Infinite-Horizon Linear-Quadratic Control by Forward Propagation of the Differential Riccati Equation [Lecture Notes]

One of the foundational principles of optimal control theory is that optimal control laws are propagated backward in time. For linear-quadratic control, this means that the solution of the Riccati equation must be obtained from backward integration from a final-time condition. These features are a direct consequence of the transversality conditions of optimal control, which imply that a free final state corresponds to a fixed final adjoint state [1], [2]. In addition, the principle of dynamic programming and the associated Hamilton-Jacobi-Bellman equation is an inherently backward-propagating methodology [3].