Linear time-invariant solutions for LQ optimal control problems with terminal-state affine constraints

We study the problem of steering the state of a system from a given initial condition towards a prescribed affine set while minimizing a quadratic cost functional. While the optimal solution is defined in terms of a time-varying openloop control law, herein the problem is solved by limiting the search for the optimal input in the space of linear time-invariant feedback control laws. This choice preserves the LTI nature of the original plant in closed loop. Within this framework, the solution of the underlying optimal control problem hinges upon the solution of a nonlinear constrained optimization problem. Constructive algorithms and comparison with the time-varying optimal control law are discussed.