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Finite-dimensional control of parabolic PDE systems using approximate inertial manifolds. (English) Zbl 0890.93051
The authors introduce a methodology for the synthesis of nonlinear finite-dimensional output feedback controllers for systems of quasilinear parabolic partial differential equations (PDEs), for which the eigenspectrum of the spatial differential operator can be partitioned into a finite-dimensional slow one and an infinite-dimensional stable fast complement. Singular perturbation methods are initially employed to establish that the discrepancy between the solutions of an ordinary differential equation (ODE) system (whose dimension is equal to the number of slow modes), obtained through Galerkin’s method, and the PDE system is proportional to the degree of separation of the fast and slow modes of the spatial operator. Then, a procedure, motivated by the theory of singular perturbations, is proposed for the construction of approximate inertial manifolds (AIMs) for the PDE system. The AIMs are used for the derivation of ODE systems of dimension equal to the number of slow modes, that yield solutions which are close, up to a desired accuracy, to the ones of the PDE system, for almost all times. These ODE systems are used as the basis for the synthesis of nonlinear output feedback controllers that guarantee stability and enforce the output of the closed-loop system to follow to desired accuracy, a prespecified response for almost all times.

MSC:
93C20 Control/observation systems governed by partial differential equations
93C73 Perturbations in control/observation systems
93D15 Stabilization of systems by feedback
93C10 Nonlinear systems in control theory
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