A Riccati equation approach to the standard infinite-dimensional H∞ problem

We solve the standard (four-block) H problem for regular well-posed linear systems (in the sense of George Weiss). The state space H , the disturbance space W , the control space U , the regulated output space Z, and the measurement output space Y are all Hilbert spaces of finite or infinite dimension. Our main result is an infinite-dimensional version of the following standard result: there exist a dynamic controller which feeds the measured output y into the control input u, makes the closed loop system exponentially stable, and also makes the norm of the mapping from the external disturbance w to the regulated output z less than a predefined constant g # 0 if and only if two algebraic Riccati equations have exponentially stabilizing solutions P X and P Y , respectively, and the spectral radius of P X P Y is less that g . Another equivalent condition which is given in terms of two nested Riccati equations is available as well. Finally, we establish a generalized version of the standard parameterization of all stabilizing solutions. The exact formulation varies depending on the regularity assumptions that we make, but our assumptions allow for roughly twice as much unboundedness of the control and observation operators as the Pritchard--Salamon class does, and they permit a countable number of pure delays in the input/output responses. Analogous discrete time results are valid as well