Abstract
This paper deals with the approximate controllability of the semilinear heat equation, when the nonlinear term depends on both the state y and its spatial gradient ∇y and the control acts on any nonempty open subset of the domain. Our proof relies on the fact that the nonlinearity is globally Lipschitz with respect to (y, ∇y). The approximate controllability is viewed as the limit of a sequence of optimal control problems. Another key ingredient is a unique continuation property proved by Fabre (Ref. 1) in the context of linear heat equations. Finally, we prove that approximate controllability can be obtained simultaneously with exact controllability over finite-dimensional subspaces.
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Fernández, L.A., Zuazua, E. Approximate Controllability for the Semilinear Heat Equation Involving Gradient Terms. Journal of Optimization Theory and Applications 101, 307–328 (1999). https://doi.org/10.1023/A:1021737526541
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DOI: https://doi.org/10.1023/A:1021737526541