Optimal Control with Irregular Performance

In this paper, we solve the long-standing fundamental problem of irregular linear--quadratic (LQ) optimal control, which has received significant attention since the 1960s. We derive the optimal controllers via the key technique of finding the analytical solutions to two different forward and backward differential equations (FBDEs). We give a complete solution to the finite-horizon irregular LQ control problem using a new `two-layer optimization' approach. We also obtain the necessary and sufficient condition for the existence of optimal and stabilizing solutions in the infinite-horizon case in terms of solutions to two Riccati equations and the stabilization of one specific system. For the first time, we explore the essential differences between irregular and standard LQ control, making a fundamental contribution to classical LQ control theory. We show that irregular LQ control is totally different from regular control as the irregular controller must guarantee the terminal state constraint of $P_1(T)x(T)=0$.