Stochastic \bfitvarepsilon -optimal linear quadratic adaptation: An alternating controls policy

This paper presents a continuous time stochastic linear quadratic (LQ) adaptive control algorithm for completely observed linear stochastic systems with unknown parameters. Based on a certainty equivalence approach, we propose to utilize an alternating controls policy, whereby the linear feedback matrix is switched between two \surd \varepsilon -apart distinct matrices K_i, i = 1, 2. The associated adaptive estimation algorithm is designed so that it drives the maximum likelihood based estimate into the sets \scrI _i, i = 1, 2, and consequently into \scrI ₁ \cap \scrI ₂, with \scrI _i corresponding to the true closed loop dynamics under the ith control. A mild geometric assumption is shown to guarantee that \scrI ₁ \cap \scrI ₂ = \theta ^\ast , the true parameter. This strongly consistent estimation, coupled with the alternating controls policy, then yields \varepsilon -optimal long-run LQ closed loop performance.