Near-optimal controls: deterministic case

Near-optimization is as sensible and important as optimization for both theory and applications. It is the ultimate purpose of this series of papers to establish a unified and in-depth theory for dynamic near-optimization, or near-optimal controls, and to apply the theory to real-world systems that cannot be otherwise settled by existing approaches. In this paper, systems governed by ordinary differential equations are considered. Necessary and sufficient conditions of near-optimal controls with any given error bound are derived in terms of near-maximum conditions of the Hamiltonian. The relationship among the adjoint functions, the value functions, and the Hamiltonian along near-optimal trajectories are investigated by using dynamic programming and viscosity solution approach. Verification theorems with which near-optimal feedback controls can be constructed are obtained