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September  2021, 10(3): 519-544. doi: 10.3934/eect.2020079

## Funnel control for boundary control systems

 1 University of Hamburg Bundesstraße 55 20146 Hamburg Germany 2 University of Twente P.O. Box 217 7500AE Enschede The Netherlands

* Corresponding author: Marc Puche

Received  March 2019 Revised  June 2020 Published  September 2021 Early access  July 2020

We study a nonlinear, non-autonomous feedback controller applied to boundary control systems. Our aim is to track a given reference signal with prescribed performance. Existence and uniqueness of solutions to the resulting closed-loop system is proved by using nonlinear operator theory. We apply our results to both hyperbolic and parabolic equations.

Citation: Marc Puche, Timo Reis, Felix L. Schwenninger. Funnel control for boundary control systems. Evolution Equations and Control Theory, 2021, 10 (3) : 519-544. doi: 10.3934/eect.2020079
##### References:
 [1] R. Adams, Sobolev Spaces, Pure and Applied Mathematics, 65, Academic press, New York, London, 1975. [2] H. Alt, Linear Functional Analysis, Universitext, Springer-Verlag, London, 2016. doi: 10.1007/978-1-4471-7280-2. [3] W. Arendt, R. Chill, C. Seifert, H. Vogt and J. Voigt, Form methods for evolution equations, and applications, 2015, Available at https://www.mat.tuhh.de/veranstaltungen/isem18/pdf/LectureNotes.pdf. [4] W. Arendt and A. ert Elst, From forms to semigroups, in Spectral Theory, Mathematical System Theory, Evolution Equations, Differential and Difference Equations (eds. W. Arendt, J. A. Ball, J. Behrndt, K.-H. Förster, V. Mehrmann and C. Trunk), Operator Theory: Advances and Applications, 221, Birkhäuser, Basel, Switzerland, 2012, 47–69. doi: 10.1007/978-3-0348-0297-0_4. [5] B. Augner, Stabilisation of Infinite-Dimensional Port-Hamiltonian Systems via Dissipative Boundary Feedback, Ph.D thesis, Bergische Universität Wuppertal, 2016. [6] B. Augner, Well-posedness and stability of infinite-dimensional linear port-Hamiltonian systems with nonlinear boundary feedback, SIAM J. Control Optim., 57 (2019), 1818-1844.  doi: 10.1137/15M1024901. [7] B. Augner and B. Jacob, Stability and stabilization of infinite-dimensional linear port-Hamiltonian systems, Evol. Equ. Control Theory, 3 (2014), 207-229.  doi: 10.3934/eect.2014.3.207. [8] G. Bastin and J.-M. Coron, Stability and Boundary Stabilization of 1-D Hyperbolic Systems, Progress in Nonlinear Differential Equations and Their Applications: Subseries in Control, 88, Birkhäuser, 2016. doi: 10.1007/978-3-319-32062-5. [9] T. Berger, L. H. Hoang and T. Reis, Funnel control for nonlinear systems with known strict relative degree, Automatica, 87 (2018), 345-357.  