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Boundary stabilization of flows in networks of open channels modeled by Saint-Venant equations

Amaury Hayat, Yating Hu, Peipei Shang

Abstract

This work investigates the boundary stabilization of flows in star-shaped and tree-shaped networks of open channels governed by the Saint-Venant equations with a friction term. Due to the existence of the friction term, the steady-states are non-uniform. We show that any such network can be stabilized with only controls at the terminal nodes of the network, even when there are no controls at the nodes inside the network. The number of control is optimal. The main tool we use is the Lyapunov approach, and the main challenge is that the state-of-the-art Lyapunov functions developed for Saint-Venant equations with source terms cannot be used. In this work, we manage to construct a new efficient and explicit Lyapunov function and, in turn, we give explicit ranges of the control tuning parameters that depend only on the values of the given non-uniform steady-states at the ends of the branches. Moreover, this Lyapunov function also improves the existing conditions found in the last decade for a single channel modelled by Saint-Venant equations.

Boundary stabilization of flows in networks of open channels modeled by Saint-Venant equations

Abstract

This work investigates the boundary stabilization of flows in star-shaped and tree-shaped networks of open channels governed by the Saint-Venant equations with a friction term. Due to the existence of the friction term, the steady-states are non-uniform. We show that any such network can be stabilized with only controls at the terminal nodes of the network, even when there are no controls at the nodes inside the network. The number of control is optimal. The main tool we use is the Lyapunov approach, and the main challenge is that the state-of-the-art Lyapunov functions developed for Saint-Venant equations with source terms cannot be used. In this work, we manage to construct a new efficient and explicit Lyapunov function and, in turn, we give explicit ranges of the control tuning parameters that depend only on the values of the given non-uniform steady-states at the ends of the branches. Moreover, this Lyapunov function also improves the existing conditions found in the last decade for a single channel modelled by Saint-Venant equations.
Paper Structure (14 sections, 10 theorems, 186 equations, 2 figures)

This paper contains 14 sections, 10 theorems, 186 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Context
  3. Contribution of this paper
  4. Main Results
  5. The Star-shaped Model
  6. Tree-shaped Network
  7. A side result for a single channel
  8. A Lyapunov Function for the Linearized System of the Star-shaped Model
  9. A Lyapunov Function for the Original Nonlinear System of the Star-shaped Model
  10. A Lyapunov Function for the Linearized System of the Tree-shaped Network
  11. Conclusion and perspectives
  12. Computations for Lemma \ref{['lem1']}
  13. Proof of Remark \ref{['remf']}
  14. Proof of Theorem \ref{['thm:single']}

Key Result

Theorem 2.1

For any boundary controls $\mathscr{B}_j$, $j\in\{2,\cdots,n\}$ satisfying where are all constants depending only on the values of $H^*_j(x)$ at two ends $x=0$ and $x=L_j$. The (steady-state $(H^{*}_{i},V^{*}_{i})$ of the) nonlinear hyperbolic system sys01, bou01 is exponentially stable for the $H^2$-norm .

Figures (2)

  • Figure 1: Star-shaped model of divergent flow.
  • Figure 2: Example diagram of a tree-shaped channels network model. In this example, $2,3,4 \in \mathcal{I}_{out}^{1}$, $\text{deg}(J_{M}^{1})=4$, and $1$, $i \in \mathcal{M}$ and $5\in \mathcal{S}$.

Theorems & Definitions (25)

  • Remark 2.1: friction model
  • Definition 2.1
  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4: Number of controls
  • Theorem 2.2
  • Theorem 2.3
  • Remark 2.5
  • Theorem 3.1
  • ...and 15 more