On information structures, convexity, and linear optimality

In 1968, Witsenhausen introduced his celebrated counterexample, which illustrated that when an information pattern is nonclassical, the controllers which optimize an expected quadratic cost may be nonlinear. For the special invited session commemorating the fortieth anniversary of the counter-example, we address one of the four follow-up questions listed in his original paper; namely, whether there is a relation between the convexity of finding the optimal affine controller, and whether that controller is in fact optimal. In particular, we discuss the connections between partially nested structures, for which linear controllers are known to be optimal, and quadratically invariant structures, for which optimal linear control is known to be convex.