doi: 10.1016/j.automatica.2017.10.017. [10] T. Berger, M. Puche and F. Schwenninger, Funnel control for a moving water tank, 2019, Submitted for publication. Available at arXiv: https://arXiv.org/abs/1902.00586. [11] T. Berger, M. Puche and F. L. Schwenninger, Funnel control in the presence of infinite-dimensional internal dynamics, Systems Control Lett., 139 (2020), 104678. doi: 10.1016/j.sysconle.2020.104678. [12] A. Cheng and K. Morris, Well-posedness of boundary control systems, SIAM J. Control Optim., 42 (2003), 1244-1265.  doi: 10.1137/S0363012902384916. [13] J. Diestel and J. Uhl, Vector Measures, Mathematical surveys and monographs, 15, American Mathematical Society, Providence, RI, 1977. [14] K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, 194, Springer, New York, 2000. [15] V. Girault and P.-A. Raviart, Finite Element Approximation of the Navier-Stokes Equations, Lecture Notes in Mathematics, 749, Springer, Berlin Heidelberg, 1979. [16] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics, 24, Pitman Advanced Publishing Program, Boston, London, Melbourne, 1985. [17] A. Ilchmann, E. Ryan and P. Townsend, Tracking with prescribed transient behaviour for non-linear systems of known relative degree, SIAM J. Control Optim., 46 (2007), 210-230.  doi: 10.1137/050641946. [18] B. Jacob and H. Zwart, Linear Port-Hamiltonian Systems on Infinite-Dimensional Spaces, Operator Theory: Advances and Applications, 223, Birkhäuser, Basel, 2012. doi: 10.1007/978-3-0348-0399-1. [19] T. Kato, Nonlinear semigroups and evolution equations, J. Math. Soc. Japan, 19 (1967), 508-520.  doi: 10.2969/jmsj/01940508. [20] T. Kato, Perturbation Theory for Linear Operators, 2$^nd$ edition, Springer, Berlin Heidelberg, Germany, 1980. [21] I. Miyadera, Nonlinear Semigroups, Translations of Mathematical Monographs, 109, American Mathematical Society, Providence, RI, 1992. [22] T. Reis and T. Selig, Funnel control for the boundary controlled heat equation, SIAM J. Control Optim., 53 (2015), 547-574.  doi: 10.1137/140971567. [23] E. P. Ryan, A. Ilchmann and C. J. Sangwin, Tracking with prescribed transient behaviour, ESAIM: Control, Optimisation and Calculus of Variations, 7 (2002), 471-493.  doi: 10.1051/cocv:2002064. [24] T. Selig, A. Ilchmann and C. Trunk, The Byrnes–Isidori form for infinite-dimensional systems, SIAM J. Control Optim., 54 (2016), 1504-1534.  doi: 10.1137/130942413. [25] R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, American Mathematical Society, Providence, RI, 1996. [26] O. J. Staffans, Well-Posed Linear Systems, 103, Cambridge University Press, 2005.  doi: 10.1017/CBO9780511543197. [27] M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Advanced Texts Basler Lehrbücher, Birkhäuser, Basel, 2009. doi: 10.1007/978-3-7643-8994-9.

