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Sampled-data integral control of multivariable linear infinite-dimensional systems with input nonlinearities
Networks of geometrically exact beams: Well-posedness and stabilization
Lehrstuhl 2 für Angewandte Mathematik, Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstr. 11, 91058 Erlangen, Germany |
In this work, we are interested in tree-shaped networks of freely vibrating beams which are geometrically exact (GEB) – in the sense that large motions (deflections, rotations) are accounted for in addition to shearing – and linked by rigid joints. For the intrinsic GEB formulation, namely that in terms of velocities and internal forces/moments, we derive transmission conditions and show that the network is locally in time well-posed in the classical sense. Applying velocity feedback controls at the external nodes of a star-shaped network, we show by means of a quadratic Lyapunov functional and the theory developed by Bastin & Coron in [
References:
[1] |
F. Alabau-Boussouira, V. Perrollaz and L. Rosier,
Finite-time stabilization of a network of strings, Math. Control Relat. Fields, 5 (2015), 721-742.
doi: 10.3934/mcrf.2015.5.721. |
[2] |
G. Bastin and J.-M. Coron, Stability and Boundary Stabilization of 1-D Hyperbolic Systems, Birkhäuser/Springer, Cham, 2016.
doi: 10.1007/978-3-319-32062-5. |
[3] |
G. Bastin and J.-M. Coron, Exponential stability of semi-linear one-dimensional balance laws, in Feedback Stabilization of Controlled Dynamical Systems, Springer, Cham, 2017,265–278.
doi: 10.1007/978-3-319-51298-3_10. |
[4] |
G. Bastin and J.-M. Coron,
A quadratic Lyapunov function for hyperbolic density-velocity systems with nonuniform steady states, Systems Control Lett., 104 (2017), 66-71.
doi: 10.1016/j.sysconle.2017.03.013. |
[5] |
G. Bastin, B. Haut, J.-M. Coron and B. D'andréa-Novel,
Lyapunov stability analysis of networks of scalar conservation laws, Netw. Heterog. Media, 2 (2007), 751-759.
doi: 10.3934/nhm.2007.2.751. |
[6] |
G. Chen, M. C. Delfour, A. M. Krall and G. Payre,
Modeling, stabilization and control of serially connected beams, SIAM J. Control Optim., 25 (1987), 526-546.
doi: 10.1137/0325029. |
[7] |
R. Dáger and E. Zuazua, Wave Propagation, Observation and Control in {$1\text{-}d$} Flexible Multi-Structures, Springer-Verlag, Berlin, 2006.
doi: 10.1007/3-540-37726-3. |
[8] |
J. de Halleux, C. Prieur, J.-M. Coron, B. d'Andréa Novel and G. Bastin,
Boundary feedback control in networks of open channels, Automatica J. IFAC, 39 (2003), 1365-1376.
doi: 10.1016/S0005-1098(03)00109-2. |
[9] |
L. C. Evans, Partial Differential Equations, 2nd edition, American Mathematical Society, Providence, RI, 2010.
doi: 10.1090/gsm/019. |
[10] |
S. Grazioso, G. Di Gironimo and B. Siciliano,
A geometrically exact model for soft continuum robots: The finite element deformation space formulation, Soft Robotics, 6 (2019), 790-811.
doi: 10.1089/soro.2018.0047. |
[11] |
M. Gugat, V. Perrollaz and L. Rosier,
Boundary stabilization of quasilinear hyperbolic systems of balance laws: Exponential decay for small source terms, J. Evol. Equ., 18 (2018), 1471-1500.
doi: 10.1007/s00028-018-0449-z. |
[12] |
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. |
[13] |
Y. N. Guo and G. Q. Xu,
Exponential stabilisation of a tree-shaped network of strings with variable coefficients, Glasg. Math. J., 53 (2011), 481-499.
doi: 10.1017/S0017089511000085. |
[14] |
Z. J. Han and G. Q. Xu,
Exponential stabilisation of a simple tree-shaped network of Timoshenko beams system, Internat. J. Control, 83 (2010), 1485-1503.
