Abstract
In this letter, we prove that almost every infinite-horizon linear quadratic regulator (LQR) control problem with single-input admits an optimal solution in the form of a feedback that is a suitable constant linear combination of the state and its first derivative—a proportional plus derivative state feedback. The only assumption that we make is that the associated Hamiltonian matrix pencil has no eigenvalues on the imaginary axis. This condition is known to be generically true. Since a regular infinite-horizon LQR problem is known to admit an optimal control in the form of a static state feedback (a proportional feedback), we concentrate in this letter only on singular LQR problems.
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