doi: 10.3934/eect.2021063
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Stabilization of port-Hamiltonian systems with discontinuous energy densities

Fraunhofer Institute for Industrial Mathematics (ITWM), Fraunhofer-Platz 1, 67663 Kaiserslautern, Germany

Received  October 2018 Revised  October 2021 Early access January 2022

We establish an exponential stabilization result for linear port-Hamiltonian systems of first order with quite general, not necessarily continuous, energy densities. In fact, we have only to require the energy density of the system to be of bounded variation. In particular, and in contrast to the previously known stabilization results, our result applies to vibrating strings or beams with jumps in their mass density and their modulus of elasticity.

Citation: Jochen Schmid. Stabilization of port-Hamiltonian systems with discontinuous energy densities. Evolution Equations and Control Theory, doi: 10.3934/eect.2021063
References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2nd edition. Elsevier, 2003.

[2]

H. Amann and J. Escher, Analysis I, II, III, Birkhäuser, 2005, 2008, 2009.

[3]

B. Augner, Stabilisation of Infinite-Dimensional Port-Hamiltonian Systems via Dissipative Boundary Feedback, PhD thesis. Available at http://elpub.bib.uni-wuppertal.de/edocs/dokumente/fbc/mathematik/diss2016/augner/dc1613.pdf.

[4]

B. Augner and B. Jacob, Stability and stabilization of infinite-dimensional linear port-Hamiltonian systems, Evol. Equ. Contr. Th., 3 (2014), 207-229.  doi: 10.3934/eect.2014.3.207.

[5]

S. Cox and E. Zuazua, The rate at which energy decays in a string damped at one end, Indiana Univ. Math. J., 44 (1995), 545-573.  doi: 10.1512/iumj.1995.44.2001.

[6]

K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer, 2000.

[7]

G. B. Folland, Real Analysis, 2nd edition, Wiley, 1999.

[8]

E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups, American Mathematical Society Colloquium Publications, 1957.

[9]

B. JacobK. Morris and H. Zwart, $C_0$-semigroups for hyperbolic partial differential equations on a one-dimensional spatial domain, J. Evol. Equ., 15 (2015), 493-502.  doi: 10.1007/s00028-014-0271-1.

[10]

B. Jacob and H. J. Zwart, Linear Port-Hamiltonian Systems on Infinite-Dimensional Spaces, Birkhäuser, 2012. doi: 10.1007/978-3-0348-0399-1.

[11]

W. Rudin, Real and Complex Analysis, 3rd edition. McGraw-Hill, 1987.

[12]

J. Schmid and H. Zwart, Stabilization of port-Hamiltonian systems by nonlinear boundary control in the presence of disturbances, ESAIM Contr. Optim. Calc. Var., 27 (2021), Paper No. 53, 37 pp. doi: 10.1051/cocv/2021051.

[13]

W. Sierpiński, Sur un problème concernant les ensembles mésurables superficiellement, Fund. Math., 1 (1920), 112-115.  doi: 10.4064/fm-1-1-112-115.

[14]

W. Sierpiński, Sur les rapports entre l'existence des intégrales $\int_0^1f(x, y)dx$, $\int_0^1f(x, y)dy$ et $\int_0^1dx\int_0^1f(x, y)dy$, Fund. Math., 1 (1920), 142-147.  doi: 10.4064/fm-1-1-142-147.

[15]

E. D. Sontag, Mathematical Control Theory. Deterministic Finite-Dimensional Systems, 2nd edition, Springer, 1998. doi: 10.1007/978-1-4612-0577-7.

[16]

M. Tucsnak and G. Weiss, Well-posed systems – the LTI case and beyond, Automatica, 50 (2014), 1757-1779.  doi: 10.1016/j.automatica.2014.04.016.

[17]

J. Villegas, A Port-Hamiltonian Approach to Distributed-Parameter Systems, Ph.D. thesis, Universiteit Twente, 2007.

[18]

J. A. VillegasH. ZwartY. Le Gorrec and B. Maschke, Exponential stability of a class of boundary control systems, IEEE Trans. Autom. Contr., 54 (2009), 142-147.  doi: 10.1109/TAC.2008.2007176.

show all references

References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2nd edition. Elsevier, 2003.

[2]

H. Amann and J. Escher, Analysis I, II, III, Birkhäuser, 2005, 2008, 2009.

[3]

B. Augner, Stabilisation of Infinite-Dimensional Port-Hamiltonian Systems via Dissipative Boundary Feedback, PhD thesis. Available at http://elpub.bib.uni-wuppertal.de/edocs/dokumente/fbc/mathematik/diss2016/augner/dc1613.pdf.

[4]

B. Augner and B. Jacob, Stability and stabilization of infinite-dimensional linear port-Hamiltonian systems, Evol. Equ. Contr. Th., 3 (2014), 207-229.  doi: 10.3934/eect.2014.3.207.

[5]

S. Cox and E. Zuazua, The rate at which energy decays in a string damped at one end, Indiana Univ. Math. J., 44 (1995), 545-573.  doi: 10.1512/iumj.1995.44.2001.

[6]

K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer, 2000.

[7]

G. B. Folland, Real Analysis, 2nd edition, Wiley, 1999.

[8]

E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups, American Mathematical Society Colloquium Publications, 1957.

[9]

B. JacobK. Morris and H. Zwart, $C_0$-semigroups for hyperbolic partial differential equations on a one-dimensional spatial domain, J. Evol. Equ., 15 (2015), 493-502.  doi: 10.1007/s00028-014-0271-1.

[10]

B. Jacob and H. J. Zwart, Linear Port-Hamiltonian Systems on Infinite-Dimensional Spaces, Birkhäuser, 2012. doi: 10.1007/978-3-0348-0399-1.

[11]

W. Rudin, Real and Complex Analysis, 3rd edition. McGraw-Hill, 1987.

[12]

J. Schmid and H. Zwart, Stabilization of port-Hamiltonian systems by nonlinear boundary control in the presence of disturbances, ESAIM Contr. Optim. Calc. Var., 27 (2021), Paper No. 53, 37 pp. doi: 10.1051/cocv/2021051.

[13]

W. Sierpiński, Sur un problème concernant les ensembles mésurables superficiellement, Fund. Math., 1 (1920), 112-115.  doi: 10.4064/fm-1-1-112-115.

[14]

W. Sierpiński, Sur les rapports entre l'existence des intégrales $\int_0^1f(x, y)dx$, $\int_0^1f(x, y)dy$ et $\int_0^1dx\int_0^1f(x, y)dy$, Fund. Math., 1 (1920), 142-147.  doi: 10.4064/fm-1-1-142-147.

[15]

E. D. Sontag, Mathematical Control Theory. Deterministic Finite-Dimensional Systems, 2nd edition, Springer, 1998. doi: 10.1007/978-1-4612-0577-7.

[16]

M. Tucsnak and G. Weiss, Well-posed systems – the LTI case and beyond, Automatica, 50 (2014), 1757-1779.  doi: 10.1016/j.automatica.2014.04.016.

[17]

J. Villegas, A Port-Hamiltonian Approach to Distributed-Parameter Systems, Ph.D. thesis, Universiteit Twente, 2007.

[18]

J. A. VillegasH. ZwartY. Le Gorrec and B. Maschke, Exponential stability of a class of boundary control systems, IEEE Trans. Autom. Contr., 54 (2009), 142-147.  doi: 10.1109/TAC.2008.2007176.

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