Small-Gain-Based Boundary Feedback Design for Global Exponential Stabilization of 1-D Semilinear Parabolic PDEs
For control theorists working on PDE stabilization, this provides a novel design framework with fundamental limitations, but the results are theoretical and incremental in nature.
This paper develops a small-gain-based methodology for designing boundary feedback stabilizers that achieve global exponential stabilization for 1-D semilinear parabolic PDEs with nonlinearities satisfying a linear growth condition, including nonlocal terms. Two types of stabilizers are constructed, and it is shown that arbitrary gain assignment is not possible via boundary feedback.
This paper presents a novel methodology for the design of boundary feedback stabilizers for 1-D, semilinear, parabolic PDEs. The methodology is based on the use of small-gain arguments and can be applied to parabolic PDEs with nonlinearities that satisfy a linear growth condition. The nonlinearities may contain nonlocal terms. Two different types of boundary feedback stabilizers are constructed: a linear static boundary feedback and a nonlinear dynamic boundary feedback. It is also shown that there are fundamental limitations for feedback design in the parabolic case: arbitrary gain assignment is not possible by means of boundary feedback. An example with a nonlocal nonlinear term illustrates the applicability of the proposed methodology.