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 LQ optimal 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 advectiondiffusion model.

In many areas of finance and of risk management it is interesting to know how to specify time-dependent correlation matrices. In this work we propose a new methodology to create valid time-dependent instantaneous correlation matrices, which we called correlation flows. In our methodology one needs only an initial correlation matrix to create these correlation flows based on isospectral flows. The tendency of the time-dependent matrices can be controlled by requirements. An application example is presented to illustrate our methodology.

In predictive geophysical model systems, uncertain initial values and model parameters jointly influence the temporal evolution of the system. This renders initial-value-only optimization by traditional data assimilation methods as insufficient. However, blindly extending the optimization parameter set jeopardizes the validity of the resulting analysis because of the increase of the ill-posedness of the inversion task. Hence, it becomes important to assess the potential observability of measurement networks for model state and parameters in atmospheric modelings in advance of the optimization. In this paper, we novelly establish the dynamic model of emission rates and extend the transport-diffusion model extended by emission rates. Considering the Kalman smoother as underlying assimilation technique, we develop a quantitative assessment method to evaluate the potential observability and the sensitivity of observation networks to initial values and emission rates jointly. This benefits us to determine the optimizable parameters to observation configurations before the data assimilation procedure and make the optimization more efficiently. For high-dimensional models in practical applications, we derive an ensemble based version of the approach and give several elementary experiments for illustrations.