New Lyapunov functions for systems with source terms

• Autorzy: Gugat Martin

Identyfikatory

DOI

10.2478/candc-2024-0008

• Języki publikacji: EN

Abstrakty

EN

Lyapunov functions with exponential weights have been used successfully as a powerful tool for the stability analysis of hyperbolic systems of balance laws. In this paper we extend the class of weight functions to a family of hyperbolic functions and study the advantages in the analysis of 2 × 2 systems of balance laws. We present cases connected with the study of the limit of stabilizability, where the new weights provide Lyapunov functions that show exponential stability for a larger set of problem parameters than classical exponential weights. Moreover, we show that sufficiently large time-delays influence the limit of stabilizability in the sense that the parameter set, for which the system can be stabilized becomes substantially smaller. We also demonstrate that the hyperbolic weights are useful in the analysis of the boundary feedback stability of systems of balance laws that are governed by quasilinear hyperbolic partial differential equations.

Słowa kluczowe

EN

Lyapunov function; exponential weights; hyperbolic weights; feedback law; stabilization; boundary feedback; Riemann feedback; hyperbolic system

• Wydawca: Systems Research Institute, Polish Academy of Sciences
• Czasopismo: Control and Cybernetics
• Rocznik: 2024
• Tom: Vol. 53, No. 1
• Strony: 163--187
• Opis fizyczny: Bibliogr. 22 poz.

Twórcy

autor

Gugat Martin

• Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), Department Mathematik, Lehrstuhl für Dynamics, Control, Machine Learning and Numerics (Alexander von Humboldt-Professur), Cauerstr. 11, 91058 Erlangen, Germany

Bibliografia

• Bastin, G. and Coron, J.-M. (2011) On boundary feedback stabilization of non-uniform linear 2 x 2 hyperbolic systems over a bounded interval. Systems & Control Letters, 60(11): 900–906.
• Bastin, G. and Coron, J.-M. (2016) Stability and Boundary Stabilization of 1-D Hyperbolic Systems. Progress in Nonlinear Differential Equations and their Applications, 88. Birkhäuser/Springer, Cham. Subseries in Control.
• Banda, M. K., Herty, M. and Klar, A. (2006) Gas flow in pipeline networks. Netw. Heterog. Media 1(1): 41–56.
• Coron, J.-M., d’Andrea Novel, B. and Bastin, G. (2007) A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws. IEEE Trans. Autom. Control, 52(1): 2–11.
• Coron, J.-M. (1999) On the null asymptotic stabilization of the two-dimensional incompressible Euler equations in a simply connected domain. SIAM J. Control Optim., 37(6): 1874–1896.
• Datko, R. (1988) Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks. SIAM Journal on Control and Optimization, 26(3): 697–713.
• Datko, R., Lagnese, J. and Polis, M. P. (1986) An example on the effect of time delays in boundary feedback stabilization of wave equations. SIAM Journal on Control and Optimization, 24(1): 152–156, 1986.
• Gugat, M. and Dick, M. (2011) Time-delayed boundary feedback stabilization of the isothermal Euler equations with friction. Math. Control Relat. Fields, 1(4): 469–491.
• Gugat, M. and Gerster, S. (2019) On the limits of stabilizability for networks of strings. Syst. Control Lett., 131:10. Id/No 104494.
• Gugat, M., Giesselmann, J. and Kunkel, T. (2021) Exponential synchronization of a nodal observer for a semilinear model for the flow in gas networks. IMA Journal of Mathematical Control and Information, 38(4): 1109–1147.
• Gugat, M. and Herty, M. (2011) Existence of classical solutions and feedback stabilization for the flow in gas networks. ESAIM, Control Optim. Calc. Var., 17(1): 28–51.
• Gugat, M., Huang, X. and Wang, Z. (2023) Limits of stabilization of a networked hyperbolic system with a circle. Control and Cybernetics, 52(1): 79–121.
• Gugat, M., Leugering, G. and Wang, K. (2017) Neumann boundary feedback stabilization for a nonlinear wave equation: A strict H2-Lyapunov function. Math. Control Relat. Fields, 7(3): 419–448.
• Gerster, S., Nagel, F., Sikstel, A. and Visconti, G. (2023) Numerical boundary control for semilinear hyperbolic systems. Mathematical Control and Related Fields, 13(4): 1344–1361.
• Gugat, M. and Tucsnak, M. (2011) An example for the switching delay feedback stabilization of an infinite dimensional system: The boundary stabilization of a string. Systems & Control Letters, 60(4): 226–233.
• Haidar, I., Chitour, Y., Mason, P. and Sigalotti, M. (2021) Lyapunov characterization of uniform exponential stability for nonlinear infinitedimensional systems. IEEE Transactions on Automatic Control, 67(4): 1685–1697.
• Hayat, A. (2021a) Boundary stabilization of 1D hyperbolic systems. Annu. Rev. Control, 52: 222–242.
• Hayat, A. (2021b) Global exponential stability and input-to-state stability of semilinear hyperbolic systems for the l2 norm. Systems & Control Letters, 148: 104848.
• Hayat, A. and Shang, P. (2021) Exponential stability of density-velocity systems with boundary conditions and source term for the h2 norm. Journal de mathématiques pures et appliquées, 153: 187–212.
• Lichtner, M. (2008) Spectral mapping theorem for linear hyperbolic systems. Proceedings of the American Mathematical Society, 136(6): 2091–2101.
• Li, T., Wang, K. and Gu, Q. (2016) Exact Boundary Controllability of Nodal Profile for Quasilinear Hyperbolic Systems. Springer.
• Wang, Z. (2006) Exact controllability for nonautonomous first order quasilinear hyperbolic systems. Chinese Annals of Mathematics, Series B, 27(6): 643–656.