Examples of Constructing Exact Reachable Sets for Infinite Dimensional Control System

Exact time-dependent reachable sets for dynamic control systems are natural control theory extensions of the exact solutions of Cauchy problem for differential equations. The reachable sets are important in many applications, including optimal control, when analytical expressions are possible. The extensions are usually based on differential-geometric approaches that enable effective methods of analysis and design of nonlinear finite-dimensional control systems. Although a theoretical development for infinite-dimensional systems, including PDEs, requires significantly more advanced mathematics, the analytical technique is of a comparable level of complexity to the finite-dimensional case. The purpose of this paper is to present a set of examples illustrating techniques of obtaining exact analytical expressions of the reachable sets for important classes of linear and nonlinear PDEs: 1-st order, hyperbolic, and parabolic.