Constrained nonlinear control of parabolic PDE systems

This paper focuses on the development of a general framework for analysis and control of parabolic PDE systems with manipulated input constraints. Initially, a nonlinear model reduction scheme based on combination of Galerkin's method with the concept of approximate inertial manifold is employed for the derivation of low-order ODE systems that yield solutions very close to those of the PDE system. These ODE systems are used as the basis for the explicit construction of bounded nonlinear optimal feedback controllers, via Lyapunov techniques, that enforce stability and provide an explicit characterization of the limitations imposed by input constraints on the allowable control actuator locations. Precise conditions that guarantee stability of the closed-loop parabolic PDE system in the presence of input constraints are provided. The proposed analysis and controller synthesis results are used to stabilize an unstable steady-state of a diffusion-reaction process.