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M. Quincampoix, V.M. Veliov:
"Metric Regularity and Stability of Optimal Control Problems for Linear Systems";
SIAM Journal on Control and Optimization, 51 (2013), 5; 4118 - 4137.

This paper studies stability properties of the solutions of optimal control problems for linear systems. The analysis is based on an adapted concept of metric regularity, the strong bi-metric regularity, which is introduced and investigated in the paper. It allows one to give a more precise description of the effect of perturbations on the optimal solutions in terms of a H¨oldertype
estimate and to investigate the robustness of this estimate. The H¨older exponent depends on a natural number k, which is known as the controllability index of the reference solution. An inverse function theorem for strongly bi-metrically regular mappings is obtained, which is used in the case k = 1 for proving stability of the solution of the considered optimal control problem under small nonlinear perturbations. Moreover, a new stability result with respect to perturbations in the matrices of the system is proved in the general case k ≥ 1.

optimal control, linear control systems, metric regularity, inverse function theorem

http://dx.doi.org/10.1137/130914383

http://publik.tuwien.ac.at/files/PubDat_223914.pdf