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. 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.