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

We discuss the equivalence of bi-H/sup /spl infin// control problems to certain problems of approximation and interpolation by analytic functions in several complex variables. In bi-H/sup /spl infin// control, the goal is to perform H/sup /spl infin// control design for a plant where part of it is known and a stable subsystem /spl delta/ is not known, i.e. the response at "frequency" s is P(s, /spl delta/(s)). We assume that once our system is running, we can identify /spl delta/ online. Thus the problem is to design a function K off-line that uses this information to produce a H/sup /spl infin// controller via the formula K(s, /spl delta/(s)). The controller should yield a closed loop system with H/sup /spl infin// gain at most /spl gamma/ no matter which /spl delta/ occurs. This is a frequency domain problem. The article shows how several bi-H/sup /spl infin// 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/sup /spl infin// 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/sup /spl infin// cases these lead to a necessary and sufficient treatment of the control problem.