# The Adaptive LQ Regulator

## TL;DR

This paper investigates the optimal adaptive control of linear systems with infinite horizon LQ costs, identifying conditions for certainty equivalence and adaptive stabilizability.

## Contribution

It characterizes when certainty equivalence holds and when linear systems are adaptively stabilizable, advancing understanding of adaptive control in partially observed systems.

## Key findings

- Certainty equivalence property holds for specific partially observed systems.
- A linear system is adaptively stabilizable iff it is uniformly stabilizable.
- Identifies classes where the certainty equivalence value function satisfies Bellman equation.

## Abstract

The optimal adaptive control of a linear system in a signal-plus-noise setting with infinite horizon LQ regulator cost is studied. The class of partially observed linear systems for which the certainty equivalence property holds is identified. It is also shown that a linear system is adaptively stabilizable if and only if it is uniformly stabilizable, and the class of partially observed linear systems for which the certainty equivalence value function is a supersolution of the Bellman equation is identified.