Recoverable and reachable zones for control systems with linear plants and bounded controller outputs

In vector-matrix representation, input constraints for a completely controllable, linear system are imposed by constraining the input vector to belong to a compact, convex subset, Ω, of its vector space. Precise definitions are given for the maximum region of recoverability and the maximum region of teachability. Both regions depend on three parameters; the final or initial state, starting time, and terminal time. General properties of both regions are given. Three theorems are presented. The first gives a necessary and sufficient condition for the maximum region of recoverability to be the whole state space. The second relates the maximum region of recoverability and the maximum region of teachability for constant systems. The third theorem relates the maximum region of recoverability (for infinite travel time) for a "totally unstable" (all eigenvalue real-parts positive) constant system with the maximum region of recoverability (for infinite travel time) for the general constant system. Computational results for the maximum region of recoverability for the constant case are presented.