Dissipative noncommutative multidimensional linear systems and robust control theory

It is well known that any contractive analytic operator-valued function on the unit disk can be realized as the transfer function of a dissipative discrete-time input/state/output linear system and conversely. We review analogues of this observation for the context of multidimensional linear systems and multivariable transfer functions. We emphasize the so-called noncommutative case where the transfer function is actually a formal power series in noncommuting indeterminates and survey recent applications of this case to problems in robust control. We conclude with suggestions for directions for future work.