Sufficiency for singular trajectories in the calculus of variations

Abstract

For calculus of variations problems of Bolza with variable end-points, nonlinear inequality and equality isoperimetric constraints and nonlinear inequality and equality mixed pointwise constraints, sufficient conditions for strong minima are derived. The main novelty of the new sufficiency results presented in this article concerns their applicability to cases in which the derivatives of the extremals to be optimal solutions are not necessarily continuous nor piecewise continuous but only essentially bounded and they do not necessarily satisfy the standard strengthened Legendre condition but only the corresponding necessary condition.

Keywords

• calculus of variations
• inequality and equality isoperimetric constraints
• inequality and equality mixed pointwise constraints
• free end-points
• sufficiency
• strong minima
• singular extremals