Stability and Local Growth near Bounded-Strong Optimal Controls

Nonlinear constrained optimal control problems as a rule suffer from the so-called two-norm discrepancy, which in particular says that under stable optimality conditions the objective functionals satisfy a quadratic local growth estimate in terms of the L2 norms but in L∞ neighborhoods of the solution only. Furthermore, in the case of weak local optima with continuous control functions, stability w.r.t. parameter changes usually can be expected to hold in L∞ sense rather than in Lp. Whenever we consider problems with discontinuous optimal control behavior, these results are too restrictive to discuss general variations of the solution including changes in the break points or switches in the active sets. In the paper we show how the use of certain integrated optimality criteria obtained via a duality approach allows for estimates also in the case of discontinuous controls. We consider L2 and L1 quadratic growth estimates and discuss consequences for the behavior of minimizing sequences. 1 Local optimality criteria in integrated form Consider first a general nonlinear constrained optimal control problem (primal problem formulation): (P) min J(x, u) = k(x(0), x(T )) + ∫ T 0 r(t, x(t), u(t)) dt s.t. ẋ = f(t, x(t), u(t)) a.e. in [0, T ], (1) β(x(0), x(T )) = 0, (2) g(t, x(t), u(t)) ≤ 0 a.e. in [0, T ] . (3) The pair (x, u) ∈ W 1 ∞(0, 1; IR)×L∞(0, 1; IR) is called admissible for (P) if the state equation (1) (including the initial condition (2)) together with the inequality constraints (3) (where g : [0, T ] × IR × IR → IR, β : [0, T ] × IR × IR → IR) is