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 locations based on the 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 the Kalman filter, is considered as the dual problem to the linear-quadratic optimal problem of the control locations. Further, the existence and convergence of optimal locations of observations for improving the estimation of the initial state, based on the Kalman smoother, is discussed. The obtained results are applied to a linear advection-diffusion model with a special extension of emission rates.