Positional solutions of Hamilton–Jacobi equations in control problems for discrete-continuous systems

We develop a canonical global optimality theory based on operating with the set of solutions for the Hamilton–Jacobi inequalities that parametrically depend on the initial (or final) position. These solutions, called positional $L$-functions (of Lyapunov type), naturally arise in the studies of control problems for discrete-continuous (hybrid, impulse) systems; an important prototype of such problems are classical optimal control problems with general end constraints on the trajectory. We analyze sufficient optimality conditions with this new class of $L$-functions and invert the maximum principle into a sufficient condition for nonlinear problems of optimal impulse control.

Presented by the member of Editorial Board: V. I. Gurman