An LMI optimization approach to Lyapunov stability analysis for linear time-invariant systems

Theorem of alternatives based on the separation theorems are proposed to provide systematic and unified proofs of necessary and sufficient conditions for solvability of Lyapunov inequality on P. As an application of the theorem of alternatives, we investigate the conditions for the existence of feasible solutions to Lyapunov inequality on P and obtain a restatement of the celebrated Lyapunov stability theorem for linear time-invariant systems. Furthermore, the effectiveness of the proposed method is illustrated by a numerical example.