Intractable problems in control theory

This paper is an attempt to understand the apparent intractability of problems in decentralized decision-making, using the concepts and methods of computational complexity. We first establish that the discrete version of an important paradigm for this area, proposed by Witsenhausen, is NP-complete, thus explaining the failures reported in the literature to attack it computationally. In the rest of the paper we show that the computational intractability of the discrete version of a control problem (the team decision problem in our particular example) can imply that there is no satisfactory (continuous) algorithm for the continuous version. To this end, we develop a theory of continuous algorithms and their complexity, and a quite general proof technique, which can prove interesting by themselves.