Local Sufficient Conditions

Abstract

We briefly mention an important topic of optimal control theory which is strictly connected to the necessary conditions framework and is a natural complement of it, namely the sufficient conditions issue. As in the (at least conceptually) related field of minimization of functions, we can distinguish between global and local conditions. Some of the most significant results for global conditions stem from the Hamilton–Jacoby theory: however they are not mentioned in the present treatment as they are rooted in a somehow unrelated context. On the contrary, we focus the attention on local conditions resulting from a variational approach closely related to the one underlying the achievements of the Maximum Principle. Indeed if we require that, once the first variation of the performance index is zero, its second variation is positive, then we end up with local sufficient conditions. We illustrate them with reference to four significant scenarios.