scholarly journals On boundary stability of inhomogeneous 2 × 2 1-D hyperbolic systems for the C1 norm

2019 ◽  
Vol 25 ◽  
pp. 82 ◽  
Author(s):  
Amaury Hayat
Keyword(s):  
Lyapunov Function ◽  
Exponential Stability ◽  
Source Term ◽  
Hyperbolic Systems ◽  
Practical Interest ◽  
Quasilinear Hyperbolic Systems ◽  
Subcritical Regime ◽  
Saint Venant Equations ◽  
Boundary Controls ◽  
Boundary Stability

We study the exponential stability for the C1 norm of general 2 × 2 1-D quasilinear hyperbolic systems with source terms and boundary controls. When the propagation speeds of the system have the same sign, any nonuniform steady-state can be stabilized using boundary feedbacks that only depend on measurements at the boundaries and we give explicit conditions on the gain of the feedback. In other cases, we exhibit a simple numerical criterion for the existence of basic C1 Lyapunov function, a natural candidate for a Lyapunov function to ensure exponential stability for the C1 norm. We show that, under a simple condition on the source term, the existence of a basic C1 (or Cp, for any p ≥ 1) Lyapunov function is equivalent to the existence of a basic H2 (or Hq, for any q ≥ 2) Lyapunov function, its analogue for the H2 norm. Finally, we apply these results to the nonlinear Saint-Venant equations. We show in particular that in the subcritical regime, when the slope is larger than the friction, the system can always be stabilized in the C1 norm using static boundary feedbacks depending only on measurements at the boundaries, which has a large practical interest in hydraulic and engineering applications.

scholarly journals Stabilization of Stochastic Differential Equations Driven by G-Brownian Motion with Aperiodically Intermittent Control

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 988
Author(s):  
Pengju Duan
Keyword(s):  
Brownian Motion ◽  
Lyapunov Function ◽  
Differential Equations ◽  
Exponential Stability ◽  
Stochastic Differential Equations ◽  
Mild Solution ◽  
Intermittent Control

The paper is devoted to studying the exponential stability of a mild solution of stochastic differential equations driven by G-Brownian motion with an aperiodically intermittent control. The aperiodically intermittent control is added into the drift coefficients, when intermittent intervals and coefficients satisfy suitable conditions; by use of the G-Lyapunov function, the p-th exponential stability is obtained. Finally, an example is given to illustrate the availability of the obtained results.

scholarly journals Inequalities and pth moment exponential stability of impulsive delayed Hopfield neural networks

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Yutian Zhang ◽  
Guici Chen ◽  
Qi Luo
Keyword(s):  
Neural Networks ◽  
Lyapunov Function ◽  
Exponential Stability ◽  
Hopfield Neural Networks ◽  
New Method ◽  
Integral Inequalities ◽  
Delay Function ◽  
Pth Moment Exponential Stability ◽  
Moment Exponential Stability

AbstractIn this paper, the pth moment exponential stability for a class of impulsive delayed Hopfield neural networks is investigated. Some concise algebraic criteria are provided by a new method concerned with impulsive integral inequalities. Our discussion neither requires a complicated Lyapunov function nor the differentiability of the delay function. In addition, we also summarize a new result on the exponential stability of a class of impulsive integral inequalities. Finally, one example is given to illustrate the effectiveness of the obtained results.

scholarly journals Conservative finite-volume forms of the Saint-Venant equations for hydrology and urban drainage

2019 ◽  
Vol 23 (3) ◽  
pp. 1281-1304 ◽  
Author(s):  
Ben R. Hodges
Keyword(s):  
Free Surface ◽  
Finite Volume ◽  
Source Term ◽  
Weighting Function ◽  
Bottom Topography ◽  
Urban Drainage ◽  
Venant Equation ◽  
Saint Venant Equations ◽  
Volume Forms ◽  
Conservative Flux

Abstract. New integral, finite-volume forms of the Saint-Venant equations for one-dimensional (1-D) open-channel flow are derived. The new equations are in the flux-gradient conservation form and transfer portions of both the hydrostatic pressure force and the gravitational force from the source term to the conservative flux term. This approach prevents irregular channel topography from creating an inherently non-smooth source term for momentum. The derivation introduces an analytical approximation of the free surface across a finite-volume element (e.g., linear, parabolic) with a weighting function for quadrature with bottom topography. This new free-surface/topography approach provides a single term that approximates the integrated piezometric pressure over a control volume that can be split between the source and the conservative flux terms without introducing new variables within the discretization. The resulting conservative finite-volume equations are written entirely in terms of flow rates, cross-sectional areas, and water surface elevations – without using the bottom slope (S0). The new Saint-Venant equation form is (1) inherently conservative, as compared to non-conservative finite-difference forms, and (2) inherently well-balanced for irregular topography, as compared to conservative finite-volume forms using the Cunge–Liggett approach that rely on two integrations of topography. It is likely that this new equation form will be more tractable for large-scale simulations of river networks and urban drainage systems with highly variable topography as it ensures the inhomogeneous source term of the momentum conservation equation is Lipschitz smooth as long as the solution variables are smooth.

Singularities of solutions for first order quasilinear hyperbolic systems

1983 ◽  
Vol 94 (1-2) ◽  
pp. 137-147 ◽  
Author(s):  
Lee Da-tsin(Li Ta-tsien) ◽  
Shi Jia-hong
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Hyperbolic Systems ◽  
Smooth Solutions ◽  
Small Initial Data ◽  
Quasilinear Hyperbolic Systems ◽  
First Order ◽  
Global Smooth Solutions ◽  
The Cauchy Problem

SynopsisIn this paper, the existence of global smooth solutions and the formation of singularities of solutions for strictly hyperbolic systems with general eigenvalues are discussed for the Cauchy problem with essentially periodic small initial data or nonperiodic initial data. A result of Klainerman and Majda is thus extended to the general case.

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