Concepts of Scattering in the Linear Optimal Control Problem

The scattering formulation characterising the effects of incident and reflected waves in a physical flow process which includes a series of obstacles is discussed. Similar concepts can be applied in the linear optimal control problem which includes terminal constraints and the defining linear and quadratic matrix differential equations are shown to be analogous in both cases. The control problem is also shown to include a series of " obstacles" which can be interconnected according to the properties of the star product which forms the basis of the scattering representation of the general physical system. The scattering matrix of electrical network theory and the associated power relations are also shown to exist analogously in the linear optimal regulator problem. The resulting "incident" and "reflected" variables possess similar properties to the scattering variables of network theory and the particular form of scattering matrix is shown to be closely related to the matrix solution of the steady-state Riccati equation. More general energy considerations are extended to the optimal control problem and discussed in terms of inequality properties of a scattering-type matrix and a particular solution of the Riccati equation is shown to be associated with the property defining the state of dissipation in the physical system.