March  2017, 6(1): 59-75. doi: 10.3934/eect.2017004

Indirect stabilization of hyperbolic systems through resolvent estimates

Radon Institute for Computational and Applied Mathematics - RICAM, Altenbergerstraße 69, 4040 - Linz, Austria

Received  February 2015 Revised  September 2016 Published  December 2016

We prove a sharp decay rate for the total energy of two classes of systems of weakly coupled hyperbolic equations. We show that we can stabilize the full system through a single damping term, in feedback form, acting on one component only of the system (\emph{indirect stabilization}). The energy estimate is achieved by means of suitable estimates of the resolvent operator norm. We apply this technique to a wave-wave system and to a wave-Petrovsky system.

Citation: Roberto Guglielmi. Indirect stabilization of hyperbolic systems through resolvent estimates. Evolution Equations & Control Theory, 2017, 6 (1) : 59-75. doi: 10.3934/eect.2017004
References:
[1]

F. Alabau, Stabilisation frontiére indirecte de systémes faiblement couplés, C. R. Acad. Sci. Paris Sér. I Math., 328 (1999), 1015-1020.  doi: 10.1016/S0764-4442(99)80316-4.  Google Scholar

[2]

F. AlabauP. Cannarsa and V. Komornik, Indirect internal stabilization of weakly coupled evolution equations, J. Evol. Equ., 2 (2002), 127-150.  doi: 10.1007/s00028-002-8083-0.  Google Scholar

[3]

F. Alabau-BoussouiraP. Cannarsa and R. Guglielmi, Indirect stabilization of weakly coupled systems with hybrid boundary conditions, Math. Control Relat. Fields, 1 (2011), 413-436.  doi: 10.3934/mcrf.2011.1.413.  Google Scholar

[4]

F. Alabau-Boussouira and M. Léautaud, Indirect stabilization of locally coupled wave-type systems, ESAIM Control Optim. Calc. Var., 18 (2012), 548-582.  doi: 10.1051/cocv/2011106.  Google Scholar

[5]

F. Alabau-Boussouira and M. Léautaud, Indirect controllability of locally coupled wave-type systems and applications, J. Math. Pures Appl. (9), 99 (2013), 544-576.  doi: 10.1016/j.matpur.2012.09.012.  Google Scholar

[6]

W. Arendt and C.J.K. Batty, Tauberian theorems and stability of one-parameter semigroups, Trans. Amer. Math. Soc., 306 (1988), 837-852.  doi: 10.1090/S0002-9947-1988-0933321-3.  Google Scholar

[7]

G. Avalos and F. Bucci, Rational rates of uniform decay for strong solutions to a fluid-structure PDE system, J. Differential Equations, 258 (2015), 4398-4423.  doi: 10.1016/j.jde.2015.01.037.  Google Scholar

[8]

G. AvalosI. Lasiecka and R. Triggiani, Beyond lack of compactness and lack of stability of a coupled parabolic-hyperbolic fluid-structure system, In Optimal control of coupled systems of partial differential equations, volume 158 of Internat. Ser. Numer. Math, (2009), 1-33.  doi: 10.1007/978-3-7643-8923-9_1.  Google Scholar

[9]

G. AvalosI. Lasiecka and R. Triggiani, Heat-wave interaction in 2-3 dimensions: Optimal rational decay rate, J. Math. Anal. Appl., 437 (2016), 782-815.  doi: 10.1016/j.jmaa.2015.12.051.  Google Scholar

[10]

G. Avalos and R. Triggiani, Rational decay rates for a PDE heat-structure interaction: A frequency domain approach, Evol. Equ. Control Theory, 2 (2013), 233-253.  doi: 10.3934/eect.2013.2.233.  Google Scholar

[11]

A. BátkaiK.-J. EngelJ. Prüss and R. Schnaubelt, Polynomial stability of operator semigroups, Math. Nachr., 279 (2006), 1425-1440.  doi: 10.1002/mana.200410429.  Google Scholar

[12]

C. J.K. Batty and T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces, J. Evol. Equ., 8 (2008), 765-780.  doi: 10.1007/s00028-008-0424-1.  Google Scholar

[13]

A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Math. Ann., 347 (2010), 455-478.  doi: 10.1007/s00208-009-0439-0.  Google Scholar

[14]

R. Dáger and E. Zuazua, Wave Propagation, Observation and Control in 1-d Flexible Multi-structures, volume 50 of Mathématiques & Applications (Berlin) [Mathematics & Applications], Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-37726-3.  Google Scholar

