Sliding Mode Boundary Control for Heat Equations with Uncertain Dynamic Actuators

The present study focuses on designing a boundary controller for an uncertain heat equation with a dynamic actuator, leading to a coupled PDE-ODE representation of the system’s dynamics. Our goal is to regulate the state of the PDE to a constant set-point, despite the presence of uncertainties in both the diffusivity parameter and the actuator time constant. The proposed design is based on sliding mode control and guarantees pointwise-in-space exponential stabilization of the closed-loop system in the second-order Sobolev space H<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup>(0,1). A Lyapunov analysis is conducted to demonstrate the stability characteristics of the closed-loop system and numerical simulation results are provided to corroborate the effectiveness of the proposed controller. The simulations also confirm the rejection of smooth time-varying disturbances acting at the controlled boundary, specifically between the actuator process and the controlled PDE boundary.