On output stabilizability for a class of infinite dimensional linear systems
by
Faouzi Haddouchi
Université des Sciences et de la Technologie d'Oran (USTO), Faculté des Sciences, Département de Physique, Algerie

This work is concerned with the output stabilizability of infinite dimensional linear system described by the following abstract differential equation \sum

× x   = Ax(t) + Bu(t), x(o) = x0, y(t) = Cx(t)

where x(t) in H , u(t) in U , y(t) in Y and H, U and Y are always intended infinite dimensional Hilbert spaces unless otherwise stated. The (state) operator A is an infinitesimal generator of a semigroup S(t) of class C₀ on H; B and C are linear and continuous operators, i.e.B in L(U, H), C in L(H, Y). As usual u, x, y represent respectively the input, state and output of the system sum. The output stabilization problem (o.s.p) is the problem of finding a feedback control u(t) = Fx(t), where F in L(H, U) such that the output y(t) = CS_A+BF(t)x₀ of the closed loop system converges to zero exponentially (strongly or weakly), for every x₀ in H. S_A+BF(t) being the semigroup generated by the operator A+BF. For finite dimensional systems, all three types of output stabilizability are equivalent. However in the infinite dimensional case only a few articles have been published and there is no equivalence between these three concepts of output stabilizability in general. The interesting case in infinite dimensions is when A is unbounded, but generates a C₀-semigroup. In this last case, we study the above mentioned problem for a class of infinite dimensional linear systems: those having a finite number of inputs and outputs but for which the state operator A is self-adjoint with compact resolvant. One of the main results that we shall give, is criterion for the exponential output stabilization. Furthermore, we shall show that for this class of systems there is no difference between exponential, strong and weak output stabilizability. Finally, we shall conclude the work with an example which shows how the theory can be applied.

Date received: May 6, 2003

Copyright © 2003 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts. Document # calh-14.