Inertia theorems for the periodic Lyapunov difference equation and periodic Riccati difference equation

Abstract The periodic Lyapunov difference equation (PLDE) and periodic Riccati difference equation (PRDE) are dealt with. The inertia (i.e., the number of positive, null, and negative eigenvalues) of any symmetric periodic solution of such equations is linked with the pattern of eigenvalues of the monodromy matrix associated with the open-loop (for PLDE) or closed-loop (for PRDE) underlying systems. Different results are obtained by imposing requirements with decreasing strength to the original system. Precisely, assumptions of observability, reconstructibility, and detectability are successively introduced. Some results are also given for the particular case of positive semidefinite periodic solutions.

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