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Metric regularity under approximations

Dontchev A. L., Veliov V. M.

Control and Cybernetics

2009

Vol. 38, no 4B

1283-1303

In this paper we show that metric regularity and strong metric regularity of a set-valued mapping imply convergence of inexact iterative methods for solving a generalized equation associated with this mapping. To accomplish this, we first focus on the question how these properties are preserved under changes of the mapping and the reference point. As an application, we consider discrete approximations in optimal control.

Error analysis of discrete approximations to bang-bang optimal control problems: the linear case

Veliov V. M.

Control and Cybernetics

2005

Vol. 34, no 3

967-982

The paper presents an error estimate for Runge-Kutta direct discretizations of terminal optimal control problems for linear systems. The optimal control for such problems is typically discontinuous, and Lipschitz stability of the solution with respect to perturbations does not necessarily hold. The estimate (in terms of the optimal controls) is of first order if certain recently obtained sufficient conditions for structural stability hold, and of fractional order, otherwise. The main tool in the proof is the established relation between the local convexity index of the reachable set and the multiplicity of zeros of appropriate switching functions associated with the problem.