Simultaneous state and input reachability for linear time invariant systems

In this paper, we give an explicit solution to the behavioral reachability problem for linear time invariant systems, which amounts to finding an explicit control law that reaches a given final input-state pair (u1, x1) in a given finite time t1. We first tackle the case of state space realizations, and we then extend the obtained results to the case of implicit realizations. For this, we use the geometric approach and some results of the viability theory. Some complements are given about the existing relationships between reachability and pole placement, as well as some notions of unicity and existence of solution.