllability of the linear system z′ = Az + Bu(t). Second, under some conditions on F, we prove that the exact controllability of the linear system is preserved by the semilinear system, in this case the control u steering an initial state z₀ to a final state z₁ at time τ > 0 is given by the following formula: u(t) = B*T*(τ − t)W⁻¹(I + K)⁻¹(z₁ − T(τ)z₀), according to Theorem 3.1. Finally, these results can be applied to the controlled damped wave equation.

Keywords: semilinear evolution equations; exact controllability; damped wave equations.

DOI: 10.1504/IJSCC.2008.019580

International Journal of Systems, Control and Communications, 2008 Vol.1 No.1, pp.1 - 12

Published online: 17 Jul 2008 *

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