Input-to-State Stability for Bilinear Feedback Systems
Abstract.
Input-to-state stability estimates with respect to small initial conditions and input functions for infinite-dimensional systems with bilinear feedback are shown. We apply the obtained results to controlled versions of a viscous Burger equation with Dirichlet boundary conditions, a Schrödinger equation, a Navier–Stokes system, and a semilinear wave equation.
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References
1.
R. Adams, Sobolev Spaces, Pure Appl. Math. 65, Academic Press, New York, London, 1975.
2.
A. Bensoussan, G. Da Prato, M. C. Delfour, and S. K. Mitter, Representation and Control of Infinite Dimensional Systems, Systems Control Found. Appl., 2nd ed., Birkhäuser Boston, Boston, 2007.
3.
R. F. Curtain and G. Weiss, Well posedness of triples of operators (in the sense of linear systems theory), in Control and Estimation of Distributed Parameter systems (Vorau, 1988), Internat. Ser. Numer. Math. 91, Birkhäuser, Basel, 1989, pp. 41–59.
4.
S. Dashkovskiy, O. Kapustyan, and J. Schmid, A local input-to-state stability result w.r.t. attractors of nonlinear reaction-diffusion equations, Math. Control Signals Systems, 32 (2020), pp. 309–326.
5.
S. Dashkovskiy and A. Mironchenko, Input-to-state stability of infinite-dimensional control systems, Math. Control Signals Systems, 25 (2013), pp. 1–35.
6.
S. Dashkovskiy and A. Mironchenko, Input-to-state stability of nonlinear impulsive systems, SIAM J. Control Optim., 51 (2013), pp. 1962–1987, https://doi.org/10.1137/120881993.
7.
E. Di Nezza, G. Palatucci, and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), pp. 521–573.
8.
D. L. Elliott, Bilinear Control Systems, Appl. Math. Sci. 169, Springer, Dordrecht, 2009.
9.
K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Grad. Texts in Math. 194, Springer-Verlag, New York, 2000.
10.
C. Guiver, H. Logemann, and M. R. Opmeer, Infinite-dimensional Lur’e systems: Input-to-state stability and convergence properties, SIAM J. Control Optim., 57 (2019), pp. 334–365, https://doi.org/10.1137/17M1150426.
11.
R. Hosfeld, B. Jacob, and F. L. Schwenninger, Integral input-to-state stability of unbounded bilinear control systems, Math. Control Signals Systems, 34 (2022), pp. 273–295.
12.
B. Jacob, R. Nabiullin, J. Partington, and F. Schwenninger, Infinite-dimensional input-to-state stability and Orlicz spaces, SIAM J. Control Optim., 56 (2018), pp. 868–889, https://doi.org/10.1137/16M1099467.
13.
B. Jacob, F. L. Schwenninger, and H. Zwart, On continuity of solutions for parabolic control systems and input-to-state stability, J. Differential Equations, 266 (2019), pp. 6284–6306.
14.
B. Jayawardhana, H. Logemann, and E. Ryan, Infinite-dimensional feedback systems: The circle criterion and input-to-state stability, Commun. Inf. Syst., 8 (2008), pp. 413–444.
15.
I. Karafyllis and M. Krstic, ISS in different norms for 1-D parabolic PDEs with boundary disturbances, SIAM J. Control Optim., 55 (2017), pp. 1716–1751, https://doi.org/10.1137/16M1073753.
16.
I. Karafyllis and M. Krstic, Input-to-State Stability for PDEs, Comm. Control Engrg. Ser., Springer, Cham, 2019.
17.
M. Krstic, On global stabilization of Burgers’ equation by boundary control, Systems Control Lett., 37 (1999), pp. 123–141.
18.
H. Li, R. Baier, L. Grüne, S. F. Hafstein, and F. R. Wirth, Computation of local ISS Lyapunov functions with low gains via linear programming, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), pp. 2477–2495.
19.
A. Lunardi, Interpolation Theory, Appunti. Sc. Norm. Super. Pisa (N. S.) 16, Edizioni della Normale, Pisa, 2018.
20.
H. V. Ly, K. D. Mease, and E. S. Titi, Distributed and boundary control of the viscous Burgers’ equation, Numer. Funct. Anal. Optim., 18 (1997), pp. 143–188.
21.
F. Mazenc and C. Prieur, Strict Lyapunov functions for semilinear parabolic partial differential equations, Math. Control Relat. Fields, 1 (2011), pp. 231–250.
22.
A. Mironchenko, Local input-to-state stability: Characterizations and counterexamples, Systems Control Lett., 87 (2016), pp. 23–28.
