STUDY OF EXPONENTIAL STABILITY OF A NUMERICAL SOLUTION OF A HYPERBOLIC SYSTEM WITH NEGATIVE NONLOCAL CHARACTERISTIC VELOCITIES AND MEASUREMENT ERROR
Keywords:
hyperbolic equation, nonlocal characteristic velocity, stability, explicit difference schemeAbstract
In this paper, we study the problem of stabilizing the equilibrium state for a hyper bolic system with negative nonlocal characteristic velocities and measurement error. The formulation of a mixed boundary control problem for the indicated hyperbolic system is given. The stability in the l2-norm with respect to a discrete perturbation of the equilib rium state of an initial-boundary difference problem is determined. A discrete Lyapunov function is constructed and a stability theorem for the equilibrium state of an initial boundary difference problem in the l2-norm with respect to a discrete perturbation is proved.
References
Coron, J.M., Wang, Z. Output Feedback Stabilization for a Scalar Conservation Law with a Nonlocal Velocity SIAM J. Math. Anal. 45,– 2013.– P. 2646–2665, doi:10.1137/120902203.
Chen, W., Liu, C., Wang, Z. Global Feedback Stabilization for a Class of Nonlocal Transport Equations: The Continuous and Discrete Case // SIAM J. Control Optim.– 2017.– Vol.55.– P. 760-784.– doi: http://dx.doi.org/10.1137/15m1048914.
G¨ottlich S., Herty M., Weldegiyorgis G. Input-to-State Stability of a Scalar Conservation Law with Nonlocal Velocity // Axioms.– 2021.– Vol. 10, No. 12.– doi: http://dx.doi.org/10.3390/axioms10010012.
Coron J.M., Kawski M. Wang Z. Analysis of a conservation law modeling a highly re entrant manufacturing system // Discret. Contin. Dyn. Syst. Ser. B.– 2010.– Vol. 14.– P. 1337-1359.– doi: http://dx.doi.org/10.3934/dcdsb.2010.14.1337.
Tanwani A., Prieur C., Tarbouriech S. Stabilization of linear hyperbolic systems of balance laws with measurement errors // Control Subject to Computational and Communication Constraints. Vol. 475.– Cham: Springer, 2018.– P. 357-374.
Zhang L., Prieur C. Necessary and Sufficient Conditions on the Exponential Stability of Positive Hyperbolic Systems // IEEE Trans. Automat. Control.– 2017.– Vol. 62.– P. 3610-3617.– doi: http://dx.doi.org/10.1109/TAC.2017.2661966.
Coron J.M. et al. Dissipative boundary conditions for one-dimensional nonlinear hyperbolic systems // SIAM J. Control Optim.– 2008.– Vol. 47.– P. 1460-1498.– doi: http://dx. doi.org/10.1137/070706847.
Bastin G., Coron J.M. Stability and Boundary Stabilization of 1-D Hyperbolic Systems. Berlin: Springer, 2016.– doi: http://dx.doi.org/10.1007/978-3-319-32062-5.
Aloev R. et al. Stability Analysis of an Upwind Difference Splitting Scheme for Two Dimensional Saint–Venant Equations // Symmetry.– 2022.– doi: http://dx.doi.org/ 10.3390/sym14101986.
Aloev R.D., Dadabaev S.U. Stability of the upwind difference splitting scheme for symmetric t-hyperbolic systems with constant coefficients // Results in Applied Mathematics.– 2022.– doi: http://dx.doi.org/10.1016/j.rinam.2022.100298.
Aloev R.D., Hudayberganov M.U. A Discrete Analogue of the Lyapunov Function for Hyperbolic Systems // Journal of Mathematical Sciences.– 2022.– doi: http://dx.doi. org/10.1007/s10958-022-06028
Aloev R.D. et al. The Difference Splitting Scheme for n-Dimensional Hyperbolic Systems // Malaysian Journal of Mathematical Sciences.– 2022.
Aloev R. et al. Development of an algorithm for calculating stable solutions of the Saint Venant equation using an upwind implicit difference scheme // Eastern-European Journal of Enterprise Technologies.– 2021.– doi: http://dx.doi.org/10.15587/1729-4061.2021. 239148.
Aloev R.D. et al. The difference splitting scheme for hyperbolic systems with variable coefficients // Mathematics and Statistics.– 2019.– doi: http://dx.doi.org/10.13189/ ms.2019.070305.
Aloev R. et al. Construction and research of adequate computational models for quasilinear hyperbolic systems // Numerical Algebra, Control and Optimization.– 2018.– doi: http: //dx.doi.org/10.3934/naco.2018017.
Aloev R.D. et al. The Difference Splitting Scheme for n-Dimensional Hyperbolic Systems // Malaysian Journal of Mathematical Sciences.– 2022.– No. 16(1).– P. 1-10 2022. doi: http://dx.doi.org/10.47836/mjms.16.1.01.
