Abstract
In this letter, we show that a constrained generalized continuous algebraic Riccati equation (CGCARE), arising in the context of a singular infinite-horizon linear quadratic regulator (LQR) problem, is generically 1 unsolvable. We achieve this result in three steps: first, we provide a set of necessary and sufficient conditions for the solvability of a CGCARE. Then, using these conditions, we show that the determinant of a certain matrix pencil being identically zero is necessary for a CGCARE to be solvable. Finally, we show that this necessary condition is generically false. It is well-known that CGCARE solvability is a necessary and sufficient condition for a singular infinite-horizon LQR problem to admit a solution that is implementable as a static state-feedback control law. Thus, our main result reveals that a singular infinite-horizon LQR problem generically disallows solution by a static state-feedback law.
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