(747e) 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.