Exact controllability of semilinear evolution equation and applications
by Hugo Leiva
International Journal of Systems, Control and Communications (IJSCC), Vol. 1, No. 1, 2008

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.

Online publication date: Thu, 17-Jul-2008

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