[15]

B. DehmanJ. Le Rousseau and M. Léautaud, Controllability of two coupled wave equations on a compact manifold, Arch. Ration. Mech. Anal., 211 (2014), 113-187.  doi: 10.1007/s00205-013-0670-4.  Google Scholar

[16]

M. Eller, I. Lasiecka and R. Triggiani, Simultaneous exact/approximate boundary controllability of thermo-elastic plates with variable transmission coefficient, In Shape optimization and optimal design (Cambridge, 1999), volume 216 of Lecture Notes in Pure and Appl. Math. , pages 109-230, Dekker, New York, 2001. Google Scholar

[17]

R. GulliverI. LasieckaW. Littman and R. Triggiani, The case for differential geometry in the control of single and coupled PDEs: the structural acoustic chamber, In Geometric methods in inverse problems and PDE control, volume 137 of IMA Vol. Math., (2004), 73-181.  doi: 10.1007/978-1-4684-9375-7_5.  Google Scholar

[18]

V. Komornik, Exact Controllability and Stabilization RAM: Research in Applied Mathematics. Masson, Paris, 1994. The multiplier method.  Google Scholar

[19]

J. Lagnese and J. -L. Lions, Modelling Analysis and Control of Thin Plates volume 6 of Recherches en Mathématiques Appliquées [Research in Applied Mathematics], Masson, Paris, 1988.  Google Scholar

[20]

G. Lebeau, Équation des ondes amorties, In Algebraic and geometric methods in mathematical physics (Kaciveli, 1993), volume 19 of Math. Phys. Stud. , pages 73-109, Kluwer Acad. Publ. , Dordrecht, 1996.  Google Scholar

[21]

Yu. I. Lyubich and Quôc Phóng Vû, Asymptotic stability of linear differential equations in Banach spaces, Studia Math., 88 (1988), 37-42.   Google Scholar

[22]

D.L. Russell, A general framework for the study of indirect damping mechanisms in elastic systems, J. Math. Anal. Appl., 173 (1993), 339-358.  doi: 10.1006/jmaa.1993.1071.  Google Scholar

[23]

L. Tebou, Energy decay estimates for some weakly coupled Euler-Bernoulli and wave equations with indirect damping mechanisms, Math. Control Relat. Fields, 2 (2012), 45-60.  doi: 10.3934/mcrf.2012.2.45.  Google Scholar

[24]

X. Zhang and E. Zuazua, Asymptotic behavior of a hyperbolic-parabolic coupled system arising in fluid-structure interaction, in Free Boundary Problems, Internat. Ser. Numer. Math., 154, Birkhäuser, Basel, (2007), 445-455.  doi: 10.1007/978-3-7643-7719-9_43.  Google Scholar

show all references

References:
[1]

F. Alabau, Stabilisation frontiére indirecte de systémes faiblement couplés, C. R. Acad. Sci. Paris Sér. I Math., 328 (1999), 1015-1020.  doi: 10.1016/S0764-4442(99)80316-4.  Google Scholar

[2]

F. AlabauP. Cannarsa and V. Komornik, Indirect internal stabilization of weakly coupled evolution equations, J. Evol. Equ., 2 (2002), 127-150.  doi: 10.1007/s00028-002-8083-0.  Google Scholar

[3]

F. Alabau-BoussouiraP. Cannarsa and R. Guglielmi, Indirect stabilization of weakly coupled systems with hybrid boundary conditions, Math. Control Relat. Fields, 1 (2011), 413-436.  doi: 10.3934/mcrf.2011.1.413.  Google Scholar

[4]

F. Alabau-Boussouira and M. Léautaud, Indirect stabilization of locally coupled wave-type systems, ESAIM Control Optim. Calc. Var., 18 (2012), 548-582.  doi: 10.1051/cocv/2011106.  Google Scholar

[5]

F. Alabau-Boussouira and M. Léautaud, Indirect controllability of locally coupled wave-type systems and applications, J. Math. Pures Appl. (9), 99 (2013), 544-576.  doi: 10.1016/j.matpur.2012.09.012.  Google Scholar

[6]

W. Arendt and C.J.K. Batty, Tauberian theorems and stability of one-parameter semigroups, Trans. Amer. Math. Soc., 306 (1988), 837-852.  doi: 10.1090/S0002-9947-1988-0933321-3.  Google Scholar

[7]

G. Avalos and F. Bucci, Rational rates of uniform decay for strong solutions to a fluid-structure PDE system, J. Differential Equations, 258 (2015), 4398-4423.  doi: 10.1016/j.jde.2015.01.037.  Google Scholar