23.
A. Mironchenko and H. Ito, Construction of Lyapunov functions for interconnected parabolic systems: An iISS approach, SIAM J. Control Optim., 53 (2015), pp. 3364–3382, https://doi.org/10.1137/14097269X.
24.
A. Mironchenko and H. Ito, Characterizations of integral input-to-state stability for bilinear systems in infinite dimensions, Math. Control Relat. Fields, 6 (2016), pp. 447–466.
25.
A. Mironchenko, I. Karafyllis, and M. Krstic, Monotonicity methods for input-to-state stability of nonlinear parabolic PDEs with boundary disturbances, SIAM J. Control Optim., 57 (2019), pp. 510–532, https://doi.org/10.1137/17M1161877.
26.
A. Mironchenko and C. Prieur, Input-to-state stability of infinite-dimensional systems: Recent results and open questions, SIAM Rev., 62 (2020), pp. 529–614, https://doi.org/10.1137/19M1291248.
27.
A. Mironchenko, C. Prieur, and F. Wirth, Local stabilization of an unstable parabolic equation via saturated controls, IEEE Trans. Automat. Control, 66 (2021), pp. 2162–2176.
28.
A. Mironchenko and F. Wirth, Characterizations of input-to-state stability for infinite-dimensional systems, IEEE Trans. Automat. Control, 63 (2018), pp. 1602–1617.
29.
P. L. Sachdev, Nonlinear Diffusive Waves, Cambridge University Press, Cambridge, 1987.
30.
F. L. Schwenninger, Input-to-state stability for parabolic boundary control: Linear and semi-linear systems, in Control Theory of Infinite-Dimensional Systems, J. Kerner, L. Laasri, and D. Mugnolo, eds., Birkhäuser, Cham, 2020, pp. 83–116.
31.
E. Sontag, Smooth stabilization implies coprime factorization, IEEE Trans. Automat. Control, 34 (1989), pp. 435–443.
32.
E. Sontag, Input to state stability: Basic concepts and results, in Nonlinear and Optimal Control Theory, Lecture Notes in Math. 1932, Springer, Berlin, 2008, pp. 163–220.
33.
E. D. Sontag and Y. Wang, New characterizations of input-to-state stability, IEEE Trans. Automat. Control, 41 (1996), pp. 1283–1294.
34.
O. Staffans, Well-Posed Linear Systems, Encyclopedia Math. Appl. 103, Cambridge University Press, Cambridge, 2005.
35.
M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 2009.
36.
M. Tucsnak and G. Weiss, Well-posed systems—the LTI case and beyond, Automatica J. IFAC, 50 (2014), pp. 1757–1779.
37.
G. Weiss, Admissibility of unbounded control operators, SIAM J. Control Optim., 27 (1989), pp. 527–545, https://doi.org/10.1137/0327028.
38.
G. Weiss, Admissible observation operators for linear semigroups, Israel J. Math., 65 (1989), pp. 17–43.
39.
G. Weiss, The representation of regular linear systems on Hilbert spaces, in Control and Estimation of Distributed Parameter Systems (Vorau, 1988), Internat. Ser. Numer. Math. 91, Birkhäuser, Basel, 1989, pp. 401–416.
40.
S. Wu, S. Mei, and X. Zhang, Estimation of LISS (local input-to-state stability) properties for nonlinear systems, Sci. China Technol. Sci., 53 (2010), pp. 909–917.
41.
J. Zheng and G. Zhu, Input-to-state stability with respect to boundary disturbances for a class of semi-linear parabolic equations, Automatica J. IFAC, 97 (2018), pp. 271–277.
42.
J. Zheng and G. Zhu, A De Giorgi iteration-based approach for the establishment of ISS properties for Burgers’ equation with boundary and in-domain disturbances, IEEE Trans. Automat. Control, 64 (2019), pp. 3476–3483.
Information & Authors
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Published In
SIAM Journal on Control and Optimization
Pages: 1369 - 1389
DOI: 10.1137/23M155788X
ISSN (online): 1095-7138
Copyright
© 2024 Society for Industrial and Applied Mathematics.
History
Submitted: 9 March 2023
Accepted: 8 February 2024
Published online: 9 May 2024
Keywords
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Authors
Funding Information
Deutsche Forschungsgemeinschaft (DFG): JA 735/18-1 / SCHW 2022/2-1.
Funding: The first three authors are supported by the German Research Foundation (DFG) via the joint grant JA 735/18-1/SCHW 2022/2-1.
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