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Approximate Controllability for the Semilinear Heat Equation Involving Gradient Terms

Author

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  • L. A. Fernández

    (Universidad de Cantabria)

  • E. Zuazua

    (Universidad Complutense)

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.

Suggested Citation

  • L. A. Fernández & E. Zuazua, 1999. "Approximate Controllability for the Semilinear Heat Equation Involving Gradient Terms," Journal of Optimization Theory and Applications, Springer, vol. 101(2), pages 307-328, May.
    • Handle: RePEc:spr:joptap:v:101:y:1999:i:2:d:10.1023_a:1021737526541
    DOI: 10.1023/A:1021737526541
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    Cited by:

    1. V. R. Cabanillas & S. B. de Menezes & E. Zuazua, 2001. "Null Controllability in Unbounded Domains for the Semilinear Heat Equation with Nonlinearities Involving Gradient Terms," Journal of Optimization Theory and Applications, Springer, vol. 110(2), pages 245-264, August.

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