On the Topology of Parametric Optimal Control

We will study one-parameter families .Pu s /s∈[Þ;þ] of differentiable optimal control problems given by: P .s/  MinI u.s; x/ := ∫ b a L.s; t; x.t/; u.s; t// dt; where ẋ = F.s; t; x; u/ .t ∈ [a; b]/ H .s; t; x; u/ = 0 .t ∈ {t1 0 ; : : : ; t 0 }/ G.s; t; x; u; y/ ≥ 0; .y ∈ Y 1.s; t; x; u/; t ∈ [a; b]/ H b a .s; a; x.a/; u.s; a/; b; x.b/; u.s; b// = 0 Gb a.s; a; x.a/; u.s; a/; b; x.b/; u.s; b/; y/ ≥ 0 .y ∈ Y 2.s; a; x.a/; u.s; a/; b; x.b/; u.s; b///: Here, at given times t the inequality constraint functions are of semi-infinite nature, the objective functional may also be of max-type. For each s ∈ Rthe problemPu s is equivalent to a one-parameter family .Ps.t//t∈[a;b] of differentiable optimization problems. From these the consideration of generalized critical trajectories, such as a local minimum trajectory, comes into our investigation. According to a concept introduced by Hettich, Jongen and Stein in optimization, we distinguish eight types of generalized critical trajectories. Under suitable continuity, compactness and integrability assumptions, those problems, which exclusively have generalized critical points being of one of these eight types, are generic. We study normal forms and characteristic examples, locally around these trajectories. Moreover, we indicate the related concept of structural stability of optimal control problems Pu due to the topological behaviour of the lower level sets under small data perturbations. Finally, we discuss the numerical consequences of our investigations for pathfollowing techniques with jumps.