Abstract
We consider the problem of controlling the solution of the heat equation with the convective boundary condition taking the heat transfer coefficient as the control. We take as our cost functional the sum of theL 2-norms of the control and the difference between the temperature attained and the desired temperature. We establish the existence of solutions of the underlying initial boundary-value problem and of an optimal control that minimizes the cost functional. We derive an optimality system by formally differentiating the cost functional with respect to the control and evaluating the result at an optimal control. We show how the solution depends in a differentiable way on the control using appropriate a priori estimates. We establish existence and uniqueness of the solution of the optimality system, and thus determine the unique optimal control in terms of the solution of the optimality system.
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Communicated by R. Rishel
This research was sponsored by the Applied Mathematical Sciences Research Program, Office of Energy Research, U.S. Department of Energy under Contract DE-AC05-84OR21400 with the Martin Marietta Energy Systems. The authors thank David R. Adams for his assistance in clarifying the proof of Proposition 2.1 and appreciate the comments of the referees for needed revisions.
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Lenhart, S., Wilson, D.G. Optimal control of a heat transfer problem with convective boundary condition. J Optim Theory Appl 79, 581–597 (1993). https://doi.org/10.1007/BF00940560
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DOI: https://doi.org/10.1007/BF00940560