Second-Order Optimality Conditions for Variable End Time Terminal Control Problems

Abstract

A set of sufficient conditions for a weak minimum is derived for the nonsingular Bolza problem of variational calculus, expressed in control notation. The initial state and time are assumed fixed; the terminal state and time are variable, subject to a set of equality constraints. Specified terminal time is considered as a special case. The sufficient conditions derived are minimal sets for normal problems. The existence of a matrix determined by backward integration of a Riccati equation takes the place of the classical conditions involving conjugate points or focal points. This condition is easier to implement than the classical conditions in optimization problems requiring numerical solution. During the backward integration, feedback gains for deviations in the state variables and terminal constraints are easily computed for use in a neighboring optimum control law. The conditions derived are shown to be easier to apply, more concise, and more generally valid than several other recently proposed sets of optimality conditions.