# Metric Regularity and Stability of Optimal Control Problems for Linear Systems

Marc Quincampoix and Vladimir Veliov^{*}

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 so-called {\em strong bi-metric regularity}, which is introduced and investigated in the paper. It allows to give a more precise description of the effect of perturbations on the optimal solutions in terms of a Hölder-type estimation, and to investigate the robustness of this estimation. The Hölder exponent depends on a natural number $k$, which is known as the \emph{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 non-linear perturbations. Moreover, a new stability result with respect to perturbations in the matrices of the system is proved in the general case $k \geq 1$.

Mathematics Subject Classification: 49K40 90C31 49N05 93C05 47J07 54C60

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

Minisymposion: Stability, Sensitivity and Error Analysis for Optimal Control Problems