The regular free-endpoint linear quadratic problem with indefinite cost

This paper studies an open problem in the context of linear quadratic optimal control, the free-endpoint regular linear quadratic problem with indefinite cost functional. It is shown that the optimal cost for this problem is given by a particular solution of the algebraic Riccati equation. This solution is characterized in terms of the geometry on the lattice of all real symmetric solutions of the algebraic Riccati equation as developed by Willems [IEEE Traps. Automat. Control, 16 (1971), pp. 621–634] and Coppel [Bull. Austral. Math. Soc., 10 (1974), pp. 377–401]. A necessary and sufficient condition is established for the existence of optimal controls. This condition is stated in terms of a subspace inclusion involving the extremal solutions of the algebraic Riccati equation. The optimal controls are shown to be generated by a feedback control law. Finally, the results obtained are compared with "classical" results on the linear quadratic regulator problem.