Abstract
In this paper, we show that singular LQR problems with zero input-cost cannot be solved using static state-feedback controllers. To this end we first show that for such problems the corresponding constrained generalized continuous algebraic Riccati equation (CGCARE) is not solvable. This is achieved by establishing that the Hamiltonian system in such a case does not admit a transfer function which is identically zero. Further, we also show that, unlike the multi-input case which admits both autonomous and non-autonomous Hamiltonian systems, a single-input singular LQR problem always admits an autonomous Hamiltonian system.
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