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September  2017, 7(3): 419-448. doi: 10.3934/mcrf.2017015

Neumann boundary feedback stabilization for a nonlinear wave equation: A strict $H^2$-lyapunov function

a. 

Lehrstuhl 2 für Angewandte Mathematik, Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstr. 11,91058 Erlangen, Germany

b. 

College of Science, Donghua University, Shanghai 201620, China

Received  July 2015 Revised  January 2017 Published  July 2017

Fund Project: This work was supported by DFG in the framework of the Collaborative Research Centre CRC/Transregio 154, Mathematical Modelling, Simulation and Optimization Using the Example of Gas Networks, project C03 and A05

For a system that is governed by the isothermal Euler equations with friction for ideal gas, the corresponding field of characteristic curves is determined by the velocity of the flow. This velocity is determined by a second-order quasilinear hyperbolic equation. For the corresponding initial-boundary value problem with Neumann-boundary feedback, we consider non-stationary solutions locally around a stationary state on a finite time interval and discuss the well-posedness of this kind of problem. We introduce a strict $H^2$-Lyapunov function and show that the boundary feedback constant can be chosen such that the $H^2$-Lyapunov function and hence also the $H^2$-norm of the difference between the non-stationary and the stationary state decays exponentially with time.

Citation: Martin Gugat, Günter Leugering, Ke Wang. Neumann boundary feedback stabilization for a nonlinear wave equation: A strict $H^2$-lyapunov function. Mathematical Control & Related Fields, 2017, 7 (3) : 419-448. doi: 10.3934/mcrf.2017015
References:
[1]

F. Alabau-BoussouiraL. Rosier and V. Perrollaz, Finite-time stabilization of a network of strings, Mathematical Control and Related Fields (MCRF), 5 (2015), 721-742. doi: 10.3934/mcrf.2015.5.721. Google Scholar

[2]

H. Alli and T. Singhj, On the feedback control of the wave equation, Journal of Sound and Vibration, 234 (2000), 625-640. doi: 10.1109/CCA.1996.558717. Google Scholar

[3]

H. Attouch, G. Buttazzo and G. Michaille, Variational Analysis in Sobolev and BV Spaces: Applications to PDEs and Optimization, , Society for Industrial and Applied Mathematics and Mathematical Programming Society, Philadelphia, 2006. Google Scholar

[4]

M. K. BandaM. Herty and A. Klar, Coupling conditions for gas networks governed by the isothermal Euler equations, Netw. Heterog. Media, 1 (2006), 295-314. doi: 10.3934/nhm.2006.1.295. Google Scholar

[5]

M. K. BandaM. Herty and A. Klar, Gas flow in pipeline networks, Netw. Heterog. Media, 1 (2006), 41-56. doi: 10.3934/nhm.2006.1.41. Google Scholar

[6]

G. Bastin and J. -M. Coron, Stability and Boundary Stabilization of 1-d Hyperbolic Systems, Birkhäuser, Basel, Switzerland, 2016. doi: 10.1007/978-3-319-32062-5. Google Scholar

[7]

R. M. CorlessG. H. GonnetD. E. G. HareD. J. Jeffrey and D. E. Knuth, On the Lambert W Function, Adv. Comp. Math., 5 (1996), 329-359. doi: 10.1007/BF02124750. Google Scholar

[8]

J. M. Coron and G. Bastin, Dissipative boundary conditions for one-dimensional quasi-linear hyperbolic systems: Lyapunov stability for the C1-Norm, SIAM J. Control Optim., 53 (2015), 1464-1483. doi: 10.1137/14097080X. Google Scholar

[9]

J. M. CoronB. d'Andréa-Novel and G. Bastin, A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws, IEEE Trans. Automat. Control, 52 (2007), 2-11. doi: 10.1109/TAC.2006.887903. Google Scholar

[10]

J. M. CoronG. Bastin and B. d'Andréa-Novel, Disspative boundary conditions for one-dimensional nonlinear hyperbolic systems, SIAM J. Control Optim., 47 (2008), 1460-1498. doi: 10.1137/070706847. Google Scholar