doi: 10.1080/00207179.2010.481767. |
[15] |
Z. J. Han and G. Q. Xu,
Dynamical behavior of networks of non-uniform Timoshenko beams system with boundary time-delay inputs, Netw. Heterog. Media, 6 (2011), 297-327.
doi: 10.3934/nhm.2011.6.297. |
[16] |
A. Hayat, Exponential stability of general 1-D quasilinear systems with source terms for the $C^1$ norm under boundary conditions, preprint, arXiv: 1801.02353. |
[17] |
A. Hayat and P. Shang, Exponential stability of density-velocity systems with boundary conditions and source term for the $H^2$ norm, preprint, hal-02190778. |
[18] |
A. Hayat and P. Shang,
A quadratic Lyapunov function for Saint-Venant equations with arbitrary friction and space-varying slope, Automatica J. IFAC, 100 (2019), 52-60.
doi: 10.1016/j.automatica.2018.10.035. |
[19] |
M. Herty and H. Yu,
Feedback boundary control of linear hyperbolic equations with stiff source term, Internat. J. Control, 91 (2018), 230-240.
doi: 10.1080/00207179.2016.1276635. |
[20] |
D. H. Hodges,
A mixed variational formulation based on exact intrinsic equations for dynamics of moving beams, Int. J. Solids Struct., 26 (1990), 1253-1273.
doi: 10.1016/0020-7683(90)90060-9. |
[21] |
D. H. Hodges,
Geometrically exact, intrinsic theory for dynamics of curved and twisted anisotropic beams, AIAA Journal, 41 (2003), 1131-1137.
doi: 10.2514/2.2054. |
[22] |
R. A. Horn and C. R. Johnson, Matrix Analysis, 2$^{nd}$ edition, CUP, Cambridge, 2013. |
[23] |
J. E. Lagnese, G. Leugering and E. J. P. G. Schmidt, Modeling, Analysis and Control of Dynamic Elastic Multi-Link Structures, Birkhäuser Boston, Inc., Boston, MA, 1994.
doi: 10.1007/978-1-4612-0273-8. |
[24] |
G. Leugering and E. J. P. G. Schmidt,
On the modelling and stabilization of flows in networks of open canals, SIAM J. Control Optim., 41 (2002), 164-180.
doi: 10.1137/S0363012900375664. |
[25] |
T. T. Li, Global Classical Solutions for Quasilinear Hyperbolic Systems, Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994. |
[26] |
T. T. Li and W. C. Yu, Boundary Value problems for Quasilinear Hyperbolic Systems, Duke University Mathematics Series, 5, Duke University, Mathematics Department, Durham, NC, 1985. |
[27] |
M. Matsuoka, T. Murakami and K. Ohnishi, Vibration suppression and disturbance rejection control of a flexible link arm, in Proceedings of IECON'95-21st Annual Conference on IEEE Industrial Electronics, IEEE, 1995, 1260–1265.
doi: 10.1109/IECON.1995.483978. |
[28] |
S. Nicaise and J. Valein,
Stabilization of the wave equation on 1-D networks with a delay term in the nodal feedbacks, Netw. Heterog. Media, 2 (2007), 425-479.
doi: 10.3934/nhm.2007.2.425. |
[29] |
R. Palacios, J. Murua and R. Cook,
Structural and aerodynamic models in nonlinear flight dynamics of very flexible aircraft, AIAA Journal, 48 (2010), 2648-2659.
doi: 10.2514/1.J050513. |
[30] |
E. Reissner, On finite deformations of space-curved beams, Zeitschrift für Angewandte Mathematik und Physik ZAMP, 32 (1981), 734–744.
doi: 10.1007/BF00946983. |
[31] |
C. Rodriguez and G. Leugering,
Boundary feedback stabilization for the intrinsic geometrically exact beam model, SIAM J. Control Optim., 58 (2020), 3533-3558.
doi: 10.1137/20M1340010. |
[32] |
J. Simo,
A finite strain beam formulation. The three-dimensional dynamic problem. Part Ⅰ, Comput. Methods in Appl. Mech. and Engrg., 49 (1985), 55-70.
