[Paper Review] Bilateral Boundary Control Design for a Cascaded Diffusion-ODE System Coupled at an Arbitrary Interior Point

Stephen Chen, Rafael Vázquez+1 more|arXiv (Cornell University)|Jun 12, 2019

Stability and Controllability of Differential Equations20 references4 citations

TL;DR

This paper presents a bilateral boundary control design for a cascaded diffusion-ODE system coupled at an arbitrary interior point using a folding transformation and backstepping. By transforming the system into a target system with exponential stability via Volterra transformations, the authors establish stabilizing boundary controllers that ensure $L^2 \times \mathbb{R}^n$ stability through Lyapunov analysis and existence of bounded kernels.

ABSTRACT

We present a methodology for designing bilateral boundary controllers for a class of systems consisting of a coupled diffusion equation with an unstable ODE at an arbitrary interior point. A folding transformation is applied about the coupling point, transforming the system into an ODE with an input channel consisting of two coupled diffusive actuation paths. A target system with an exponentially stable trivial solution in the sense of L^2 X R^n is proposed, and the stability property is shown via the Lyapunov method. The stabilizing control laws are formulated via tiered Volterra transformations of the second kind, establishing an equivalence relation between the stable target system and the original plant under boundary feedback. Stability properties of the plant under feedback is inferred from the equivalence relation. The well-posedness of the backstepping transformations involved are studied, and the existence of bounded Volterra kernels is shown, constituting a sufficient condition for the invertibility of the Volterra transformations.

Motivation & Objective

• To address the stabilization of a parabolic PDE coupled with an unstable ODE at an arbitrary interior point, a problem not fully resolved under unilateral control.
• To extend the folding framework to bilateral boundary control, enabling two-point actuation for enhanced controllability and robustness.
• To design a stabilizing feedback law using infinite-dimensional backstepping with Volterra transformations of the second kind.
• To prove the existence and invertibility of Volterra kernels through fixed-point arguments and kernel boundedness.
• To establish exponential stability of the closed-loop system in the $L^2 \times \mathbb{R}^n$ sense using Lyapunov methods.

Proposed method

• Apply a folding transformation about the interior coupling point to convert the original system into a cascaded PDE-ODE configuration with two diffusive actuation paths.
• Construct a target system with a trivial solution that is exponentially stable in the $L^2 \times \mathbb{R}^n$ norm using Lyapunov-based analysis.
• Design stabilizing control laws via tiered Volterra transformations of the second kind, establishing equivalence between the original plant and the stable target system under boundary feedback.
• Prove well-posedness of the backstepping transformations by showing existence and boundedness of Volterra kernels using the Schauder fixed point theorem and Weierstrass M-test.
• Establish uniform convergence of kernel sequences through recursive bounds and kernel norm estimates, ensuring invertibility of the transformation.
• Verify regularity of the solution to the ODE component via convolution with smooth matrix exponentials, ensuring $C^\infty$ regularity of the transformation kernels.

Experimental results

Research questions

• RQ1 How can bilateral boundary control be designed for a PDE-ODE system coupled at an interior point, where traditional unilateral control is insufficient?
• RQ2 Can the folding transformation technique be extended to enable bilateral control in 1-D cascaded diffusion-ODE systems?
• RQ3 What conditions ensure the existence and invertibility of Volterra kernels in the backstepping design for such systems?
• RQ4 How can exponential stability in the $L^2 \times \mathbb{R}^n$ norm be proven for the closed-loop system using Lyapunov methods?
• RQ5 What is the role of the ODE’s coupling point in shaping the structure and controllability of the transformed system?

Key findings

• The proposed bilateral control design achieves exponential stability of the trivial solution in the $L^2 \times \mathbb{R}^n$ norm via a Lyapunov-based analysis of the target system.
• The existence of bounded Volterra kernels is proven using the Schauder fixed point theorem, with uniform convergence established via the Weierstrass M-test.
• The kernel sequence for the transformation is shown to converge uniformly in $C(\mathcal{T})$, ensuring the invertibility of the backstepping transformation.
• The solution to the ODE component is shown to be $C^\infty([0,1])$, resulting from convolution with a smooth matrix exponential, which acts as a mollifier.
• The transformation maps the original unstable system to a stable target system, and stability of the original plant under feedback is inferred from the equivalence relation.
• The method provides a systematic framework for stabilizing a class of sandwiched systems where an ODE is coupled between two parabolic PDEs at an interior point.