Some adaptive control problems which convert to a "classical" problem in several complex variables

We discuss the equivalence of bi-H^∞ control problems to certain problems of approximation and interpolation by analytic functions in several complex variables. In bi-H^∞ control, the goal is to perform H^∞ control design for a plant where part of it is known and a stable subsystem δ is not known, i.e. the response at "frequency" s is P(s, δ(s)). We assume that once our system is running, we can identify δ online. Thus the problem is to design a function K off-line that uses this information to produce a H^∞ controller via the formula K(s, δ(s)). The controller should yield a closed loop system with H^∞ gain at most γ no matter which δ occurs. This is a frequency domain problem. The article shows how several bi-H^∞ control problems convert to two complex variable interpolation problems. These precisely generalize the classical (one complex variable) interpolation (AAK-commutant lifting) problems which lay at the core of H^∞ control. These problems are hard, but the last decade has seen substantial success on them in the operator theory community. In the most ideal of bi-H^∞ cases these lead to a necessary and sufficient treatment of the control problem