show all references

##### References:
 [1] R. Adams, Sobolev Spaces, Pure and Applied Mathematics, 65, Academic press, New York, London, 1975. [2] H. Alt, Linear Functional Analysis, Universitext, Springer-Verlag, London, 2016. doi: 10.1007/978-1-4471-7280-2. [3] W. Arendt, R. Chill, C. Seifert, H. Vogt and J. Voigt, Form methods for evolution equations, and applications, 2015, Available at https://www.mat.tuhh.de/veranstaltungen/isem18/pdf/LectureNotes.pdf. [4] W. Arendt and A. ert Elst, From forms to semigroups, in Spectral Theory, Mathematical System Theory, Evolution Equations, Differential and Difference Equations (eds. W. Arendt, J. A. Ball, J. Behrndt, K.-H. Förster, V. Mehrmann and C. Trunk), Operator Theory: Advances and Applications, 221, Birkhäuser, Basel, Switzerland, 2012, 47–69. doi: 10.1007/978-3-0348-0297-0_4. [5] B. Augner, Stabilisation of Infinite-Dimensional Port-Hamiltonian Systems via Dissipative Boundary Feedback, Ph.D thesis, Bergische Universität Wuppertal, 2016. [6] B. Augner, Well-posedness and stability of infinite-dimensional linear port-Hamiltonian systems with nonlinear boundary feedback, SIAM J. Control Optim., 57 (2019), 1818-1844.  doi: 10.1137/15M1024901. [7] B. Augner and B. Jacob, Stability and stabilization of infinite-dimensional linear port-Hamiltonian systems, Evol. Equ. Control Theory, 3 (2014), 207-229.  doi: 10.3934/eect.2014.3.207. [8] G. Bastin and J.-M. Coron, Stability and Boundary Stabilization of 1-D Hyperbolic Systems, Progress in Nonlinear Differential Equations and Their Applications: Subseries in Control, 88, Birkhäuser, 2016. doi: 10.1007/978-3-319-32062-5. [9] T. Berger, L. H. Hoang and T. Reis, Funnel control for nonlinear systems with known strict relative degree, Automatica, 87 (2018), 345-357.  doi: 10.1016/j.automatica.2017.10.017. [10] T. Berger, M. Puche and F. Schwenninger, Funnel control for a moving water tank, 2019, Submitted for publication. Available at arXiv: https://arXiv.org/abs/1902.00586. [11] T. Berger, M. Puche and F. L. Schwenninger, Funnel control in the presence of infinite-dimensional internal dynamics, Systems Control Lett., 139 (2020), 104678. doi: 10.1016/j.sysconle.2020.104678. [12] A. Cheng and K. Morris, Well-posedness of boundary control systems, SIAM J. Control Optim., 42 (2003), 1244-1265.  doi: 10.1137/S0363012902384916. [13] J. Diestel and J. Uhl, Vector Measures, Mathematical surveys and monographs, 15, American Mathematical Society, Providence, RI, 1977. [14] K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, 194, Springer, New York, 2000. [15] V. Girault and P.-A. Raviart, Finite Element Approximation of the Navier-Stokes Equations, Lecture Notes in Mathematics, 749, Springer, Berlin Heidelberg, 1979. [16] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics, 24, Pitman Advanced Publishing Program, Boston, London, Melbourne, 1985. [17] A. Ilchmann, E. Ryan and P. Townsend, Tracking with prescribed transient behaviour for non-linear systems of known relative degree, SIAM J. Control Optim., 46 (2007), 210-230.  doi: 10.1137/050641946. [18] B. Jacob and H. Zwart, Linear Port-Hamiltonian Systems on Infinite-Dimensional Spaces, Operator Theory: Advances and Applications, 223, Birkhäuser, Basel, 2012. doi: 10.1007/978-3-0348-0399-1. [19] T. Kato, Nonlinear semigroups and evolution equations, J. Math. Soc. Japan, 19 (1967), 508-520.  doi: 10.2969/jmsj/01940508. [20] T. Kato, Perturbation Theory for Linear Operators, 2$^nd$ edition, Springer, Berlin Heidelberg, Germany, 1980. [21] I. Miyadera, Nonlinear Semigroups, Translations of Mathematical Monographs, 109, American Mathematical Society, Providence, RI, 1992. [22] T. Reis and T. Selig, Funnel control for the boundary controlled heat equation, SIAM J. Control Optim., 53 (2015), 547-574.  doi: 10.1137/140971567. [23] E. P. Ryan, A. Ilchmann and C. J. Sangwin, Tracking with prescribed transient behaviour, ESAIM: Control, Optimisation and Calculus of Variations, 7 (2002), 471-493.  doi: 10.1051/cocv:2002064. [24] T. Selig, A. Ilchmann and C. Trunk, The Byrnes–Isidori form for infinite-dimensional systems, SIAM J. Control Optim., 54 (2016), 1504-1534.  doi: 10.1137/130942413. [25] R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, American Mathematical Society, Providence, RI, 1996. [26] O. J. Staffans, Well-Posed Linear Systems, 103, Cambridge University Press, 2005.  doi: 10.1017/CBO9780511543197. [27] M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Advanced Texts Basler Lehrbücher, Birkhäuser, Basel, 2009. doi: 10.1007/978-3-7643-8994-9.
Error evolution in a funnel $\mathcal{F}_{\varphi}$ with boundary $\varphi(t)^{-1}$
Left: Norm of the error within the funnel boundary followed by the two reference signals and the respective outputs. Right: Inputs obtained from the feedback law
Performance funnel with the error, reference signal with the output of the closed-loop system and input of the closed-loop
From left to right, top to bottom, the temperature of the plate for different increasing times
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