[11]

J. M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs, 136, American Mathematical Society, Providence, RI, 2007. Google Scholar

[12]

M. DickM. Gugat and G. Leugering, A Strict $H^1$-Lyapunov function and feedback stabilization for the isothermal Euler equations with friction, Numerical Algebra Control and Optimization, 1 (2011), 225-244. doi: 10.3934/naco.2011.1.225. Google Scholar

[13]

M. DickM. Gugat and G. Leugering, Classical solutions and feedback stabilization for the gas flow in a sequence of pipes, Netw. Heterog. Media, 5 (2010), 691-709. doi: 10.3934/nhm.2010.5.691. Google Scholar

[14]

J. M. Greenberg and T. T. Li, The effect of boundary damping for the quasilinear wave equation, J. Differential Equations, 52 (1984), 66-75. doi: 10.1016/0022-0396(84)90135-9. Google Scholar

[15]

M. Gugat, Boundary feedback stabilization by time delay for one-dimensional wave equations, IMA J. Math. Control Inform., 27 (2010), 189-203. doi: 10.1093/imamci/dnq007. Google Scholar

[16]

M. Gugat and M. Dick, Time-delayed boundary feedback stabilization of the isothermal Euler equations with friction, Math. Control and Related Fields, 1 (2011), 469-491. doi: 10.3934/mcrf.2011.1.469. Google Scholar

[17]

M. GugatF. HanteM. Hirsch-Dick and G. Leugering, Stationary states in gas networks, Netw. Heterog. Media, 10 (2015), 295-320. doi: 10.3934/nhm.2015.10.295. Google Scholar

[18]

M. GugatM. HertyA. KlarG. Leugering and V. Schleper, Well-posedness of networked hyperbolic systems of balance laws, International Series of Numerical Mathematics, 160 (2012), 123-146. doi: 10.1007/978-3-0348-0133-1_7. Google Scholar

[19]

M. GugatG. LeugeringS. Tamasoiu and K. Wang, $H^2$-stabilization of the isothermal Euler equations: A Lyapunov function approach, Chinese Annals of Mathematics, Series B, 33 (2012), 479-500. doi: 10.1007/s11401-012-0727-y. Google Scholar

[20]

M. Gugat and M. Sigalotti, Stars of vibrating strings: Switching boundary feedback stabilization, Netw. Heterog. Media, 5 (2010), 299-314. doi: 10.3934/nhm.2010.5.299. Google Scholar

[21]

M. Gugat and M. Tucsnak, An example for the switching delay feedback stabilization of an infinite dimensional system: The boundary stabilization of a string, Syst. Cont. Lett., 60 (2011), 226-233. doi: 10.1016/j.sysconle.2011.01.004. Google Scholar

[22]

M. Gugat and S. Ulbrich, The isothermal Euler equations for ideal gas with source term: Product solutions, flow reversal and no blow up, J. Math. Anal. Appl., 454 (2017), 439-452. doi: 10.1016/j.jmaa.2017.04.064. Google Scholar

[23]

M. Gugat, Optimal Boundary Control and Boundary Stabilization of Hyperbolic Systems, SpringerBriefs in Control, Automation and Robotics, Springer, New York, New York, 2015. doi: 10.1007/978-3-319-18890-4. Google Scholar

[24]

L. Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations, Springer Verlag, Berlin, 1997. Google Scholar

[25]

T. J. R. HughesT. Kato and J. E. Marsden, Well-posed quasi-linear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity, Arch. Rational Mech. Anal., 63 (1977), 273-294. doi: 10.1007/BF00251584. Google Scholar

[26]

V. Komornik, Rapid boundary stabilization of the wave equation, SIAM J. Control and Optimization, 29 (1991), 197-208. doi: 10.1137/0329011. Google Scholar

[27]

J. H. Lambert, Observationes variae in mathesin puram, Acta Helvetica, physico-mathematico-anatomico-botanico-medica, 3 (1758), 128-168. Google Scholar

[28]

