Optimal adaptive control of an LQG system

We consider the problem of adaptively controlling a linear system so as to minimize a long-term average quadratic cost criterion. It is well known that certainty equivalent controllers based on standard parameter estimators run into an identifiability problem which leads to a strictly suboptimal performance. In this contribution, a cost-biased parameter estimator is introduced to overcome this difficulty. The corresponding adaptive scheme is proven to be stable and optimal when the unknown system parameter lies in an infinite, yet compact, parameter set