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Error bounds for Euler approximation of linear-quadratic control problems with bang-bang solutions. (English) Zbl 1259.49003

In the paper, the authors investigate a class of linear-quadratic optimal control problems. The aim of the article is to derive error estimates for the Euler approximations of the considered problems. Such estimates were previously known only in the case of Lipschitz continuous controls. Since an optimal control for the given problems typically has a bang-bang structure, the assumption that the optimal control is Lipschitz continuous is not satisfied. After the Euler discretization for the linear-quadratic control problems, the authors show convergence of order \(h\) for the optimal values of the objective function, where \(h\) is the mesh size of the discretization. Assuming that the optimal control is of bang-bang type, it is shown then that the discrete and the continuous controls coincide except on a set of measure \(O(\sqrt{h})\). Here the authors use an approach based on a second-order optimality condition known from the stability analysis of bang-bang control. Finally, under a slightly stronger assumption on the smoothness of the coefficients of the system equation the error estimates for the discretized solutions are improved to order \(O(h)\). The authors also give a numerical example which confirms the error estimates.

MSC:

49J15 Existence theories for optimal control problems involving ordinary differential equations
49M25 Discrete approximations in optimal control
49N10 Linear-quadratic optimal control problems
49J30 Existence of optimal solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.)
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