Finite-time stabilization in optimal time of homogeneous quasilinear hyperbolic systems in one dimensional space

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.

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Finite-time stabilization of weak solutions for a class of non-local Lipschitzian stochastic nonlinear systems with inverse dynamics

Automatica ◽

10.1016/j.automatica.2018.07.015 ◽

2018 ◽

Vol 98 ◽

pp. 285-295 ◽

Cited By ~ 1

Author(s):

Gui-Hua Zhao ◽

Jian-Chao Li ◽

Shu-Jun Liu

Keyword(s):

Nonlinear Systems ◽

Weak Solutions ◽

Finite Time ◽

Inverse Dynamics ◽

Stochastic Nonlinear Systems ◽

Non Local ◽

Finite Time Stabilization

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Semilinear Hyperbolic Systems with Singular Non-Local Boundary Conditions: Reflection of Singularities and Delta Waves

Zeitschrift für Analysis und ihre Anwendungen ◽

10.4171/zaa/1036 ◽

2001 ◽

Vol 20 (3) ◽

pp. 637-659 ◽

Cited By ~ 1

Author(s):

Irina Kmit ◽

Günther Hörmann

Keyword(s):

Boundary Conditions ◽

Hyperbolic Systems ◽

Local Boundary ◽

Delta Waves ◽

Non Local ◽

Semilinear Hyperbolic Systems

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Null controllability and finite-time stabilization in minimal time of one-dimensional first-order 2×2 linear hyperbolic systems

ESAIM Control Optimisation and Calculus of Variations ◽

10.1051/cocv/2021091 ◽

2021 ◽

Author(s):

Long Hu ◽

Guillaume Olive

Keyword(s):

Finite Time ◽

Hyperbolic Systems ◽

Null Controllability ◽

Convolution Theorem ◽

Backstepping Method ◽

Technical Point ◽

One Dimensional ◽

First Order ◽

Minimal Time ◽

Finite Time Stabilization

The goal of this article is to present the minimal time needed for the null controllability and finite-time stabilization of one-dimensional first-order 2×2 linear hyperbolic systems. The main technical point is to show that we cannot obtain a better time. The proof combines the backstepping method with the Titchmarsh convolution theorem.

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The exceptional condition and unboundedness of solutions of hyperbolic systems of conservation type

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500017996 ◽

1977 ◽

Vol 77 (1-2) ◽

pp. 1-8 ◽

Cited By ~ 2

Author(s):

A. Jeffrey

Keyword(s):

Finite Time ◽

Space Dimension ◽

Hyperbolic Systems ◽

General System ◽

Quasilinear Hyperbolic Systems ◽

Characteristic Field ◽

One Space Dimension

SynopsisThis paper is concerned with quasilinear hyperbolic systems of conservation type and establishes two main results. The first is that when a general system is considered in one space dimension and time, then the exceptional nature of a characteristic field implies the coincidence of a shock with one of the characteristics of that field. The second result involves the demonstration by example that quasilinear hyperbolic systems of conservation type may possess solutions that become unbounded after only a finite time, even though they are exceptional.

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