A Note on Damped Algebraic Riccati Equations∗

In a recent paper, an algorithm that produces dampening controllers based on a periodic Hamiltonian was proposed. Central to this algorithm is the formulation of symmetric and skew-symmetric damped algebraic Riccati equations. It was shown that solutions to these two Riccati equations lead to a dampening feedback, i.e., a stable closed–loop system for which the real parts of the eigenvalues are larger in modulus than the imaginary parts. In this paper, we extend these results to include a broader class of Hermitian and skew-Hermitian solutions and show that every convex combination of these solutions produces a dampening feedback. This property can be used to vary the feedback with two parameters and thus obtain more flexibility in the controller design process.