The Continuous-Time Singular LQR Problem and the Riddle of Nonautonomous Hamiltonian Systems: A Behavioral Solution

In this article, we deal with the continuous-time singular linear quadratic regulator (LQR) problems, which give rise to nonautonomous Hamiltonian systems. This case arises when the system's transfer function matrix is not left-invertible. A special case of this problem can be solved using the constrained generalized continuous algebraic Riccati equation (CGCARE), when a certain condition on the input-cardinality of the Hamiltonian is satisfied. However, this condition is only a special case among many other possible cases. On the other hand, singular LQR problems with autonomous Hamiltonian systems have been well studied in the literature. In this article, we apply behavioral theoretic techniques to show that the general case of the singular LQR problem with nonautonomous Hamiltonian can be solved by a direct sum decomposition of the plant behavior, where one of the direct summands can be solved via CGCARE, while the other gives rise to an autonomous Hamiltonian system.