Sufficiency for Purely Essentially Bounded Optimal Controls

Abstract

For optimal control problems of Bolza with variable and free end-points, nonlinear dynamics, nonlinear isoperimetric inequality and equality restrictions, and nonlinear pointwise mixed time-state-control inequality and equality constraints, sufficient conditions for strong minima are derived. The algorithm used to prove the main theorem of the paper includes a crucial symmetric inequality, making this technique an independent self-contained method of classical concepts such as embedding theorems from ordinary differential equations, Mayer fields, Riccati equations, or Hamilton−Jacobi theory. Moreover, the sufficiency theory given in this article is able to detect discontinuous solutions, that is, solutions which need to be neither continuous nor piecewise continuous but only essentially bounded.

Keywords

• calculus of variations
• optimal control
• symmetric inequalities
• isoperimetric and mixed constraints
• inequality and equality constraints
• free end-points
• sufficiency
• strong minima
• purely essentially bounded optimal controls