A simple constructive proof of Kronecker's Density Theorem

Bridges, Douglas; Schuster, Peter

doi:10.4171/em/45

Cited by 6 publications

(2 citation statements)

References 3 publications

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“…For such systems, it follows from a theorem by Silkowski [40] (quoted in [20, Chapter 9, Theorem 6.1]; see also [29]), which relies on the Kronecker density theorem (e.g., [6]), that the boundary conditions (2.5) are, for any L p -norm, robustly dissipative with respect to arbitrary small perturbations on the Λ i 's if and only if (2.6)ρ(K) max{ρ(diag(e ιθ1 , . As can be expected, the analysis of dissipative boundary conditions is both simpler and more comprehensive for linear than for quasi-linear systems.…”

Section: Introductionmentioning

confidence: 99%

Dissipative Boundary Conditions for One-Dimensional Quasi-linear Hyperbolic Systems: Lyapunov Stability for the $C^1$-Norm

Coron¹,

Bastin²

2015

SIAM J. Control Optim.

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This paper is concerned with boundary dissipative conditions that guarantee the exponential stability of classical solutions of one-dimensional quasi-linear hyperbolic systems. We present a comprehensive review of the results that are available in the literature. The main result of the paper is then to supplement these previous results by showing how a new Lyapunov stability approach can be used for the analysis of boundary conditions that are known to be dissipative for the C 1 -norm.

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Section: Introductionmentioning

confidence: 99%

Dissipative Boundary Conditions for One-Dimensional Quasi-linear Hyperbolic Systems: Lyapunov Stability for the $C^1$-Norm

Coron¹,

Bastin²

2015

SIAM J. Control Optim.

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“…Note that this proof could be made constructive by using an effective version of Kronecker's Theorem, as studied in [7] and [17], although we do not make use of this fact in the present paper.…”

Section: Laurent Polynomialsmentioning

confidence: 99%

On the Polytope Escape Problem for Continuous Linear Dynamical Systems

Ouaknine

Sousa-Pinto

Worrell

2017

Proceedings of the 20th International Conference on Hybrid Systems: Computation and Control

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The Polyhedral Escape Problem for continuous linear dynamical systems consists of deciding, given an affine function $f: \mathbb{R}^{d} \rightarrow \mathbb{R}^{d}$ and a convex polyhedron $\mathcal{P} \subseteq \mathbb{R}^{d}$, whether, for some initial point $\boldsymbol{x}_{0}$ in $\mathcal{P}$, the trajectory of the unique solution to the differential equation $\dot{\boldsymbol{x}}(t)=f(\boldsymbol{x}(t))$, $\boldsymbol{x}(0)=\boldsymbol{x}_{0}$, is entirely contained in $\mathcal{P}$. We show that this problem is decidable, by reducing it in polynomial time to the decision version of linear programming with real algebraic coefficients, thus placing it in $\exists \mathbb{R}$, which lies between NP and PSPACE. Our algorithm makes use of spectral techniques and relies among others on tools from Diophantine approximation.Comment: Accepted to HSCC 201

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Systems of Linear Conservation Laws

Bastin

Coron

2016

Stability and Boundary Stabilization of 1-D Hyperbolic Systems

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A simple constructive proof of Kronecker's Density Theorem

Cited by 6 publications

References 3 publications

Dissipative Boundary Conditions for One-Dimensional Quasi-linear Hyperbolic Systems: Lyapunov Stability for the $C^1$-Norm

Dissipative Boundary Conditions for One-Dimensional Quasi-linear Hyperbolic Systems: Lyapunov Stability for the $C^1$-Norm

On the Polytope Escape Problem for Continuous Linear Dynamical Systems

Systems of Linear Conservation Laws

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