Some Insights on Synthesizing Optimal Linear Quadratic Controllers Using Krotov Sufficient Conditions

This letter revisits the problem of synthesizing the optimal control laws for linear systems with a quadratic cost. Traditionally, these laws are computed using the Hamilton-Jacobi-Bellman method, where the solution to the original problem is obtained by solving the Riccati equation, which hinges upon a priori information of the optimal cost function. Within the general Krotov global optimal control framework, though being less explored in the literature, that information is no longer needed. However, utilizing this framework, the original optimization problem is translated into a non-convex problem, which is solved by using iterative methods. In this letter, we propose a new method to compute a direct (non-iterative) solution by transforming the resulting non-convex optimization problem into a convex problem. It turns out that the proposed method naturally leads to the Riccati inequality as the crucial intermediate step, of which the origin was not well understood, although it serves as a strong backbone to address linear quadratic problems and other significant linear system theoretic results. Numerical results and future directions, particularly for solving the optimal control problem for bilinear systems, are also provided to demonstrate the usability of the proposed method.