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Exact controllability of semilinear evolution equation and applications

Exact controllability of semilinear evolution equation and applications 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png International Journal of Systems, Control and Communications Inderscience Publishers

Exact controllability of semilinear evolution equation and applications

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Publisher
Inderscience Publishers
Copyright
Copyright © Inderscience Enterprises Ltd. All rights reserved
ISSN
1755-9340
eISSN
1755-9359
DOI
10.1504/IJSCC.2008.01958
Publisher site
See Article on Publisher Site

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.

Journal

International Journal of Systems, Control and CommunicationsInderscience Publishers

Published: Jan 1, 2008

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