Oblique projections for state stabilization and estimation

The project is concerned with the state stabilization and state estimation for partial differential equations. Such equations are known to model real world phenomena, including state dynamics of populations, heat, fluids velocity, and traffic flow. The design of a robust stabilizing feedback controller is important, because such controller plays a crucial role in the suppression of instabilities that can occur in the dynamics of the state modeled by the equation. A feedback controller is essentially an operator that computes the appropriate control, at a given instant of time, from the knowledge of state of the equation at the same instant of time. Thus, even if we have such stabilizing feedback at our disposal we still need the state to compute the control. This shows the importance of designing a state estimator for our equation. We are interested in the case the control (the input) is a linear combination a finite number of actuators at our disposal, and the observation (the output) is the set of measurements received from a finite number of sensors. We are particularly interested in the case of nonautonomous systems (time-varying dynamics). Though, the expected results will hold for autonomous systems as well. We look for stabilizing feedback controls and state estimators (observers) which are simple as possible to implement in applications (e.g., numerical simulations). In particular, we will investigate the class of explicit oblique projection based feedback controllers which has been proposed recently for stabilization of parabolic-like equations. The explicitness of such feedback makes it easy to compute numerically, at least when compared to the numerical computation of a classical Riccati based feedback, or to the numerical computation of a Hamilton- Jacobi-Bellman based feedback. The project aims to contribute with new stabilizability results for other types of models as wave-like equations. Another aim is the construction of explicit oblique projection based Luenberger-type observers for estimation of the full state of parabolic-like and wave-like equations. The theoretical findings will be accompanied by their validation through the results of numerical simulations.

This project contributed with results on the stabilization of mathematical models for real-world evolution phenomena. We may think of models for the evolving state of populations, of heat, of fluid velocity, and of traffic flow. For example, the state y(t) could be the temperature in a room at time t>0, and we would like to stabilize it towards a given targeted value Y(t) as time t increases. Stabilizability: The stabilization "y(t)-converges-to-Y(t)" is achieved by feedback inputs, namely, once we know the state y(t) at a given time t, we look for an input as u(t)=K(y(t)-Y(t)) depending on the difference to the targeted state Y(t). This input is then fed back into the system by tuning a set of actuators at our disposal. For example, the input could contain the values to set the temperature in radiators (actuators) placed in the room. Results were derived for a general class of models. A particularity of the contributions is that the feedback-input operator K is given explicitly, can be straightforwardly implemented in numerical simulations, and can be computed in real time, which are important features for applications. Furthermore, we found operators K the design of which do not require exact knowledge of the model, depending only on suitable norms of the terms involved in the model. This means that the proposed design of K is robust against some model uncertainties. This is also an important feature for applications because mathematical models are usually just approximations for the dynamics of real-world evolution processes. Detectability: Once we find such a stabilizing feedback-input operator K, the computation of the input u(t) requires the knowledge of the state y(t) at time t. This is not possible in many applications; for example, we cannot measure the temperature y(t)=y(x,t) at every single point x in a room. So, in applications we will need an estimate z(t) for y(t), and then we will use the (approximated) input u(t)=K(z(t)-T(t)) instead. The stimates z(t) were constructed by designing dynamic observers, consisting of a copy of the model for the controlled dynamics plus an extra forcing/correction term of the form L(W(z)-w), where w=W(y) is the output of sensor measurements, for example, the output W(y(t)) of the partial measurements, of the state y(t) at time t, made by thermostats placed at some locations within the room. Dynamic observers providing estimates so that "z(t)-converges-to- y(t)" as time increases were obtained for a general class of models, with the output-injection operator L given explicitly. Numerical validation: The stabilizing performance of the proposed feedback-input operator K and the detecting/estimating performance of the output-injection operator L were validated by numerical simulations for parabolic-like models, including named models as the Schloegl, the Kuramoto-Sivashinsky, and the Cahn-Hilliard equations.