doi: 10.1016/0045-7825(85)90050-7. |
[33] |
C. Strohmeyer, Networks of Nonlinear Thin Structures - Theory and Applications, Ph.D thesis, FAU University Press, 2018.
doi: 10.25593/978-3-96147-138-6.![]() ![]() |
[34] |
M. Uchiyama and A. Konno, Computed acceleration control for the vibration suppression of flexible robotic manipulators, in Fifth International Conference on Advanced Robotics' Robots in Unstructured Environments, IEEE, 1991,126–131.
doi: 10.1109/ICAR.1991.240464. |
[35] |
J. Valein and E. Zuazua,
Stabilization of the wave equation on 1-D networks, SIAM J. Control Optim., 48 (2009), 2771-2797.
doi: 10.1137/080733590. |
[36] |
A. H. von Flotow,
Traveling wave control for large spacecraft structures, Journal of Guidance, Control, and Dynamics, 9 (1986), 462-468.
doi: 10.2514/3.20133. |
[37] |
L. Wang, X. Liu, N. Renevier, M. Stables and G. M. Hall,
Nonlinear aeroelastic modelling for wind turbine blades based on blade element momentum theory and geometrically exact beam theory, Energy, 76 (2014), 487-501.
doi: 10.1016/j.energy.2014.08.046. |
[38] |
H. Weiss, Zur Dynamik Geometrisch Nichtlinearer Balken, Ph.D thesis, Technische Universität Chemnitz, 1999. |
[39] |
G. Q. Xu, Z. J. Han and S. P. Yung,
Riesz basis property of serially connected Timoshenko beams, Internat. J. Control, 80 (2007), 470-485.
doi: 10.1080/00207170601100904. |
[40] |
K. T. Zhang, G. Q. Xu and N. E. Mastorakis,
Stability of a complex network of Euler-Bernoulli beams, WSEAS Trans. Syst., 8 (2009), 379-389.
|
[41] |
Y. Zhang and G. Xu,
Exponential and super stability of a wave network, Acta Appl. Math., 124 (2013), 19-41.
doi: 10.1007/s10440-012-9768-1. |
[42] |
E. Zuazua, Control and stabilization of waves on 1-d networks, in Modelling and Optimisation of Flows on Networks, Lecture Notes in Math., 2062, Springer, Heidelberg, 2013,463–493.
doi: 10.1007/978-3-642-32160-3_9. |
show all references
References:
[1] |
F. Alabau-Boussouira, V. Perrollaz and L. Rosier,
Finite-time stabilization of a network of strings, Math. Control Relat. Fields, 5 (2015), 721-742.
doi: 10.3934/mcrf.2015.5.721. |
[2] |
G. Bastin and J.-M. Coron, Stability and Boundary Stabilization of 1-D Hyperbolic Systems, Birkhäuser/Springer, Cham, 2016.
doi: 10.1007/978-3-319-32062-5. |
[3] |
G. Bastin and J.-M. Coron, Exponential stability of semi-linear one-dimensional balance laws, in Feedback Stabilization of Controlled Dynamical Systems, Springer, Cham, 2017,265–278.
doi: 10.1007/978-3-319-51298-3_10. |
[4] |
G. Bastin and J.-M. Coron,
A quadratic Lyapunov function for hyperbolic density-velocity systems with nonuniform steady states, Systems Control Lett., 104 (2017), 66-71.
doi: 10.1016/j.sysconle.2017.03.013. |
[5] |
G. Bastin, B. Haut, J.-M. Coron and B. D'andréa-Novel,
Lyapunov stability analysis of networks of scalar conservation laws, Netw. Heterog. Media, 2 (2007), 751-759.
doi: 10.3934/nhm.2007.2.751. |
[6] |
G. Chen, M. C. Delfour, A. M. Krall and G. Payre,
Modeling, stabilization and control of serially connected beams, SIAM J. Control Optim., 25 (1987), 526-546.
doi: 10.1137/0325029. |
[7] |
R. Dáger and E. Zuazua, Wave Propagation, Observation and Control in {$1\text{-}d$} Flexible Multi-Structures, Springer-Verlag, Berlin, 2006.
