On a Constrained Approximate Controllability Problem for the Heat Equation

J. H. Ortega and E. Zuazua
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J. H. Ortega: Universidad del Bío-Bío
E. Zuazua: Universidad Complutense de Madrid

Journal of Optimization Theory and Applications, 2001, vol. 108, issue 1, No 2, 29-64

Abstract: Abstract In this work, we study an approximate control problem for the heatequation, with a nonstandard but rather natural restriction on thesolution. It is well known that approximate controllability holds. On theother hand, the total mass of the solutions of the uncontrolled system isconstant in time. Therefore, it is natural to analyze whether approximatecontrollability holds supposing the total mass of the solution to be a givenconstant along the trajectory. Under this additional restriction,approximate controllability is not always true. For instance, this propertyfails when Ω is a ball. We prove that the system is genericallycontrollable; that is, given an open regular bounded domain Ω, thereexists an arbitrarily small smooth deformation u, such that the system isapproximately controllable in the new domain Ω+u underthis constraint. We reduce our control problem to a nonstandard uniquenessproblem. This uniqueness property is shown to hold generically with respectto the domain.

Keywords: boundary controllability; heat equation; spectral theory; shape differentiation (search for similar items in EconPapers)
Date: 2001
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Citations: View citations in EconPapers (1)

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DOI: 10.1023/A:1026409821088

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