This research uses the central solution for the commutant lifting theorem along with some state space techniques to solve mixed L2 - Hoo control problems. Many classical linear quadratic control problems can be converted to an L2 optimization problem. It is well known that these L2 or linear quadratic controllers may not be robust. To get around this problem many researchers turned to Hoo optimization. Although, the Hoo controllers are robust, they produce systems with a large bandwidth. This research will use the commutant lifting theorem to find controllers which exhibit the appropriate trade off between the standard L2 and Hoo controllers. By solving certain mixed L2 - Hoo (four block) problems this work will obtain robust controllers with strong linear quadratic performance. The proofs are based on the central solution for the commutant lifting theorem. In the rational case the L2 - Hoo controllers will be computed by using standard state space techniques. In the nonrational case these controllers will be computed by using the commutant lifting theorem, along with some skew-Toeplitz techniques. This work will also be extended to certain nonlinear systems involving Volterra series.
