Abstract
The existence and numerical estimation of a boundary control for then-dimensional linear diffusion equation is considered. The problem is modified into one consisting of the minimization of a linear functional over a set of Radon measures. The existence of an optimal measure corresponding to the above problem is shown, and the optimal measure is approximated by a finite convex combination of atomic measures. This construction gives rise to a finite-dimensional linear programming problem, whose solution can be used to construct the combination of atomic measures, and thus a piecewise-constant control function which approximates the action of the optimal measure, so that the final state corresponding to the above control function is close to the desired final state, and the value it assigns to the performance criterion is close to the corresponding infimum. A numerical procedure is developed for the estimation of these controls, entailing the solution of large, finite-dimensional linear programming problems. This procedure is illustrated by several examples.
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Communicated by L. Cesari
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Kamyad, A.V., Rubio, J.E. & Wilson, D.A. Optimal control of the multidimensional diffusion equation. J Optim Theory Appl 70, 191–209 (1991). https://doi.org/10.1007/BF00940511
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DOI: https://doi.org/10.1007/BF00940511