Networks of geometrically exact beams: Well-posedness and stabilization

Charlotte Rodriguez

doi:10.3934/mcrf.2021002

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March 2022, 12(1): 49-80. doi: 10.3934/mcrf.2021002

Networks of geometrically exact beams: Well-posedness and stabilization

Charlotte Rodriguez ^,

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

^* Corresponding author: Charlotte Rodriguez

Received September 2020 Revised December 2020 Published March 2022 Early access January 2021

Fund Project: This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No.765579-ConFlex

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 [2] that the zero steady state of this network is exponentially stable for the $ H^1 $ and $ H^2 $ norms. The major obstacles to overcome in the intrinsic formulation of the GEB network, are that the governing equations are semilinar, containing a quadratic nonlinearity, and that linear lower order terms cannot be neglected.

Keywords: Geometrically exact beam, intrinsic beam, one-dimensional first-order semilinear hyperbolic systems, tree-shaped network, well-posedness, exponential stabilization, boundary feedback.

Mathematics Subject Classification: 35L50, 35R02, 93D15.

Citation: Charlotte Rodriguez. Networks of geometrically exact beams: Well-posedness and stabilization. Mathematical Control & Related Fields, 2022, 12 (1) : 49-80. doi: 10.3934/mcrf.2021002

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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[9]	L. C. Evans, Partial Differential Equations, 2^nd edition, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[22]	R. A. Horn and C. R. Johnson, Matrix Analysis, 2$^{nd}$ edition, CUP, Cambridge, 2013. Google Scholar
[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. Google Scholar
[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. Google Scholar
[25]	T. T. Li, Global Classical Solutions for Quasilinear Hyperbolic Systems, Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[38]	H. Weiss, Zur Dynamik Geometrisch Nichtlinearer Balken, Ph.D thesis, Technische Universität Chemnitz, 1999. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[9]	L. C. Evans, Partial Differential Equations, 2^nd edition, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[22]	R. A. Horn and C. R. Johnson, Matrix Analysis, 2$^{nd}$ edition, CUP, Cambridge, 2013. Google Scholar
[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. Google Scholar
[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. Google Scholar
[25]	T. T. Li, Global Classical Solutions for Quasilinear Hyperbolic Systems, Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[38]	H. Weiss, Zur Dynamik Geometrisch Nichtlinearer Balken, Ph.D thesis, Technische Universität Chemnitz, 1999. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

Figure 1. Left: tree-shaped network with $ N = 8 $ edges, $ \mathcal{N}_S = \{0, 3, 5, 6, 7, 8\} $ and $ \mathcal{N}_M = \{ 1, 2, 4 \} $. Right: star-shaped network with $ N = 4 $ edges, $ \mathcal{N}_S = \{0, 2, 3, 4\} $ and $ \mathcal{N}_M = \{ 1 \} $

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Figure 2. A multiple $ n $ and the incident edges and nodes

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Figure 3. The $ i $-th beam in its different configurations $ \Omega_s^i $, $ \Omega_c^i $ and $ \Omega_t^i $

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Figure 4. Characteristic curves $ (\mathbf{x}(t), t) $ with $ \frac{\mathrm{d}\mathbf{x}}{\mathrm{d}t}(t) = \lambda(\mathbf{x}(t)) $, where either $ \lambda(s)>0 $ or $ \lambda(s)<0 $ for all $ s \in [0, \ell] $

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Figure 5. Example of choice of $ \rho $ and the weight functions

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Figure 6. Recovering the single beam case: example of choice of $ \rho $ and weight functions

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