[8]

G. AvalosI. Lasiecka and R. Triggiani, Beyond lack of compactness and lack of stability of a coupled parabolic-hyperbolic fluid-structure system, In Optimal control of coupled systems of partial differential equations, volume 158 of Internat. Ser. Numer. Math, (2009), 1-33.  doi: 10.1007/978-3-7643-8923-9_1.  Google Scholar

[9]

G. AvalosI. Lasiecka and R. Triggiani, Heat-wave interaction in 2-3 dimensions: Optimal rational decay rate, J. Math. Anal. Appl., 437 (2016), 782-815.  doi: 10.1016/j.jmaa.2015.12.051.  Google Scholar

[10]

G. Avalos and R. Triggiani, Rational decay rates for a PDE heat-structure interaction: A frequency domain approach, Evol. Equ. Control Theory, 2 (2013), 233-253.  doi: 10.3934/eect.2013.2.233.  Google Scholar

[11]

A. BátkaiK.-J. EngelJ. Prüss and R. Schnaubelt, Polynomial stability of operator semigroups, Math. Nachr., 279 (2006), 1425-1440.  doi: 10.1002/mana.200410429.  Google Scholar

[12]

C. J.K. Batty and T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces, J. Evol. Equ., 8 (2008), 765-780.  doi: 10.1007/s00028-008-0424-1.  Google Scholar

[13]

A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Math. Ann., 347 (2010), 455-478.  doi: 10.1007/s00208-009-0439-0.  Google Scholar

[14]

R. Dáger and E. Zuazua, Wave Propagation, Observation and Control in 1-d Flexible Multi-structures, volume 50 of Mathématiques & Applications (Berlin) [Mathematics & Applications], Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-37726-3.  Google Scholar

[15]

B. DehmanJ. Le Rousseau and M. Léautaud, Controllability of two coupled wave equations on a compact manifold, Arch. Ration. Mech. Anal., 211 (2014), 113-187.  doi: 10.1007/s00205-013-0670-4.  Google Scholar

[16]

M. Eller, I. Lasiecka and R. Triggiani, Simultaneous exact/approximate boundary controllability of thermo-elastic plates with variable transmission coefficient, In Shape optimization and optimal design (Cambridge, 1999), volume 216 of Lecture Notes in Pure and Appl. Math. , pages 109-230, Dekker, New York, 2001. Google Scholar

[17]

R. GulliverI. LasieckaW. Littman and R. Triggiani, The case for differential geometry in the control of single and coupled PDEs: the structural acoustic chamber, In Geometric methods in inverse problems and PDE control, volume 137 of IMA Vol. Math., (2004), 73-181.  doi: 10.1007/978-1-4684-9375-7_5.  Google Scholar

[18]

V. Komornik, Exact Controllability and Stabilization RAM: Research in Applied Mathematics. Masson, Paris, 1994. The multiplier method.  Google Scholar

[19]

J. Lagnese and J. -L. Lions, Modelling Analysis and Control of Thin Plates volume 6 of Recherches en Mathématiques Appliquées [Research in Applied Mathematics], Masson, Paris, 1988.  Google Scholar

[20]

G. Lebeau, Équation des ondes amorties, In Algebraic and geometric methods in mathematical physics (Kaciveli, 1993), volume 19 of Math. Phys. Stud. , pages 73-109, Kluwer Acad. Publ. , Dordrecht, 1996.  Google Scholar

[21]

Yu. I. Lyubich and Quôc Phóng Vû, Asymptotic stability of linear differential equations in Banach spaces, Studia Math., 88 (1988), 37-42.   Google Scholar

[22]

D.L. Russell, A general framework for the study of indirect damping mechanisms in elastic systems, J. Math. Anal. Appl., 173 (1993), 339-358.  doi: 10.1006/jmaa.1993.1071.  Google Scholar

[23]

L. Tebou, Energy decay estimates for some weakly coupled Euler-Bernoulli and wave equations with indirect damping mechanisms, Math. Control Relat. Fields, 2 (2012), 45-60.  doi: 10.3934/mcrf.2012.2.45.  Google Scholar

[24]

X. Zhang and E. Zuazua, Asymptotic behavior of a hyperbolic-parabolic coupled system arising in fluid-structure interaction, in Free Boundary Problems, Internat. Ser. Numer. Math., 154, Birkhäuser, Basel, (2007), 445-455.  doi: 10.1007/978-3-7643-7719-9_43.  Google Scholar

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