[Paper Review] Finite-time stabilization in optimal time of homogeneous quasilinear hyperbolic systems in one dimensional space

Hoài-Minh Nguyên, Jean‐Michel Coron|arXiv (Cornell University)|May 4, 2020

Stability and Controllability of Differential Equations21 references25 citations

TL;DR

This paper presents time-independent feedback controls that achieve finite-time stabilization in optimal time for homogeneous quasilinear hyperbolic systems in one space dimension with one-sided controls and nonlinear boundary conditions at the other end. By introducing auxiliary dynamics and leveraging characteristic methods, the authors establish local well-posedness and prove that solutions vanish identically in time T > T_opt for sufficiently small initial data, achieving optimal controllability time T_opt = max{τ_k+1, τ_k + τ_m+1} under a non-degeneracy condition on the boundary map ∇B(0).

ABSTRACT

We consider the finite-time stabilization of homogeneous quasilinear hyperbolic systems with one side controls and with nonlinear boundary condition at the other side. We present time-independent feedbacks leading to the finite-time stabilization in any time larger than the optimal time for the null controllability of the linearized system if the initial condition is sufficiently small. One of the key technical points is to establish the local well-posedness of quasilinear hyperbolic systems with nonlinear, non-local boundary conditions.

Motivation & Objective

• To achieve finite-time stabilization in the optimal time T_opt for quasilinear hyperbolic systems with one-sided controls.
• To establish local well-posedness for quasilinear hyperbolic systems with nonlinear, non-local boundary conditions.
• To construct time-independent feedback laws that stabilize the system in any time T > T_opt for small initial data.
• To prove optimality of the time T_opt by showing it cannot be reduced further under the given conditions.
• To extend the backstepping approach to quasilinear systems with nonlinear boundary dynamics.

Proposed method

• Introduces auxiliary dynamics (4m additional state variables) to satisfy compatibility conditions at x=1 without imposing them on the original system.
• Constructs feedback laws recursively using characteristic curves, starting from the rightmost variable and propagating backward through time delays.
• Employs a characteristic method to trace the influence of boundary states backward in time to determine the required feedback at x=1.
• Uses a nonlinear feedback structure involving functions M_i that depend on the state at points determined by characteristic trajectories.
• Implements dynamic extensions for the feedback variables ζ_j and η_j via ODEs with nonlinear damping terms to ensure smoothness and convergence.
• Applies a continuity argument and implicit function theorem to ensure the existence of initial conditions for the auxiliary dynamics.

Experimental results

Research questions

• RQ1 Can finite-time stabilization in the optimal time be achieved for quasilinear hyperbolic systems with nonlinear boundary conditions?
• RQ2 What conditions on the boundary map B ensure that the optimal time T_opt is achievable via time-independent feedback?
• RQ3 How can compatibility conditions at x=1 be satisfied without imposing them directly on the original system?
• RQ4 What is the role of auxiliary dynamics in enabling well-posedness and finite-time stabilization?
• RQ5 Can the backstepping framework be extended to quasilinear systems with non-local, nonlinear boundary conditions?

Key findings

• For any T > T_opt, there exist time-independent feedback controls that stabilize the system in finite time T, provided the initial data are sufficiently small.
• The optimal time T_opt is given by T_opt = max{τ_k+1, τ_k + τ_m+1} when m ≥ k, and T_opt = max{τ_k+1−m + τ_k+1, ..., τ_k + τ_k+m} when m < k.
• The feedback laws are well-defined and lead to C1-solutions of the closed-loop system, ensuring finite-time stability.
• The construction relies on auxiliary dynamics to satisfy compatibility conditions at x=1, avoiding direct imposition on the original state.
• The system achieves exact null state in time T > T_opt, with w(T, ·) = 0 for all x ∈ [0,1], under the condition that ∇B(0) ∈ B.
• The optimality of T_opt is confirmed by showing that no smaller time allows stabilization under the given assumptions.