[Paper Review] Finite-Time Stabilization Of Systems Of Conservation Laws On Networks

Vincent Perrollaz, Lionel Rosier|arXiv (Cornell University)|Feb 23, 2013

Stability and Controllability of Differential Equations22 references3 citations

TL;DR

This paper presents a finite-time stabilization strategy for 1-D quasilinear hyperbolic systems in diagonal form on networks using non-Lipschitz boundary feedback laws. It proves that solutions with small Lipschitz-continuous initial data reach the origin in finite time—approximately $1/c$ for both boundaries controlled, and roughly $2/c$ when only one is controlled—via a fixed-point argument and energy estimates.

ABSTRACT

We investigate the finite-time boundary stabilization of a 1-D first order quasilinear hyperbolic system of diagonal form on [0,1]. The dynamics of both boundary controls are governed by a finite-time stable ODE. The solutions of the closed-loop system issuing from small initial data in Lip([0,1]) are shown to exist for all times and to reach the null equilibrium state in finite time. When only one boundary feedback law is available, a finite-time stabilization is shown to occur roughly in a twice longer time. The above feedback strategy is then applied to the Saint-Venant system for the regulation of water flows in a network of canals.

Motivation & Objective

• To establish finite-time stabilization for 1-D quasilinear hyperbolic systems in diagonal form on networks.
• To extend the finite-time extinction property of the wave equation to more general hyperbolic systems with Riemann invariants.
• To design boundary feedback laws based on finite-time stable ODEs that ensure rapid convergence to equilibrium.
• To analyze the stabilization of the Saint-Venant system for water flow regulation in canal networks.
• To prove existence, uniqueness, and finite-time decay of solutions under minimal regularity assumptions on initial data.

Proposed method

• Transform the quasilinear hyperbolic system into diagonal form using Riemann invariants $u$ and $v$.
• Apply non-Lipschitz boundary feedback laws: $\frac{d}{dt}u(t,0) = -K \text{sgn}(u(t,0))|u(t,0)|^\gamma$ and similarly for $v(t,1)$, with $K>0$, $\gamma \in (0,1)$.
• Use a fixed-point argument based on Schauder's theorem to prove existence and uniqueness of solutions in the space of Lipschitz continuous functions.
• Apply energy estimates to control the growth of solutions and ensure finite-time decay.
• Extend the result to networked systems via induction on tree-structured canal networks, using compatibility conditions at junctions.
• Utilize an extension operator $\Pi$ to extend solutions periodically and preserve Lipschitz regularity for flow estimates.

Experimental results

Research questions

• RQ1 Can finite-time stabilization be achieved for 1-D quasilinear hyperbolic systems in diagonal form with non-Lipschitz boundary feedback?
• RQ2 What is the minimal time required for solutions to reach zero under such feedback laws, and how does it depend on system parameters?
• RQ3 How does finite-time stabilization behave when only one boundary is controlled versus both?
• RQ4 Can this strategy be extended to networked systems, such as a network of canals governed by the Saint-Venant equations?
• RQ5 What conditions on initial data and system structure ensure finite-time convergence to equilibrium?

Key findings

• Solutions to the closed-loop system with both boundaries controlled exist globally in time and vanish in finite time, approximately $T \approx 1/c$, for small initial data in $\text{Lip}([0,1])$.
• When only one boundary is controlled, finite-time stabilization occurs in roughly twice the time, $T \approx 2/c$, due to wave reflection and propagation delay.
• The finite-time stabilization is achieved via a non-Lipschitz feedback law of the form $\dot{u} = -K \text{sgn}(u)|u|^\gamma$, which ensures finite-time convergence of the ODE dynamics at the boundary.
• The existence and uniqueness of solutions are established using Schauder's fixed-point theorem and energy estimates in the space of Lipschitz continuous functions.
• The method extends to tree-structured canal networks by induction on the depth of the network, ensuring finite-time decay at each junction.
• The Saint-Venant system for open-channel flow is shown to be finite-time stabilizable using this feedback strategy, enabling rapid regulation of water levels and flows.