Sufficiency and sensitivity for nonlinear optimal control problems on time scales via coercivity

Abstract

The main focus of this paper is to develop a sufficiency criterion for optimality in nonlinear optimal control problems defined on time scales. In particular, it is shown that the coercivity of the second variation together with the controllability of the linearized dynamic system are sufficient for the weak local minimality. The method employed is based on a direct approach using the structure of this optimal control problem. The second aim pertains to the sensitivity analysis for parametric control problems defined on time scales with separately varying state endpoints. Assuming a slight strengthening of the sufficiency criterion at a base value of the parameter, the perturbed problem is shown to have a weak local minimum and the corresponding multipliers are shown to be continuously differentiable with respect to the parameter. A link is established between (i) a modification of the shooting method for solving the associated boundary value problem, and (ii) the sufficient conditions involving the coercivity of the accessory problem, as opposed to the Riccati equation, which is also used for this task. This link is new even for the continuous time setting.