Group Invariance and Symmetries in Nonliear Control and Estimation
J. S. Baras
Proceedings of the Second Nonlinear Control Network (NCN) Workshop, Paris, France, June 5-9, 2000.
We consider nonlinear filtering problems, nonlinear robust control problems and the partial differential equations that characterize their solutions. These include the Zakai equation, and in the robust control case two coupled Dynamic Programming equations. We then characterize equivalence between two such problems when we can compute the solution of one from the solution of the other using change of dependent, independent variables and solving an ordinary differential equation. We characterize the resulting transformation groups via their Lie Algebras. We illustrate the relationship of these results to symmetries and invariances in physics, Noether's theorem, and calculus of variations. We show how using these techniques one can solve nonlinear problems by reduction to linear ones.