Finite-dimensional methods for computing the information state in nonlinear robust control

In nonlinear output robust control and in nonlinear risk-sensitive partially observed stochastic control, the optimal control is a memoryless function of the information state. The information state dynamics are directly influenced by the control performance metric, thus displaying a direct linkage between control objectives and sufficient statistics for control. It has been observed in several examples that by modifying the control performance metric one can render the dynamics of the information state finite dimensional. In linear robust control, it is well known that the information state can be computed by a simple finite-dimensional formula. Using the Lie theory for transformations of this basic solution, we show how the information state for certain nonlinear control problems can also be obtained using finite-dimensional calculations, e.g. via the solution of systems of ODE. This is explained using fundamental results on the invariance groups of the equations involved. An intuitive interpretation of the significance of the result is also provided.