Modeling of laser beam control systems using projections onto constraint sets

Traditional linear control theory has only limited utility in the description of near-term new approaches for laser beam control for propagation through turbulence. In this paper, a sample of the foundation work providing an alternate framework for analysis of the stability of a laser beam control system is described. An important class of laser beam control systems are well described by a pair of systems coupled by generalized projections onto constraint sets - where the constraint sets can be convex or non-convex, and the resultant generalized projection operators linear or nonlinear. The theory of sequential projections onto constraint sets is modified to construct a theory of simultaneous projections onto constraint sets, which is suitable for analysis of the interactions of systems with components whose behavior is described by generalized projection operators. The basic element of the theory of simultaneous projections onto constraint sets is reviewed. Numerical examples related to control of laser beams for propagation through turbulence are given.