doi: 10.3934/mcrf.2021002
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Networks of geometrically exact beams: Well-posedness and stabilization

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 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.

Citation: Charlotte Rodriguez. Networks of geometrically exact beams: Well-posedness and stabilization. Mathematical Control & Related Fields, doi: 10.3934/mcrf.2021002
References:
[1]

F. Alabau-BoussouiraV. 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. BastinB. HautJ.-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. ChenM. C. DelfourA. 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 HalleuxC. PrieurJ.-M. CoronB. 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, 2nd edition, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019.  Google Scholar

[10]

S. GraziosoG. 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. GugatV. 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. PalaciosJ. 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. WangX. LiuN. RenevierM. 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. XuZ. 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. ZhangG. 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-BoussouiraV. 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. BastinB. HautJ.-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. ChenM. C. DelfourA. 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 HalleuxC. PrieurJ.-M. CoronB. 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, 2nd edition, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019.  Google Scholar

[10]

S. GraziosoG. 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. GugatV. 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. PalaciosJ. 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. WangX. LiuN. RenevierM. 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. XuZ. 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. ZhangG. 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 \} $
Figure 2.  A multiple $ n $ and the incident edges and nodes
Figure 3.  The $ i $-th beam in its different configurations $ \Omega_s^i $, $ \Omega_c^i $ and $ \Omega_t^i $
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] $
Figure 5.  Example of choice of $ \rho $ and the weight functions
Figure 6.  Recovering the single beam case: example of choice of $ \rho $ and weight functions
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