  The choice of the location of controllers and observations is of great
importance for designing control systems and improving the estimations in
various practical problems. For time-varying systems in Hilbert spaces, the
existence and convergence of the optimal location based on linear-quadratic
control on a finite-time horizon is studied. The optimal location of
observations for improving the estimation of the state at the final time, based
on Kalman filter, is considered as the dual problem to the LQoptimal problem of
the control locations. Further, the existence and convergence of optimal
locations of observations for improving the estimation at the initial time,
based on Kalman smoother is discussed. The obtained results are applied to a
linear advection-diffusion model.
