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 Hölder-type estimate and to investigate the robustness of this estimate. The Hölder 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 $k\geq 1$.