Control of Unstable Heat Equation On Two-Dimensional Time-Varying Domain Using Empirical Eigenfunctions

Many industrial transport-reaction nonlinear 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. The mathematical models of such processes are obtained from conservation laws, such as mass, momentum and/or energy, and usually form nonlinear partial differential equations (PDEs).

The approach of modal decomposition, which implies that the dominant behavior of PDE systems can be approximately described by finite-dimensional systems, is extensively used for the controller synthesis of PDE systems. A well-known methodology in the extraction of eigenfunctions of nonlinear PDEs 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 nonlinear reduced-order approximations of many diffusion-reaction systems and fluid flows.

Compared to the extensive research efforts on the order-reduction of distributed parameter systems modelled by PDEs, there are only few studies to address model-reduction and control of PDE systems with spatially time-varying domain. Assuming that the evolution of domain is known a priori, 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 appropriate 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 [3]. 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. 

To find the control law to stabilize the unstable steady-state of a nonlinear heat-equation on a two-dimensional spatially time-varying domain, we first map a set of the solutions of the nonlinear PDE describing the system behavior to a reference configuration while preserving the invariant property of thermal energy. A 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 method to derive the reduced-order ODE model that accurately captures the dominant dynamics of the PDE system. Finally, the ODE system is used for the synthesis of nonlinear controller.

References:

1. 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.

2. 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.

3. 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.