Input-to-State Stability of Nonlinear Parabolic PDEs with Dirichlet Boundary Disturbances

TL;DR

This paper develops a monotonicity-based approach to analyze input-to-state stability of nonlinear parabolic PDEs with boundary disturbances, simplifying the stability analysis by relating boundary disturbances to distributed ones.

Contribution

It introduces a novel monotonicity method for ISS analysis of nonlinear parabolic PDEs with boundary inputs, linking boundary disturbances to distributed disturbances for easier analysis.

Findings

Monotone control systems are ISS iff ISS with constant inputs.

Nonlinear parabolic equations with boundary disturbances are monotone control systems.

ISS with boundary disturbances reduces to ISS with distributed disturbances.

Abstract

We introduce a monotonicity-based method for studying input-to-state stability (ISS) of nonlinear parabolic equations with boundary inputs. We first show that a monotone control system is ISS if and only if it is ISS w.r.t. constant inputs. Then we show by means of classical maximum principles that nonlinear parabolic equations with boundary disturbances are monotone control systems. With these two facts, we establish that ISS of the original nonlinear parabolic PDE with constant boundary disturbances is equivalent to ISS of a closely related nonlinear parabolic PDE with constant distributed disturbances and zero boundary condition. The last problem is conceptually much simpler and can be handled by means of various recently developed techniques.