[Paper Review] Feedback stabilization and boundary controllability of the Korteweg-de Vries equation on a star-shaped network

Kaïs Ammari, Emmanuelle Crépeau|arXiv (Cornell University)|Jun 16, 2017

Advanced Mathematical Physics Problems12 references24 citations

TL;DR

This paper studies the feedback stabilization and boundary controllability of the Korteweg-de Vries (KdV) equation on a star-shaped network of N edges. It establishes well-posedness, proves exponential decay of energy for the dissipative system, and demonstrates local exact boundary controllability using (N+1) controls at the external and central nodes under non-critical length conditions.

ABSTRACT

We propose a model using the Korteweg-de Vries $(KdV)$ equation on a finite star-shaped network. We first prove the well-posedness of the system and give some regularity results. Then we prove that the energy of the solutions of the dissipative system decays exponentially to zero when the time tends to infinity. Lastly we show an exact boundary controllability result.

Motivation & Objective

• To model and analyze the Korteweg-de Vries (KdV) equation on a star-shaped network for applications in arterial hemodynamics.
• To establish well-posedness and regularity of the KdV system on such networks, ensuring mathematical consistency.
• To prove exponential decay of the system's energy under a dissipative boundary condition at the central node.
• To achieve exact boundary controllability of the linearized and nonlinear KdV systems on the network using (N+1) boundary controls.

Proposed method

• Formulates the KdV equation on a star-shaped network with N edges, each of finite length ℓj, using a system of coupled PDEs with continuity and Kirchhoff-type conditions at the central node.
• Defines the natural energy E(t) = (1/2)∑‖uj(t,⋅)‖L²(0,ℓj)² and shows it is non-increasing due to the dissipative term −(α − N/2)|u₁(t,0)|².
• Applies an observability inequality for the adjoint system to prove controllability, leveraging the Hilbert Uniqueness Method (HUM).
• Uses a fixed-point argument to extend local controllability results from the linearized system (LKdV) to the nonlinear KdV system.
• Imposes boundary conditions: zero Dirichlet and Neumann conditions at external nodes, and a nonlinear or linear condition at the central node involving the second derivative sum.
• Requires non-critical network lengths, i.e., at most one edge length in the set 𝒩 of critical lengths, to ensure observability.

Experimental results

Research questions

• RQ1 Can the Korteweg-de Vries equation be well-posed and regular on a star-shaped network of finite edges?
• RQ2 Does the energy of the KdV system on a star-shaped network decay exponentially under a dissipative boundary condition at the central node?
• RQ3 Is the linearized KdV system on a star-shaped network exactly controllable using boundary controls at the external nodes and the central node?
• RQ4 Can local exact controllability be extended to the nonlinear KdV system on the same network structure?

Key findings

• The KdV system on a star-shaped network is well-posed in the space L²(𝒯) with solutions possessing sufficient regularity under appropriate initial data.
• The energy E(t) of the dissipative system decays exponentially to zero as t → ∞, provided α > N/2.
• An observability inequality holds for the adjoint system: ‖φᵀ‖²_{L²(𝒯)} ≤ C(∑‖∂ₓφⱼ(⋅,ℓⱼ)‖²_{L²(0,T)} + ∫₀ᵀ φ₁²(t,0) dt), valid for T > 0 and non-critical lengths.
• Exact boundary controllability is achieved for the linearized system (LKdV_control) when at most one edge length is critical.
• Local exact controllability is established for the nonlinear KdV system (KdV_control) in a small neighborhood of the origin, with initial and target states in L²(𝒯) with small norm.
• The controllability result relies on (N+1) controls: one at the central node and one at each external node, with the possibility of reducing the number in future work.