T. T. Li, Global Classical Solutions for Quasilinear Hyperbolic Systems, RAM Res. Appl. Math. 32, Masson, Paris, 1994. Google Scholar

[29]

T. T. Li and W. C. Yu, Boundary Value Problems for Quasilinear Hyperbolic Systems, Duke Univ. Math. Ser. V, 1985. Google Scholar

[30]

T. T. LiB. Rao and Z. Wang, A note on the one-side exact boundary observability for quasilinear hyperbolic systems, Georgian Mathematical Journal, 15 (2008), 571-580. Google Scholar

[31]

Z. H. Luo, B. Z. Guo and O. Morgul, Stability and Stabilization of Infinite Dimensional Systems with Applications, Communications and Control Engineering Series. Springer-Verlag London Ltd. , London, 1999. doi: 10.1007/978-1-4471-0419-3. Google Scholar

[32]

F. Mazenc and C. Prieur, Strict Lyapunov functions for semilinear parabolic partial differential equations, Math. Control and Related Fields, 1 (2011), 231-250. doi: 10.3934/mcrf.2011.1.231. Google Scholar

[33]

S. NicaiseJ. Valein and E. Fridman, Stability of the heat and of the wave equations with boundary time-varying delays, Disc. Contin. Dyn. Syst. Ser. S, 2 (2009), 559-581. doi: 10.3934/dcdss.2009.2.559. Google Scholar

[34]

M. Slemrod, Boundary feedback stabilization for a quasilinear wave equation, in Control Theory for Distributed Parameter Systems and Applications, Lecture Notes in Control and Inform. Sci., Springer-Verlag, Berlin, 54 (1983), 221-237. doi: 10.1007/BFb0043951. Google Scholar

[35]

M. E. Taylor, Partial Differential Equations, Tome Ⅲ, Nonlinear equations, Applied Mathematical Sciences, Vol. 117, Springer-Verlag, New-York, 1996. doi: 10.1007/978-1-4684-9320-7. Google Scholar

[36]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser, Basel-Boston-Berlin, 2009. doi: 10.1007/978-3-7643-8994-9. Google Scholar

[37]

Z. Q. Wang, Exact boundary controllability for nonautonomous quasilinear wave equations, Math. Meth. Appl. Sci., 30 (2007), 1311-1327. doi: 10.1002/mma.843. Google Scholar

[38]

G. P. ZouN. Cheraghi and F. Taheri, Fluid-induced vibration of composite natural gas pipelines, International Journal of Solids and Structures, 42 (2005), 1253-1268. doi: 10.1016/j.ijsolstr.2004.07.001. Google Scholar

show all references

References:
[1]

F. Alabau-BoussouiraL. Rosier and V. Perrollaz, Finite-time stabilization of a network of strings, Mathematical Control and Related Fields (MCRF), 5 (2015), 721-742. doi: 10.3934/mcrf.2015.5.721. Google Scholar

[2]

H. Alli and T. Singhj, On the feedback control of the wave equation, Journal of Sound and Vibration, 234 (2000), 625-640. doi: 10.1109/CCA.1996.558717. Google Scholar

[3]

H. Attouch, G. Buttazzo and G. Michaille, Variational Analysis in Sobolev and BV Spaces: Applications to PDEs and Optimization, , Society for Industrial and Applied Mathematics and Mathematical Programming Society, Philadelphia, 2006. Google Scholar

[4]

M. K. BandaM. Herty and A. Klar, Coupling conditions for gas networks governed by the isothermal Euler equations, Netw. Heterog. Media, 1 (2006), 295-314. doi: 10.3934/nhm.2006.1.295. Google Scholar

[5]

M. K. BandaM. Herty and A. Klar, Gas flow in pipeline networks, Netw. Heterog. Media, 1 (2006), 41-56. doi: 10.3934/nhm.2006.1.41. Google Scholar

[6]

G. Bastin and J. -M. Coron, Stability and Boundary Stabilization of 1-d Hyperbolic Systems, Birkhäuser, Basel, Switzerland, 2016. doi: 10.1007/978-3-319-32062-5. Google Scholar

[7]

