Witsenhausen counterexample

Witsenhausen's counterexample, shown in the figure below, is a deceptively simple toy problem in decentralized . It was formulated by Hans Witsenhausen in 1968. It is a counterexample to a natural conjecture that one can generalize a key result of centralized linear–quadratic–Gaussian control systems—that in a system with linear dynamics, Gaussian disturbance, and quadratic cost, affine (linear) control laws are optimal—to decentralized systems. Witsenhausen constructed a two-stage linear quadratic Gaussian system where two decisions are made by decision makers with decentralized information and showed that for this system, there exist nonlinear control laws that outperform all linear laws. The problem of finding the optimal control law remains unsolved.

The statement of the counterexample is simple: two controllers attempt to control the system by attempting to bring the state close to zero in exactly two time steps. The first controller observes the initial state ${\displaystyle x_{0}.}$ There is a cost on the input ${\displaystyle u_{1}}$ of the first controller, and a cost on the state ${\displaystyle x_{2}}$ after the input of the second controller. The input ${\displaystyle u_{2}}$ of the second controller is free, but it is based on noisy observations ${\displaystyle y_{1}=x_{1}+z}$ of the state ${\displaystyle x_{1}}$ after the first controller's input. The second controller cannot communicate with the first controller and thus cannot observe either the original state ${\displaystyle x_{0}}$ or the input ${\displaystyle u_{1}}$ of the first controller. Thus the system dynamics are

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