Sufficient Variational Conditions for Isoperimetric Control Problems

For optimal control problems involving isoperimetric constraints, by using a two-norm approach, a new sufficiency theorem for a proper strong minimum is obtained. It is applicable to processes that satisfy the Legendre-Clebsch necessary condition but its strict version is not imposed, that is, the processes may be singular. The conditions are expressed explicitly in terms of the second variation along nonnull admissible variations, as well as the Weierstrass excess functions of the constraints delimiting the problem. The proof we provide corresponds to a generalization, in several respects, of a sufficiency proof due to Hestenes for the fixed-endpoint isoperimetric problem in the calculus of variations.