Optimality and Characteristics of Hamilton-Jacobi-Bellman Equations

We study the Bolza problem arising in nonlinear optimal control and investigate under what circumstances the necessary conditions for optimality of Pontryagin’s type are also sufficient. This leads to the question when shocks do not occur in the method of characteristics applied to the associated Hamilton-Jacobi-Bellman equation. In this case the value function is its (unique) continuously differentiable solution and can be obtained from the canonical equations. In optimal control this corresponds to the case when the optimal trajectory of the Bolza problem is unique for every initial state and the optimal feedback is an upper semicontinuous set-valued map with convex, compact images.