# Monotonicity Methods for Input-to-State Stability of Nonlinear Parabolic PDEs with Boundary Disturbances

## TL;DR

This paper develops a monotonicity-based approach to analyze input-to-state stability (ISS) of nonlinear parabolic PDEs with boundary disturbances, simplifying stability analysis and demonstrating robustness of boundary controllers.

## Contribution

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

## Key findings

- ISS of boundary-disturbed PDEs is equivalent to ISS of related PDEs with distributed disturbances.
- The method simplifies stability analysis using maximum principles.
- Boundary control of reaction-diffusion equations is robust to actuator 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 \textit{boundary disturbances} is equivalent to ISS of a closely related nonlinear parabolic PDE with constant \textit{distributed disturbances} and zero boundary condition. The last problem is conceptually much simpler and can be handled by means of various recently developed techniques. As an application of our results, we show that the PDE backstepping controller which stabilizes linear reaction-diffusion equations from the boundary is robust with respect to additive actuator disturbances.