In this report, we present a new Linear-Quadratic Semistabilizers (LQS) theory for linear network systems. This new semistable H2 control framework is developed to address the robust and optimal semistable control issues of network systems while preserving network topology subject to white noise. Two new notions of semistabilizability and semicontrollability are introduced as a means to connecting semistability with the Lyapunov equation based technique. With these new notions, we first develop a semistable H2 control theory for network systems by exploiting the properties of semistability. A new series of necessary and sufficient conditions for semistability of the closed-loop system have been derived in terms of the Lyapunov equation. Based on these results, we propose a constrained optimization technique to solve the semistable H2 network-topology-preserving control design for network systems over an admissible set. Then optimization analysis and the development of numerical algorithms for the obtained constrained optimization problem are conducted. We establish the existence of optimal solutions for the obtained nonconvex optimization problem over some admissible set. Next, we propose a heuristic swarm optimization based numerical algorithm towards efficiently solving this nonconvex, nonlinear optimization problem. Finally, several numerical examples will be provided.
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