The mathematical models of many industrial transport-reaction processes are obtained from conservation laws, such as mass, momentum and/or energy in the form of partial differential equations (PDEs). In some applications, such processes involve the change in the shape of the material and domain of analysis as a result of phase change, chemical reaction, external forces, and mass transfer. It is well established that, the dominant behavior of dissipative PDE systems can be approximately described by finite-dimensional systems. The reduced-order model of a parabolic PDE system can be obtained by Galerkin’s method with the use of the eigenfunctions of the spatial differential operator. However, the analytic solution to the operator eigenvalue problem does not exist in general, examples are nonlinear operators or PDEs with nontrivial geometry. A well-known methodology in the extraction of eigenfunctions of these types of problems is the use of Karhunen-Loeve (KL) decomposition on an ensemble of solutions obtained from numerical or experimental resolution of the system. These modes, known as empirical eigenfunctions, are used in the derivation of accurate reduced-order approximations of many diffusion-reaction systems and fluid flows.
There are few studies on the order-reduction and control of PDE systems with spatially time-varying domain compared to the extensive research efforts on the order-reduction of distributed parameter systems with fixed domains. Assuming that the evolution of domain is known (which can be measured in many processes) KL decomposition cannot be directly applied to the solutions of PDEs with time-varying domain. Armaou and Christofides used a mathematical transformation to represent the nonlinear PDE on an appro- priate time-invariant domain and applied KL decomposition to obtain the set of eigenfunctions on the fixed domain [1,2]. In the study of the internal combustion engine flows by Fogleman et al., the velocity fields are stretched in one dimension to obtain data on a fixed grid such that the divergence of the original velocity field (continuity) is preserved . Following these contributions, one way to deal with the aforementioned problem is to map the set of the solutions on a time-invariant domain and then apply KL decomposition, however, different mappings could change the energy content of the solutions. The idea here is to map the solutions of the PDE system on a fixed reference geometry while preserving the invariance of physical properties (energy) of the solutions.
In this work, we find the control law to stabilize the heat-equation on a two-dimensional spatially time-varying domain. The set of the solutions of the PDE describing the system behavior is mapped to a fixed reference configuration while preserving the invariant property of thermal energy. A function basis can be found by using the KL decomposition on the mapped solutions, and by applying the inverse mapping, a set of time-varying empirical eigenfunctions are obtained that capture the most energy of the system. Subsequently, the empirical eigenfunctions are used as a basis for Galerkin’s method to derive the reduced-order ODE model that accurately captures the dominant dynamics of the PDE system. The reduced-order model is in the form of a linear time-varying system and the LQR control synthesis is considered.
. A. Armaou, P. D. Christofides, Nonlinear Feedback Control of Parabolic Partial Differential Equation Systems with Time-dependent Spatial Domains, J. Math. Anal. Appl. 239 (1), 1999, 124-157.
. A. Armaou, P. D. Christofides, Finite-Dimensional Control of Nonlinear Parabolic PDE Systems With Time-Dependent Spatial Domains Using Empirical Eigenfunctions, Int. J. Appl. Math. Comput. Sci. 11(2), 2001, 287-317.
. M. Fogleman, J. Lumley, D. Rempfer, D. Haworth, Application of the Proper Orthogonal Decomposition to Datasets of Internal Combustion Engine Flows, J. Turbul. 5, 2004, 023.