In this note, we discuss the equivalence of what we refer to as bi-H-infinity control problems to certain problems of approximation and interpolation by analytic functions in several complex variables. We now introduce what we mean by bi-H-infinity control. The goal is to perform an H-infinity control design for a plant where part of it is known and a stable subsystem delta is not known, that is, the response of the plant at "frequency" s is P(s, delta(s)). We assume that once our control (closed loop) system is running, we can identify the subsystem delta on line. Thus the problem is to design a function K offline that uses this information to produce a H-infinity controller via the formula K(s, delta(s)). The challenge is to pick K so that the controller yields a closed loop system with H-infinity gain at most gamma no matter which delta occurs. This is entirely a frequency domain problem, which has the flavor of some types of gain scheduling or control which adapts to slow variations. The bulk of this article is devoted to showing how several bi-H-infinity 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-infinity 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-infinity cases these lead to a necessary and sufficient treatment of the control problem.