On Lipschitz Stable Switching Behavior of Bang-bang Extremals in Parametric Control

In the paper, we consider ODE driven optimal control problems with bang-bang type extremals. The specific nature of bang-bang controls causes some difficulties in optimality and stability analysis. Usual convexity arguments like (strong) Legendre-Clebsch condition fail to hold, and the control discontinuity has to be taken into account. In recent years, substantial results on secondorder optimality conditions have been obtained in [8], also [6], [1], or [7]. The solution stability under parameter perturbation was investigated e.g. in [5] and [2], [3]. Optimality and stability conditions therein are: (i) bang-bang regularity assumptions (finite number of switches, excluding e.g. endpoints), (ii) strict bang-bang properties (nonvanishing time derivatives of switching functions at switching points e.g.), (iii) assumption of simple switches (switch of no more than one control component at each time), (iv) appropriate second-order conditions (positive definiteness of related quadratic forms e.g.).