Forty years ago in 1968, Hans S. Witsenhausen published ``A Counterexample in Stochastic Optimum Control''. This paper showed that linear controllers are not always optimal for linear systems, even when the usual simplifying assumptions of quadratic cost and Gaussian noise are made. This was a true watershed that has driven vast amounts of research in the decades since.

The paper made this point with a stunningly simple counterexample - a two-stage control problem, where the controller at the first stage acts on a random initial state, and the second controller acts on a corrupted subsequent state. The subtle difference between this and a standard LQG problem, for which optimal controllers would indeed be linear, is that the second controller cannot access the information that the first controller can access. This usual implicit assumption of centralized control, or of a classical information structure, was thus shown to be crucial to the applicability of the major successes of LQG control, such as separation of estimation and control, and linear optimality. Furthermore, the best linear controller was nowhere close to achieving what nonlinear controllers could. The fact that so much standard theory broke down, and broke down badly, for even the simplest possible decentralized / nonclassical problem, made it painfully clear that much research would be needed in this area. It also opened up a plethora of interesting fundamental questions regarding when linear controllers are optimal.

The time is ripe to focus on the role of the Witsenhausen Counterexample in our understanding of decentralized control, not only because it is a milestone year for it, but also because there has been a recent resurgence of activity and fundamental results in this area, partially driven by the rapidly increasing need to understand the convergence of communication and control.

This session gathers people who have contributed enormously to our understanding of the issues raised by the Witsenhausen Counterexample, and people who have answered some of the most fundamental questions emanating from the Witsenhausen Counterexample, to commemorate its 40th anniversary by giving their perspectives on decentralized control and linear optimality, taking stock of what we have learned since, discussing the important problems which remain open, and presenting new results on the topic.