On the cost of controlling unstable systems: The case of boundary controls |
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Authors: | Jacques-Louis Lions Enrique Zuazua |
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Affiliation: | (1) Collège de France, 3 rue d’Ulm, 75231 Paris Cedex 05, France;(2) Departamento de Matemática Aplicada, Universidad Complutense, 28040 Madrid, Spain |
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Abstract: | We discuss the cost of controlling parabolic equations of the formy t + δ2 y +kδy = 0 in a bounded smooth domain Ώ ofℝ d by means of a boundary control. More precisely, we are interested in the cost of controlling from zero initial state to a given final state (in a suitable approximate sense) at timeT > 0 and in the behavior of this cost ask → ∞. We introduce finite-dimensional Galerkin approximations and prove that they are exactly controllable. Moreover, we also prove that the cost of controlling converges exponentially to zero ask → ∞. This proves, roughly speaking, that when the system becomes more unstable it is easier to control. The system under consideration does not admit a variational formulation. Thus, in order to introduce its Galerkin approximation, we first approximate it by means of a singular perturbation. We also develop a method for the construction of special Galerkin bases well adapted to the control problem. Dedicated to John E. Lagnese on his 60th Birthday. Supported by project PB93-1203 of the DGICYT (Spain) and grant CHRX-CT94-0471 of the European Union. |
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