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Existence of optimal controls for the diffusion equation

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The existence is considered of a boundary control which drives a system governed by the one-dimensional diffusion equation from the zero state to a given final state, and at the same time minimizes a given functional. The problem is first modified to one in which the minimum is sought of a functional defined on a set of Radon measures. The existence of a minimizing measure is demonstrated, and it is shown that this measure may be approximated by a piecewise constant control. Finally, conditions are given under which a minimizing measurable control exists for the unmodified problem.

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Communicated by G. Leitmann

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Wilson, D.A., Rubio, J.E. Existence of optimal controls for the diffusion equation. J Optim Theory Appl 22, 91–101 (1977).

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Key Words

  • Optimal control
  • Radon measures
  • existence theory
  • diffusion equation