Numerical boundary feedback stabilisation of non-uniform hyperbolic systems of balance laws

ABSTRACT

In this paper, numerical boundary stabilisation of a non-uniform hyperbolic system of balance laws is studied. For the numerical discretisation of the balance laws, a first order explicit upwind scheme is used for the spatial discretisation; and for the temporal discretisation a splitting technique is employed. A discrete $L^2$-Lyapunov function is employed to investigate conditions for the stability of the system. After constructing discrete numerical Lyapunov functionals, we prove an asymptotic exponential stability result for a class of non-uniform linear hyperbolic systems of balance laws. Convergence of the solution to its equilibrium is proved. Further application of the approach to practical problems through concrete examples is presented together with suggestions for numerical implementation. The numerical computations also demonstrate the stability of the numerical scheme with parameters chosen to satisfy the stability requirements.

Keywords:

• Conservation laws
• systems of non-uniform balance laws
• controllability
• feedback stabilisation
• Lyapunov functions

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Mapundi K. Banda http://orcid.org/0000-0003-4330-7355

Gediyon Y. Weldegiyorgis http://orcid.org/0000-0002-5467-0451

Additional information

Funding

G. Weldegiyorgis thanks Department of Mathematics and Applied Mathematics of University of Pretoria for partial funding for his research. M.K. Banda is also grateful to African Institute for Mathematical Sciences - South Africa and for hosting him while finalising this work. He is also grateful for DAAD funding (ID: 57314019) for a visit at RWTH-Aachen. This work is also supported in part by the National Research Foundation of South Africa (Grant number: 93099 and 93476).