Continuous-time optimal control theory for cost functionals including discrete state penalty terms

Abstract

In this short paper, continuous-time optimal control problems are treated, where the cost functional includes additional discrete state penalty terms <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K_{j}(x(t_{j}))</tex> at specified times <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t_{j}(j = 1,..., N - 1)</tex> in the interior of the time interval <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[t_{0}, t_{N}]</tex> of the problem. Necessary conditions for a control to be optimal are given in the form of a minimum principle. The costate is necessarily discontinuous at the times <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t_{j}(j = 1,..., N - 1)</tex> . Consequently, the optimal control is discontinuous at t <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">j</inf> , in general. A numerical example is given.