Optimal semistable control for continuous-time linear systems

In this paper, we develop a new H2H2 semistability theory for linear dynamical systems. Specifically, necessary and sufficient conditions based on the new notion of weak semiobservability for the existence of solutions to the semistable Lyapunov equation are derived. Unlike the standard H2H2 optimal control problem, a complicating feature of the H2H2 optimal semistable control problem is that the semistable Lyapunov equation can admit multiple solutions. We characterize all the solutions using matrix analysis tools. With this theory, we present a new framework to design H2H2 optimal semistable controllers for linear coupled systems by converting the original optimal control problem into a convex optimization problem.