R. M. CorlessG. H. GonnetD. E. G. HareD. J. Jeffrey and D. E. Knuth, On the Lambert W Function, Adv. Comp. Math., 5 (1996), 329-359. doi: 10.1007/BF02124750. Google Scholar

[8]

J. M. Coron and G. Bastin, Dissipative boundary conditions for one-dimensional quasi-linear hyperbolic systems: Lyapunov stability for the C1-Norm, SIAM J. Control Optim., 53 (2015), 1464-1483. doi: 10.1137/14097080X. Google Scholar

[9]

J. M. CoronB. d'Andréa-Novel and G. Bastin, A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws, IEEE Trans. Automat. Control, 52 (2007), 2-11. doi: 10.1109/TAC.2006.887903. Google Scholar

[10]

J. M. CoronG. Bastin and B. d'Andréa-Novel, Disspative boundary conditions for one-dimensional nonlinear hyperbolic systems, SIAM J. Control Optim., 47 (2008), 1460-1498. doi: 10.1137/070706847. Google Scholar

[11]

J. M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs, 136, American Mathematical Society, Providence, RI, 2007. Google Scholar

[12]

M. DickM. Gugat and G. Leugering, A Strict $H^1$-Lyapunov function and feedback stabilization for the isothermal Euler equations with friction, Numerical Algebra Control and Optimization, 1 (2011), 225-244. doi: 10.3934/naco.2011.1.225. Google Scholar

[13]

M. DickM. Gugat and G. Leugering, Classical solutions and feedback stabilization for the gas flow in a sequence of pipes, Netw. Heterog. Media, 5 (2010), 691-709. doi: 10.3934/nhm.2010.5.691. Google Scholar

[14]

J. M. Greenberg and T. T. Li, The effect of boundary damping for the quasilinear wave equation, J. Differential Equations, 52 (1984), 66-75. doi: 10.1016/0022-0396(84)90135-9. Google Scholar

[15]

M. Gugat, Boundary feedback stabilization by time delay for one-dimensional wave equations, IMA J. Math. Control Inform., 27 (2010), 189-203. doi: 10.1093/imamci/dnq007. Google Scholar

[16]

M. Gugat and M. Dick, Time-delayed boundary feedback stabilization of the isothermal Euler equations with friction, Math. Control and Related Fields, 1 (2011), 469-491. doi: 10.3934/mcrf.2011.1.469. Google Scholar

[17]

M. GugatF. HanteM. Hirsch-Dick and G. Leugering, Stationary states in gas networks, Netw. Heterog. Media, 10 (2015), 295-320. doi: 10.3934/nhm.2015.10.295. Google Scholar

[18]

M. GugatM. HertyA. KlarG. Leugering and V. Schleper, Well-posedness of networked hyperbolic systems of balance laws, International Series of Numerical Mathematics, 160 (2012), 123-146. doi: 10.1007/978-3-0348-0133-1_7. Google Scholar

[19]

M. GugatG. LeugeringS. Tamasoiu and K. Wang, $H^2$-stabilization of the isothermal Euler equations: A Lyapunov function approach, Chinese Annals of Mathematics, Series B, 33 (2012), 479-500. doi: 10.1007/s11401-012-0727-y. Google Scholar

[20]

M. Gugat and M. Sigalotti, Stars of vibrating strings: Switching boundary feedback stabilization, Netw. Heterog. Media, 5 (2010), 299-314. doi: 10.3934/nhm.2010.5.299. Google Scholar

[21]

M. Gugat and M. Tucsnak, An example for the switching delay feedback stabilization of an infinite dimensional system: The boundary stabilization of a string, Syst. Cont. Lett., 60 (2011), 226-233. doi: 10.1016/j.sysconle.2011.01.004. Google Scholar

[22]

M. Gugat and S. Ulbrich, The isothermal Euler equations for ideal gas with source term: Product solutions, flow reversal and no blow up, J. Math. Anal. Appl., 454 (2017), 439-452. doi: 10.1016/j.jmaa.2017.04.064. Google Scholar

[23]

