Optimal Control of Autonomous Linear Processes with Singular Matrices in the Quadratic Cost Functional

The optimal control of the autonomous linear process $\dot x = Ax + Bu$ with quadratic cost functional is studied. The initial and terminal times and positions are fixed. The matrices in the cost functional are allowed to be singular. An assumption, weaker than invertibility, is placed on the coefficient matrices. Under this assumption, necessary and sufficient conditions are given for the existence of an optimal control in terms of the initial and final position of the process. A closed form for the optimal control is given.