Regularity and Lyapunov stabilization of weak entropy solutions to scalar conservation laws

Sébastien Blandin 1 Xavier Litrico 2, 3 Maria Laura Delle Monache 4, 5 Benedetto Piccoli 6, 4 Alexandre Bayen 7
1 IBM Research Collaboratory [Singapore]
2 SUEZ ENVIRONNEMENT (FRANCE)
3 LyRE - Centre recherche et développement
4 Rutgers University, Department of Mathematics
Department of Mathematics - Rutgers School of Arts and Sciences
5 Rutgers University [Camden]
6 CCIB - Center for Computational and Integrative Biology [Camden]
7 University of California [Berkeley]
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.
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IEEE Transactions on Automatic Control, Institute of Electrical and Electronics Engineers, 2017, 62 (4), pp.1620-1635
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Sébastien Blandin, Xavier Litrico, Maria Laura Delle Monache, Benedetto Piccoli, Alexandre Bayen. Regularity and Lyapunov stabilization of weak entropy solutions to scalar conservation laws. IEEE Transactions on Automatic Control, Institute of Electrical and Electronics Engineers, 2017, 62 (4), pp.1620-1635. 〈hal-01267722〉

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