Christofides and P Daoutidis, "Finite-dimensional control of parabolic PDE systems using approximate inertial manifolds," Journal of Mathematical Analysis and Applications, vol.

Christofides, "Robust control of parabolic PDE systems," Chemical Engineering Science, vol.

Christofides, "Analysis and control of

parabolic PDE systems with input constraints," Automatica, vol.

2.2.1 Elliptic and Parabolic PDE Solution Frameworks.

Nonlinear Elliptic Sequential DE Solution Segment Description Declarations, Options Space for saving solution(s) Equation, CBs DE problem definition Grid/Mesh Domain discretization Triple Initial guess Fortran Newton loop start Linearized Elliptic Solver Elliptic problem discretization, indexing, solution Output Format for solution output Fortran Convergence test Fortran Newton loop end Subprograms Initial guess, Jacobian, and other support functions Similarly, there is a framework for implementing semidiscrete parabolic PDE solvers which utilizes the available PELLPACK elliptic PDE solvers.

The main focus of this paper is to study the current profile tracking of tokamak plasma which can be modeled by a parabolic PDE. In our previous work, we applied PDE-constrained optimization techniques to compute plasma discharge sequences minimizing a given cost function [7].

Thus, by defining a derivation dynamic system, we can formulate a feedback control problem of the derivation system which is governed by a linear parabolic PDE system.

In Section 3, we employ the optimal QSC method for the space discretization of parabolic PDE, which results in an ODEs system.

In this section, we take the following parabolic PDE as an example to present the advantages of the QSC-PDC algorithm:

In this situation, the study of robust [H.sub.[infinity]] control design for nonlinear parabolic PDE systems is of theoretical and practical importance.

Wang, Wu, and Li have established sufficient conditions of distributed exponential stabilization via simple fuzzy proportional state feedback controllers for first-order hyperbolic PDE systems [15-17] and via a fuzzy proportionalspatial derivative (P-sD) for semi-linear parabolic PDE system [18].

The fourth - order

parabolic PDEs of the form Equations arise in the physical phenomena of undamped transverse vibrations of a flexible straight beam in such a way that its support does not contribute to the strain energy of the system (Shahid and Arshed 2013).In equation (1) represents the transverse displacement of the beam and are time and displacement variables respectively and is dynamic driving force per unit mass.