The method of multiple maxima and optimality conditions for singular extremals

Abstract

We consider optimal control problems that can have singular or sliding modes as solutions or parts of solutions [1; 2, p. 26]; in these modes, the Pontryagin function is independent of some control variables or has a nonunique maximum on the set of control variables. For such problems with specific degeneration properties [3, p. 47], the classical optimality conditions (if they apply at all) are noneffective: the Lagrange–Pontryagin necessary conditions are satisfied and can produce infinitely many spurious solutions, while the sufficient conditions fail to hold. One needs effective conditions, which are lacking in the theory yet. The aim of the present paper is to fill the gap to a certain extent by suggesting a reasonably complete set of necessary and sufficient local optimality conditions obtained as a generalization of the classical second-order conditions in terms of the corresponding Clebsch and Riccati inequalities and the maximum principle. To this end, we use a special construction (known as the method of multiple maxima [3–5]) of the Krotov function in the Krotov sufficient conditions.