Applications of nonautonomous infinite-dimensional systems control theory for parabolic PDEs

Abstract

Parabolic partial differential equations (PDEs) are used as models of transport-reaction phenomena in a variety of different industrial chemical and materials engineering processes, and can yield precise descriptions of process variables with complex temporal and spatially dependent system dynamics. In many cases, the process dynamics are also affected by time-dependent features of the system which arise from the underlying physical characteristics of the process or the methods utilized in the formation and treatment of materials which may result in phase transitions, deformations or a combination of these behaviours. The dynamical analysis of these processes provides a fundamental basis for development of model based control strategies through a number of approaches including from within the framework of infinite-dimensional systems control theory. However, each class of transport-reaction system presents its own unique challenges and requires the development of new strategies within the existing framework.

The focus of this thesis is the systematic treatment and realization of the feedback control design for two general classes of problems. The first class deals with the optimal boundary control problem for unstable parabolic PDEs with nonautonomous and nonhomogeneous infinite-dimensional system representation, and is considered within the context of a lithium-ion battery thermal regulation problem. The key challenges addressed include the time-dependence of system parameters, system instability, the restriction of the input along a portion of the battery domain boundary, the observer based optimal boundary control design, and the realization of the outback feedback control problem based on state measurement and interpolation methods. The second class of problems is the optimal distributed and boundary control of parabolic PDEs on time-varying spatial domains with nonautonomous infinite-dimensional system representation. The key challenges addressed include the development of an appropriate function space setting to handle the time-dependence of the spatial domain, the formulation of the infinite-dimensional system representation of the PDE control problem within this function space setting, and the realization of the optimal distributed and boundary control problems within the context of the Czochralski crystal temperature stabilization problem.