Some Partial Differential Equations Arising in Nonlinear Control and Optimization

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Recently, a combination of methods drawn from geometric nonlinear control theory and from nonlinear dynamics was developed to give a local soLution to the nonlinear regulator problem, yielding necessary and sufficient conditions for nonlinear regulation for the class of detectable and stabilizable nonlinear systems[1][2]In section 2, we review the basic nonlinear regulator problem and give conditions, derived in[1] [2] [2], for solvability of the problem in terms of the solvability of a system of nonlinear partial differential equations. In the linear case, these “regulator equations” coincide with the linear equations derived by Francis [3]in his rather complete treatment of the linear multivariable regulator problem. In the nonlinear case, solvability of the regulator equations can also be cast in terms of the solvability of a more intrinsic “Sylvester PDE”, for which stable, unstable and center manifold theory provides a local existence theory. In the linear case, this Sylvester equation was studied by Hautus [7] who proved that well-posedness is equivalent to the assertion that no system Transmission zero be a natural frequency of the exosystem which produces signals to be tracked or disturbanced to be attenuated, a rather intuitive frequency domain criterion. In an example, we ilLustrate the fact that, for an interesting class of problems, the controller designed by this method has the familiar form of a feedback and a feedforward term, both of which depend however on the “off-line” soLution of the Sylvester PDE.