Regularity and Lyapunov Stabilization of Weak Entropy Solutions to Scalar Conservation Laws

Sébastien Blandin, Xavier Litrico, Maria Laura Delle Monache, Benedetto Piccoli, Alexandre Bayen

Research output: Contribution to journalArticlepeer-review

24 Scopus citations

Abstract

We consider the problem of Lyapunov boundary stabilization of the weak entropy solution to a scalar conservation law with strictly convex flux in one dimension of space, around a uniform equilibrium. We show that for a specific class of boundary conditions, the solution to the initial-boundary value problem for an initial condition with bounded variations can be approximated arbitrarily closely in the L1 norm by a piecewise smooth solution with finitely many discontinuities. The constructive method we present designs explicit boundary conditions in this class, which guarantee Lyapunov stability of the weak entropy solution to the initial-boundary value problem. We show how the greedy control, obtained by maximizing the decrease of the natural Lyapunov function, may fail to asymptotically stabilize and a brute force control generates unbounded variation of traces. We then design a stabilizing control, which avoid oscillations, and propose a nonlocal technique (depending on time and the whole initial datum) which optimizes the convergence time. Controllers performance is illustrated on numerical benchmarks using the Godunov scheme.

Original languageEnglish (US)
Article number7509658
Pages (from-to)1620-1635
Number of pages16
JournalIEEE Transactions on Automatic Control
Volume62
Issue number4
DOIs
StatePublished - Apr 2017

All Science Journal Classification (ASJC) codes

  • Control and Systems Engineering
  • Computer Science Applications
  • Electrical and Electronic Engineering

Keywords

  • Boundary value problems
  • Lyapunov methods
  • distributed parameter systems
  • partial differential equations

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