Hamiltonian Trajectories and Duality in the Optimal Control of Linear Systems with Convex Costs

A duality theorem is proved for problems of optimal control of linear dynamical systems in continuous time subject to linear constraints and convex costs, such as penalties. Optimality conditions are stated in terms of a “minimaximum principle” in which the primal and dual control vectors satisfy a saddle point condition at almost every instant of time. This principle is shown to be equivalent to a generalized Hamiltonian differential equation in the primal and dual state variables, along with a transversality condition which likewise is in Hamiltonian form.