doi: 10.1007/3-540-37726-3. |
[8] |
J. de Halleux, C. Prieur, J.-M. Coron, B. d'Andréa Novel and G. Bastin,
Boundary feedback control in networks of open channels, Automatica J. IFAC, 39 (2003), 1365-1376.
doi: 10.1016/S0005-1098(03)00109-2. |
[9] |
L. C. Evans, Partial Differential Equations, 2nd edition, American Mathematical Society, Providence, RI, 2010.
doi: 10.1090/gsm/019. |
[10] |
S. Grazioso, G. Di Gironimo and B. Siciliano,
A geometrically exact model for soft continuum robots: The finite element deformation space formulation, Soft Robotics, 6 (2019), 790-811.
doi: 10.1089/soro.2018.0047. |
[11] |
M. Gugat, V. Perrollaz and L. Rosier,
Boundary stabilization of quasilinear hyperbolic systems of balance laws: Exponential decay for small source terms, J. Evol. Equ., 18 (2018), 1471-1500.
doi: 10.1007/s00028-018-0449-z. |
[12] |
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. |
[13] |
Y. N. Guo and G. Q. Xu,
Exponential stabilisation of a tree-shaped network of strings with variable coefficients, Glasg. Math. J., 53 (2011), 481-499.
doi: 10.1017/S0017089511000085. |
[14] |
Z. J. Han and G. Q. Xu,
Exponential stabilisation of a simple tree-shaped network of Timoshenko beams system, Internat. J. Control, 83 (2010), 1485-1503.
doi: 10.1080/00207179.2010.481767. |
[15] |
Z. J. Han and G. Q. Xu,
Dynamical behavior of networks of non-uniform Timoshenko beams system with boundary time-delay inputs, Netw. Heterog. Media, 6 (2011), 297-327.
doi: 10.3934/nhm.2011.6.297. |
[16] |
A. Hayat, Exponential stability of general 1-D quasilinear systems with source terms for the $C^1$ norm under boundary conditions, preprint, arXiv: 1801.02353. |
[17] |
A. Hayat and P. Shang, Exponential stability of density-velocity systems with boundary conditions and source term for the $H^2$ norm, preprint, hal-02190778. |
[18] |
A. Hayat and P. Shang,
A quadratic Lyapunov function for Saint-Venant equations with arbitrary friction and space-varying slope, Automatica J. IFAC, 100 (2019), 52-60.
doi: 10.1016/j.automatica.2018.10.035. |
[19] |
M. Herty and H. Yu,
Feedback boundary control of linear hyperbolic equations with stiff source term, Internat. J. Control, 91 (2018), 230-240.
doi: 10.1080/00207179.2016.1276635. |
[20] |
D. H. Hodges,
A mixed variational formulation based on exact intrinsic equations for dynamics of moving beams, Int. J. Solids Struct., 26 (1990), 1253-1273.
doi: 10.1016/0020-7683(90)90060-9. |
[21] |
D. H. Hodges,
Geometrically exact, intrinsic theory for dynamics of curved and twisted anisotropic beams, AIAA Journal, 41 (2003), 1131-1137.
doi: 10.2514/2.2054. |
[22] |
R. A. Horn and C. R. Johnson, Matrix Analysis, 2$^{nd}$ edition, CUP, Cambridge, 2013. |
[23] |
J. E. Lagnese, G. Leugering and E. J. P. G. Schmidt, Modeling, Analysis and Control of Dynamic Elastic Multi-Link Structures, Birkhäuser Boston, Inc., Boston, MA, 1994.
doi: 10.1007/978-1-4612-0273-8. |
[24] |
G. Leugering and E. J. P. G. Schmidt,
On the modelling and stabilization of flows in networks of open canals, SIAM J. Control Optim., 41 (2002), 164-180.
