A Riccati based approach is proposed to solve Linear Quadratic Optimal control problems subject to linear equality path constraints including mixed state-control and state-only constraints. The proposed algorithm requires computations that scale linearly with the horizon length. It can be used as the key sub-problem to build effective iterative methodologies that tackle general inequality constrained and nonlinear optimal control problems.
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Sideris et al. (2010) studied this question.