Exponential stability of a general slope limiter scheme for scalar conservation laws subject to a dissipative boundary condition

Dus, Mathias

doi:10.1007/s00498-021-00301-2

Cited by 6 publications

(6 citation statements)

References 35 publications

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“…Proof. We use a Lyapunov argument using the Lyapunov functional firstly introduced in [6] in a continuous setting and then adapted to a discrete framework [14]:…”

Section: Convergence Properties Of the Kernelmentioning

confidence: 99%

The discretized backstepping method: An application to a general system of $ 2\times 2 $ linear balance laws

Dus¹

2023

MCRF

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<p style='text-indent:20px;'>In this paper, we introduce the numerical backstepping method by applying it to a problem of finite-time stabilization for a system of <inline-formula><tex-math id="M2">\begin{document}$ 2 \times 2 $\end{document}</tex-math></inline-formula> balance laws discretized thanks to the upwind scheme. On the one hand, we illustrate on an example that the scheme used to compute the feedback control cannot be chosen arbitrarily. On the other hand, an algorithm is given to construct this control properly and an approached finite-time stabilization result is proven.</p>

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“…Proof. We use a Lyapunov argument using the Lyapunov functional firstly introduced in [6] in a continuous setting and then adapted to a discrete framework [14]:…”

Section: Convergence Properties Of the Kernelmentioning

confidence: 99%

The discretized backstepping method: An application to a general system of $ 2\times 2 $ linear balance laws

Dus¹

2023

MCRF

Self Cite

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“…This completes the proof. Suppose, in Theorem 3.1, if control gain K is unknown then which affects the linearity of (8). In this case, by letting K = P K, one can preserve the linearity and the corresponding exponential stability results can be presented as follows.…”

Section: Now With Help Of Itö Formula We Get Thatmentioning

confidence: 99%

“…In this connection, there are numerous results addressed for disturbances in the literature. For example, Dus [8] studied the exponential stability of general systems of discretized scalar conservation laws using boundary feedback laws. Wei et al [34] introduced a multiple disturbances for stochastic systems and subsequently designed an observer to estimate the disturbance.…”

Section: Introductionmentioning

confidence: 99%

Exponential Stability Analysis of Stochastic Semi-Linear Systems With Lèvy Noise

et al. 2022

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The exponential stability of semi-linear stochastic partial differential equations (SPDEs) involving Lèvy type noise is investigated in this paper. By constructing an appropriate Lyapunov function, a new set of sufficient conditions are established in terms of linear matrix inequalities (LMIs) which ensures the mean-square exponentially stability (MSES) of given system with Neumann boundary conditions. Then the H ∞ performance index is introduced to eliminate the disturbance which occurs in the considered system. The boundary control gain is obtained by solving the LMI conditions using the standard MATLAB software. Finally, a numerical example is provided to demonstrate the usefulness of the proposed methods.

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“…Finally, we remark that the regularity assumption (12), which has been used to deduce the previous results, can be stated in terms of L 2 -solutions for general linear balance laws [2, Sec. 2.1.3] and for semilinear systems with locally Lipschitz continuous source term [35,Th.…”

Section: Proposition 2 the Semilinear Boundary Value Problemmentioning

confidence: 99%

“…In particular, first-order upwind discretizations of linear balance laws are used to construct discretized Lyapunov functions that decay exponentially fast [14,28,29,32]. Moreover, a second-order scheme applied to scalar nonlinear conservation laws with dissipative feedback boundary conditions is analyzed in [12].…”

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confidence: 99%

Numerical boundary control for semilinear hyperbolic systems

Gerster¹,

Nagel²,

Sikstel³

et al. 2022

Preprint

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This work is devoted to the design of boundary controls of physical systems that are described by semilinear hyperbolic balance laws. A computational framework is presented that yields sufficient conditions for a boundary control to steer the system towards a desired state. The presented approach is based on a Lyapunov stability analysis and a CWENO-type reconstruction.Introduction. Systems of hyperbolic partial differential equations model fluid flow, chemotaxis and viscoplastic material dynamics [2,23,26,34]. Boundary stabilization of these problems has been studied intensively in the past years [2,5]. An underlying tool for the study of these problems are Lyapunov functions that yield upper bounds on the deviation from steady states in suitable norms. The virtue of this approach are control rules that do not require the solution of the whole system, but take only measurements at the boundaries into account. So-called dissipative boundary conditions [6][7][8][9] ensure exponential decay of a continuous Lyapunov function, which in turn guarantees that the solution converges exponentially fast to a desired steady state.More precisely, a general theory for the stabilization of linear conservation laws with respect to the L 2 -norm is available [2, Sec. 3]. For nonlinear systems, however, results are still partial. A problem is posed by the fact that Lyapunov's indirect method [24] does not necessarily hold for hyperbolic systems. Furthermore, solutions to systems of conservation laws exist in the classical sense only for a finite time due to formation of shocks [31]. To this end, stability results are typically stated in terms of the Sobolev H 2 -norm or in the C 1 -norm [6,7,19,21] and restrictive smoothness assumptions on the H 2 -norm of the initial data may be needed [2, Sec. 4]. Furthermore, most analytical results are based on the assumption that the influence of the source term is small or in intuitive terms, the considered balance laws are viewed as perturbations of conservation laws [8]. On the other

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Exponential stability of a general slope limiter scheme for scalar conservation laws subject to a dissipative boundary condition

Cited by 6 publications

References 35 publications

The discretized backstepping method: An application to a general system of $ 2\times 2 $ linear balance laws

The discretized backstepping method: An application to a general system of $ 2\times 2 $ linear balance laws

Exponential Stability Analysis of Stochastic Semi-Linear Systems With Lèvy Noise

Numerical boundary control for semilinear hyperbolic systems

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