doi: 10.1137/S0363012900375664. |
[25] |
T. T. Li, Global Classical Solutions for Quasilinear Hyperbolic Systems, Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994. |
[26] |
T. T. Li and W. C. Yu, Boundary Value problems for Quasilinear Hyperbolic Systems, Duke University Mathematics Series, 5, Duke University, Mathematics Department, Durham, NC, 1985. |
[27] |
M. Matsuoka, T. Murakami and K. Ohnishi, Vibration suppression and disturbance rejection control of a flexible link arm, in Proceedings of IECON'95-21st Annual Conference on IEEE Industrial Electronics, IEEE, 1995, 1260–1265.
doi: 10.1109/IECON.1995.483978. |
[28] |
S. Nicaise and J. Valein,
Stabilization of the wave equation on 1-D networks with a delay term in the nodal feedbacks, Netw. Heterog. Media, 2 (2007), 425-479.
doi: 10.3934/nhm.2007.2.425. |
[29] |
R. Palacios, J. Murua and R. Cook,
Structural and aerodynamic models in nonlinear flight dynamics of very flexible aircraft, AIAA Journal, 48 (2010), 2648-2659.
doi: 10.2514/1.J050513. |
[30] |
E. Reissner, On finite deformations of space-curved beams, Zeitschrift für Angewandte Mathematik und Physik ZAMP, 32 (1981), 734–744.
doi: 10.1007/BF00946983. |
[31] |
C. Rodriguez and G. Leugering,
Boundary feedback stabilization for the intrinsic geometrically exact beam model, SIAM J. Control Optim., 58 (2020), 3533-3558.
doi: 10.1137/20M1340010. |
[32] |
J. Simo,
A finite strain beam formulation. The three-dimensional dynamic problem. Part Ⅰ, Comput. Methods in Appl. Mech. and Engrg., 49 (1985), 55-70.
doi: 10.1016/0045-7825(85)90050-7. |
[33] |
C. Strohmeyer, Networks of Nonlinear Thin Structures - Theory and Applications, Ph.D thesis, FAU University Press, 2018.
doi: 10.25593/978-3-96147-138-6.![]() ![]() |
[34] |
M. Uchiyama and A. Konno, Computed acceleration control for the vibration suppression of flexible robotic manipulators, in Fifth International Conference on Advanced Robotics' Robots in Unstructured Environments, IEEE, 1991,126–131.
doi: 10.1109/ICAR.1991.240464. |
[35] |
J. Valein and E. Zuazua,
Stabilization of the wave equation on 1-D networks, SIAM J. Control Optim., 48 (2009), 2771-2797.
doi: 10.1137/080733590. |
[36] |
A. H. von Flotow,
Traveling wave control for large spacecraft structures, Journal of Guidance, Control, and Dynamics, 9 (1986), 462-468.
doi: 10.2514/3.20133. |
[37] |
L. Wang, X. Liu, N. Renevier, M. Stables and G. M. Hall,
Nonlinear aeroelastic modelling for wind turbine blades based on blade element momentum theory and geometrically exact beam theory, Energy, 76 (2014), 487-501.
doi: 10.1016/j.energy.2014.08.046. |
[38] |
H. Weiss, Zur Dynamik Geometrisch Nichtlinearer Balken, Ph.D thesis, Technische Universität Chemnitz, 1999. |
[39] |
G. Q. Xu, Z. J. Han and S. P. Yung,
Riesz basis property of serially connected Timoshenko beams, Internat. J. Control, 80 (2007), 470-485.
doi: 10.1080/00207170601100904. |
[40] |
K. T. Zhang, G. Q. Xu and N. E. Mastorakis,
Stability of a complex network of Euler-Bernoulli beams, WSEAS Trans. Syst., 8 (2009), 379-389.
|
[41] |
Y. Zhang and G. Xu,
Exponential and super stability of a wave network, Acta Appl. Math., 124 (2013), 19-41.
doi: 10.1007/s10440-012-9768-1. |
[42] |
E. Zuazua, Control and stabilization of waves on 1-d networks, in Modelling and Optimisation of Flows on Networks, Lecture Notes in Math., 2062, Springer, Heidelberg, 2013,463–493.
doi: 10.1007/978-3-642-32160-3_9. |






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