Class Of Nonlinear Reaction-diffusion Systems Research Articles

An interpolatory directional splitting-local discontinuous Galerkin method with application to pattern formation in 2D/3D

An efficient computational method to approximate the solution of a general class of nonlinear reaction-diffusion systems in Cartesian grids is presented. The proposed scheme uses the Local Discontinuous Galerkin (LDG) method as spatial discretization and the symmetric Strang operator splitting as time marching scheme. As a result, not only diffusion and reaction are decoupled; but the reaction sub-step reduces to a set of completely independent small local nonlinear systems of ordinary differential equations. To reduce their computational cost a Lagrange interpolatory technique is used. Diffusion sub-steps are approximated by a directional splitting method derived from an algebraic decomposition of the discrete Laplacian operator, which amounts to solve sets of completely independent small block tridiagonal linear systems; one set per direction. Stability of the diffusive step is analyzed. Finally the efficiency and accuracy of the proposed scheme is numerically investigated on well known Turing pattern formation models in 2D/3D.

Intermittent distributed control for a class of nonlinear reaction-diffusion systems with spatial point measurements

This paper addresses the intermittent distributed stabilization problems of a class of nonlinear reaction-diffusion systems, where the state measurements are available at a finite number of sampling spatial points. By developing piecewise switching-time-dependent Lyapunov function methods combined with the descriptor system approach, L2-norm and H1-norm stability criteria are established, respectively. For the case of Dirichlet boundary condition, it is shown that the H1-norm stability implies the pointwise-in-space stability. The obtained stability conditions establish a quantitative relationship among the upper bound of the spatial sampling intervals, control width, and rest width. In the framework of linear matrix inequalities, sufficient conditions for the existence of intermittent distributed static output-feedback controllers are derived. A comparison between the two stability criteria is given by means of a numerical example.

Exponential attractors for abstract parabolic systems with bounded delay

We show that a suitable adaptation of the so-called method of trajectories can be used to construct an exponential attractor for a very general class of nonlinear reaction-diffusion systems with a bounded delay.In particular, we assume that the dependence on the past history is controlled via convolution with a possibly singular measure. Assuming a priori that the solutions are bounded, a simple proof of the existence of an exponential attractor is given under very little regularity requirements.

Distributed Control Design for Parabolic Evolution Equations: Application to Compressor Stall Control

We present a theoretical study of distributed control design for semilinear parabolic nonlocal evolution equations with an application to axial compressor stall control using air injection. By taking advantage of the spatial invariance of the equations, a linear controller is constructed (following Bamieh et al.) via linear quadratic control of each Fourier mode. We derive sufficient conditions for the linear controller to stabilize the full nonlinear system. Concepts such as controller decentralization, finite-dimensional implementation, inverse-optimality, and beneficial nonlinearities are discussed. In the second part of this paper, these developments are applied to a model of axial compressor control with air injection. The unactuated model is derived following the work of Moore and Greitzer and the model coupled with air injection actuation follows the works of Behnken et al. and Weigl et al. A numerical study of the control designs is pursued and the comparison of controller performance is discussed. The techniques presented here are expected to be useful for distributed control design of a much broader class of nonlinear reaction-diffusion systems.

Automatic control problems for reaction-diffusion systems

A class of nonlinear reaction-diffusion systems is considered. We formulate some automatic control problems based on feedback devices located on the boundary. Two different types of devices are analyzed: relay switch and Preisach hysteresis operator. The resulting models lead to a nonlinear integrodifferential parabolic system with nonlinear and nonlocal boundary conditions. We prove global existence and uniqueness of solutions in both the cases considered.

Monotone method and convergence acceleration for finite‐difference solutions of parabolic problems with time delays

AbstractIn this article we study a finite‐difference system, which is a discrete version of a class of nonlinear reaction‐diffusion systems with time delays. Existence‐comparison and uniqueness theorem are first established for the discretized problem by the method of upper and lower solutions. A monotone iterative scheme is also developed for the solution of the finite‐difference system. Under a convergence acceleration scheme, it is shown that the monotone sequences converge quadratically to the solution of the finite‐difference system. At last, numerical results on some model problems are demonstrated to substantiate our theorems. © 1995 John Wiley & Sons, Inc.