Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Sufficient Stability Conditions for Time-varying Networks of Telegrapher's Equations or Difference Delay Equations

Laurent Baratchart, Sébastien Fueyo, Gilles Lebeau, Jean-Baptiste Pomet

Abstract

We give a sufficient condition for exponential stability of a network of lossless telegrapher's equations, coupled by linear time-varying boundary conditions. The sufficient conditions is in terms of dissipativity of the couplings, which is natural for instance in the context of microwave circuits. Exponential stability is with respect to any $L^p$-norm, $1\leq p\leq\infty$. This also yields a sufficient condition for exponential stability to a special class of linear time-varying difference delay systems which is quite explicit and tractable. One ingredient of the proof is that $L^p$ exponential stability for such difference delay systems is independent of $p$, thereby reproving in a simpler way some results from [Y. Chitour, G. Mazanti, and M. Sigalotti, $\it {Netw. Heterog. Media}$, 11 (2016), pp. 563--601].

Sufficient Stability Conditions for Time-varying Networks of Telegrapher's Equations or Difference Delay Equations

Abstract

We give a sufficient condition for exponential stability of a network of lossless telegrapher's equations, coupled by linear time-varying boundary conditions. The sufficient conditions is in terms of dissipativity of the couplings, which is natural for instance in the context of microwave circuits. Exponential stability is with respect to any -norm, . This also yields a sufficient condition for exponential stability to a special class of linear time-varying difference delay systems which is quite explicit and tractable. One ingredient of the proof is that exponential stability for such difference delay systems is independent of , thereby reproving in a simpler way some results from [Y. Chitour, G. Mazanti, and M. Sigalotti, , 11 (2016), pp. 563--601].

Paper Structure

This paper contains 16 sections, 11 theorems, 76 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Problem statement
  3. A time-varying network of hyperbolic equations
  4. Well-posedness of the evolution problem in the $L^p$ and $C^0$ cases
  5. Difference-delay equations and their relation with networks of telegrapher's equations
  6. Exponential stability: definitions
  7. Results
  8. Known results in the time-invariant case
  9. Sufficient stability condition in the time-varying case
  10. Proofs
  11. A technical lemma
  12. Proof of Proposition \ref{['prop_cas_constant']}
  13. Proof of Theorem \ref{['theorem_L2_stability']} via a Lyapunov functional approach
  14. Sketch of an alternative proof of Theorem \ref{['theorem_L2_stability']} via difference-delay systems exclusively
  15. Proof of Theorem \ref{['th:L2impliesC0']}
  16. ...and 1 more sections

Key Result

Theorem 2.8

\newlabelexistence_sol_tele0 Let $\mathbf{A}:[0,\infty)\to\mathbb{R}^{2N\times 2N}$ meet Assumption ass:dissip and $1\leq p\leq\infty$. $\ $ I) If $i^{0}_k,v^{0}_k\in L^p([0,1])$, $1\leq k\leq N$, there is a unique map $(t,x)\mapsto(v_1(t,x),\ldots,$$v_N(t,x),i_1(t,x),\ldots,i_N(t,x))$ from $\Omeg $\bullet$$(t,x)\mapsto(v_1(t,x),\ldots,v_N(t,x),i_1(t,x),\ldots,i_N(t,x))$ is a solution of eq_tel-e

Figures (1)

  • Figure 1: A graph that induces coupling boundary conditions for \ref{['eq_tel']} with $N=4$.

Theorems & Definitions (33)

  • Example 2.1
  • Remark 2.2: On the minus signs in the vector $I_p$ in \ref{['eq:defVI']}
  • Remark 2.3: On the normalization of line lengths
  • Remark 2.4: On the possibility of loops
  • Definition 2.6
  • Remark 2.7
  • Theorem 2.8: Well-posedness
  • Remark 2.9
  • Proposition 2.10
  • Proof 1
  • ...and 23 more