Strict Dissipativity for LQ Problem with Left-invertible Underlying Semigroups

The paper analyzes the strict dissipativity property of generalized linear-quadratic infinite dimensional optimal control systems. For dynamics described by left-invertible semigroups, we first characterize exponential detectability of the system in terms of an analytic condition. Then, under a stabilizability assumption, we establish the equivalence beween strict dissipativity and exponential detectability of the system.