Reduced basis methods for the analysis, simulation, and control of noncoercive parabolic systems
O'Connor, Robert Gerard; Grepl, Martin Alexander (Thesis advisor); Stamm, Benjamin (Thesis advisor); Volkwein, Stefan (Thesis advisor)
Dissertation / PhD Thesis
In this thesis we present new methods for the analysis, simulation, and control of parameter-dependent parabolic problems. These methods are all closely related to the reduced basis method and greatly reduce the computational cost of solving various types of problems for a large number of parameter values. The main advantage of our methods is that they can handle systems with non coercive operators. Previous reduced basis methods for the optimal control of parabolic systems were only valid for coercive systems. We present the first method that can also handle non coercive problems. That is done by extending space-time methods that were proposed for the simulation of parabolic problems. We also introduce the use of Lyapunov's stability theory in reduced basis modeling. This opens many new possibilities in the area of control and systems theory. To make such applications possible we present an extension of the successive constraint method to linear matrix inequalities. Such inequalities play an essential role in many applications involving Lyapunov stability. The first method that we present allows for many new applications but is limited in that the decision variable needs to be low dimensional. We then show how that method can be extended and used in constructing Lyapunov functions for non coercive systems.As a final application we demonstrate how Lyapunov functions can be used to derive reduced basis error bounds for non coercive parabolic systems. Such error bounds are often more cost efficient than space-time bounds and benefit from a more accurate understanding of the system's underlying dynamics.