LQ optimal control for infinite-dimensional passive systems

We study the Linear-Quadratic optimal control problem for a general class of infinite-dimensional passive systems, allowing for unbounded input and output operators. We show that under mild assumptions, the finite cost condition is always satisfied. Moreover, we give an explicit bound on the norm of the optimal cost operator. In the case where the system is energy preserving, the unique optimal control is given together with the corresponding optimal cost operator. In this case, we derive an explicit solution to an adapted operator Riccati equation. We apply our results to boundary control systems, first-order port-Hamiltonian systems and an Euler–Bernoulli beam with shear force control.