M. Gugat, Optimal Boundary Control and Boundary Stabilization of Hyperbolic Systems, SpringerBriefs in Control, Automation and Robotics, Springer, New York, New York, 2015. doi: 10.1007/978-3-319-18890-4. Google Scholar

[24]

L. Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations, Springer Verlag, Berlin, 1997. Google Scholar

[25]

T. J. R. HughesT. Kato and J. E. Marsden, Well-posed quasi-linear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity, Arch. Rational Mech. Anal., 63 (1977), 273-294. doi: 10.1007/BF00251584. Google Scholar

[26]

V. Komornik, Rapid boundary stabilization of the wave equation, SIAM J. Control and Optimization, 29 (1991), 197-208. doi: 10.1137/0329011. Google Scholar

[27]

J. H. Lambert, Observationes variae in mathesin puram, Acta Helvetica, physico-mathematico-anatomico-botanico-medica, 3 (1758), 128-168. Google Scholar

[28]

T. T. Li, Global Classical Solutions for Quasilinear Hyperbolic Systems, RAM Res. Appl. Math. 32, Masson, Paris, 1994. Google Scholar

[29]

T. T. Li and W. C. Yu, Boundary Value Problems for Quasilinear Hyperbolic Systems, Duke Univ. Math. Ser. V, 1985. Google Scholar

[30]

T. T. LiB. Rao and Z. Wang, A note on the one-side exact boundary observability for quasilinear hyperbolic systems, Georgian Mathematical Journal, 15 (2008), 571-580. Google Scholar

[31]

Z. H. Luo, B. Z. Guo and O. Morgul, Stability and Stabilization of Infinite Dimensional Systems with Applications, Communications and Control Engineering Series. Springer-Verlag London Ltd. , London, 1999. doi: 10.1007/978-1-4471-0419-3. Google Scholar

[32]

F. Mazenc and C. Prieur, Strict Lyapunov functions for semilinear parabolic partial differential equations, Math. Control and Related Fields, 1 (2011), 231-250. doi: 10.3934/mcrf.2011.1.231. Google Scholar

[33]

S. NicaiseJ. Valein and E. Fridman, Stability of the heat and of the wave equations with boundary time-varying delays, Disc. Contin. Dyn. Syst. Ser. S, 2 (2009), 559-581. doi: 10.3934/dcdss.2009.2.559. Google Scholar

[34]

M. Slemrod, Boundary feedback stabilization for a quasilinear wave equation, in Control Theory for Distributed Parameter Systems and Applications, Lecture Notes in Control and Inform. Sci., Springer-Verlag, Berlin, 54 (1983), 221-237. doi: 10.1007/BFb0043951. Google Scholar

[35]

M. E. Taylor, Partial Differential Equations, Tome Ⅲ, Nonlinear equations, Applied Mathematical Sciences, Vol. 117, Springer-Verlag, New-York, 1996. doi: 10.1007/978-1-4684-9320-7. Google Scholar

[36]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser, Basel-Boston-Berlin, 2009. doi: 10.1007/978-3-7643-8994-9. Google Scholar

[37]

Z. Q. Wang, Exact boundary controllability for nonautonomous quasilinear wave equations, Math. Meth. Appl. Sci., 30 (2007), 1311-1327. doi: 10.1002/mma.843. Google Scholar

[38]

G. P. ZouN. Cheraghi and F. Taheri, Fluid-induced vibration of composite natural gas pipelines, International Journal of Solids and Structures, 42 (2005), 1253-1268. doi: 10.1016/j.ijsolstr.2004.07.001. Google Scholar

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Markus Dick, Martin Gugat, Günter Leugering. A strict $H^1$-Lyapunov function and feedback stabilization for the isothermal Euler equations with friction. Numerical Algebra, Control & Optimization, 2011, 1 (2) : 225-244. doi: 10.3934/naco.2011.1.225

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Martin Gugat, Markus Dick. Time-delayed boundary feedback stabilization of the isothermal Euler equations with friction. Mathematical Control & Related Fields, 2011, 1 (4) : 469-491. doi: 10.3934/mcrf.2011.1.469

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