The Riccati Equation for Optimal Control Problems with Mixed State-Control Constraints: Necessity and Sufficiency

The goal of this paper is to conduct a complete study of second-order conditions for the optimal control problem with mixed state-control constraints. The conjugate point theory is presented and a necessary condition in terms of the corresponding Riccati equation is obtained. Sufficiency criteria are developed in terms of strengthened necessary conditions, including the Riccati equation. The results generalize the known ones for pure control constraints as well as for the mixed state-control constraints.