Adaptive Design for Estimation of Unknown Parameters in Linear Systems

We consider a linear autonomous system of differential equations, where the coefficients depend on an unknown parameter $\theta$, and the input signal has a constrained specific power. We observe the solution perturbed by an additive white noise. In this case we study the asymptotic (for large observation time) design problem of the input signal, which gives us the optimal estimator of $\theta$. We suggest an adaptive algorithm for the design and an asymptotically optimal estimator of $\theta$ with respect to the quadratic risk.