Lowest-rank solutions of continuous and discrete Lyapunov equations over symmetric cone

The low-rank solutions of continuous and discrete Lyapunov equations are of great importance but generally difficult to compute in control system analysis and design. Fortunately, Mesbahi and Papavassilopoulos (1997) [30] showed that with the semidefinite cone constraint, the lowest-rank solutions of the discrete Lyapunov inequality can be efficiently solved by a linear semidefinite programming. In this paper, we further show that the lowest-rank solutions of both the continuous and discrete Lyapunov equations over symmetric cone are unique and can be exactly solved by their convex relaxations, the symmetric cone linear programming problems. Therefore, they are polynomial-time solvable. Since the underlying symmetric cone is a more general algebraic setting which contains the semidefinite cone as a special case, our results also answer an open question proposed by Recht, Fazel and Parrilo (2010) [36].