Deterministic Near-Optimal Controls. Part II: Dynamic Programming and Viscosity Solution Approach

Near-optimization is as sensible and important as optimization for both theory and applications. This paper concerns dynamic near-optimization, or near-optimal controls, for systems governed by deterministic ordinary differential equations, and uses dynamic programming to study the near-optimality. Since nonsmoothness is inherent in this subject, the viscosity solution approach is employed to investigate the problem. The dynamic programming equation is derived in terms of ϵ-superdifferential/subdifferential. The relationships among the adjoint functions, the value functions, and the Hamiltonian along near-optimal trajectories are revealed. Verification theorems with which near-optimal feedback controls can be constructed are obtained.