# The regular indefinite linear quadratic optimal control problem: stabilizable case

## TL;DR

This paper extends the theory of indefinite linear quadratic optimal control to stabilizable systems, providing explicit solutions and conditions for optimal control existence in a more general setting.

## Contribution

It generalizes previous controllable case results to stabilizable systems, explicitly characterizing the unique algebraic Riccati equation solution.

## Key findings

- Explicit characterization of the unique Riccati solution.
- Necessary and sufficient conditions for optimal control existence.
- Extension of control theory to stabilizable systems.

## Abstract

This paper addresses an open problem in the area of linear quadratic optimal control. We consider the regular, infinite-horizon, stability-modulo-a-subspace, indefinite linear quadratic problem under the assumption that the dynamics are stabilizable. Our result generalizes previous works dealing with the same problem in the case of controllable dynamics. We explicitly characterize the unique solution of the algebraic Riccati equation that gives the optimal cost and optimal feedback control, as well as necessary and sufficient conditions for the existence of optimal controls.