This paper provides novel Input-to-State Stability (ISS)-style maximum principle estimates for classical solutions of nonlinear 1-D parabolic Partial Differential Equations (PDEs). The derivation of the ISS-style maximum principle estimates is performed in two ways: by using an ISS Lyapunov Functional for the sup norm and by exploiting well-known maximum principles. The estimates provide fading memory ISS estimates in the sup norm of the state with respect to distributed and boundary inputs. The obtained results can handle parabolic PDEs with nonlinear and non-local in-domain terms/boundary conditions. Three illustrative examples show the efficiency of the proposed methodology for the derivation of ISS estimates in the sup norm of the state.
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Keywords: ISS, parabolic PDEs, maximum principles, boundary disturbances
@article{COCV_2021__27_1_A59_0,
author = {Karafyllis, Iasson and Krstic, Miroslav},
editor = {Buttazzo, G. and Casas, E. and de Teresa, L. and Glowinski, R. and Leugering, G. and Tr\'elat, E. and Zhang, X.},
title = {ISS estimates in the spatial sup-norm for nonlinear {1-D} parabolic {PDEs}},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2021},
publisher = {EDP-Sciences},
volume = {27},
doi = {10.1051/cocv/2021053},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2021053/}
}
TY - JOUR AU - Karafyllis, Iasson AU - Krstic, Miroslav ED - Buttazzo, G. ED - Casas, E. ED - de Teresa, L. ED - Glowinski, R. ED - Leugering, G. ED - Trélat, E. ED - Zhang, X. TI - ISS estimates in the spatial sup-norm for nonlinear 1-D parabolic PDEs JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2021 VL - 27 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2021053/ DO - 10.1051/cocv/2021053 LA - en ID - COCV_2021__27_1_A59_0 ER -
%0 Journal Article %A Karafyllis, Iasson %A Krstic, Miroslav %E Buttazzo, G. %E Casas, E. %E de Teresa, L. %E Glowinski, R. %E Leugering, G. %E Trélat, E. %E Zhang, X. %T ISS estimates in the spatial sup-norm for nonlinear 1-D parabolic PDEs %J ESAIM: Control, Optimisation and Calculus of Variations %D 2021 %V 27 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2021053/ %R 10.1051/cocv/2021053 %G en %F COCV_2021__27_1_A59_0
Karafyllis, Iasson; Krstic, Miroslav. ISS estimates in the spatial sup-norm for nonlinear 1-D parabolic PDEs. ESAIM: Control, Optimisation and Calculus of Variations, Tome 27 (2021), article no. 57. doi: 10.1051/cocv/2021053
[1] , and , Some recent results on feedback regularization of Navier-Stokes equations., Proceedings of the 36th Conference on Decision & Control, San Diego, California USA (1997) 2231–2236.
[2] and , An Introduction to Semilinear Evolution Equations. Oxford Lecture Series in Mathematics and its Applications, Oxford University Press (1980).
[3] and , Input-to-state stability of infinite-dimensional control systems. Math. Control Signals Syst. 25 (2013) 1–35.
[4] , Partial Differential Equations of Parabolic Type, Dover (2008).
[5] , , , and , Analysis of nonlinear spectral Eddy-viscosity models of turbulence. J. Sci. Comput. 45 (2010) 294–332.
[6] , Nonlocal expression for scalar flux in turbulent shear flow. Phys. Fluids 16 (2004) 1493–1508.
[7] , , and , Infinite-dimensional input-to-state stability and Orlicz spaces. SIAM J. Control Optim. 56 (2018) 868–889.
[8] , and , On continuity of solutions for parabolic control systems and input-to-state stability. J. Differ. Equ. 266 (2019) 6284–6306.
[9] , , and , Non-coercive Lyapunov functions for input-to-state stability of infinite-dimensional systems. SIAM J. Control Optim. 58 (2020) 2952–2978.
[10] , Partial Differential Equations, 4th Edition, Springer-Verlag, New York (1982).
[11] and , Stability and Stabilization of Nonlinear Systems. Springer-Verlag, London (Series: Communications and Control Engineering) (2011).
[12] and , ISS in different norms for 1-D parabolic PDEs with boundary disturbances. SIAM J. Control Optim. 55 (2017) 1716–1751.
[13] and , Sampled-data boundary feedback control of 1-D parabolic PDEs. Automatica 87 (2018) 226–237.
[14] and , Decay estimates for 1-D parabolic PDEs with boundary disturbances. ESAIM: COCV 24 (2018) 1511–1540.
[15] and , Small-gain-based boundary feedback design for global exponential stabilization of 1-D semilinear parabolicPDEs. SIAM J. Control Optim. 57 (2019) 2016–2036.
[16] and , Input-to-State Stability for PDEs. Springer-Verlag, London (Series: Communications and Control Engineering) (2019).
[17] , and , Sampled-data observers for 1-D parabolic PDEs with non-local outputs. Syst. Control Lett. 133 (2019) 104553.
[18] , and , Linear and Quasilinear Equations of Parabolic Type. Trans. AMS 23 (1968).
[19] , New equations for the description of the motions of viscous incompressible fluids, and global solvability for their boundary value problems. Bound. Value Probl. Math. Phys. Part 5, Trudy Matematicheskogo Instituta imeni V.A. Steklova 102 (1967) 85–104 and Proc. Steklov Inst. Math. 102 (1967) 95–118.
[20] and , Characterizations of input-to-state stability for infinite-dimensional systems. IEEE Trans. Autom. Control 63 (2018) 1602–1617.
[21] , and , Monotonicity methods for input-to-state stability of nonlinear parabolic PDEs with boundary disturbances. SIAM J. Control Optim. 57 (2019) 510–532.
[22] and , Input-to-state stability of infinite-dimensional systems: recent results and open questions. SIAM Rev. 62 (2020) 529–614.
[23] and , Maximum Principles in Differential Equations. Springer-Verlag, New York (1984).
[24] , Nonmetallic crystals with high thermal conductivity. J. Phys. Chem. Solids 34 (1973) 321–335.
[25] , Shock Waves and Reaction-Diffusion Equations, 2nd edition, Springer-Verlag, New York (1994).
[26] , and , Disturbance-to-state stabilization and quantized control for linear hyperbolic systems. | arXiv
[27] , Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd edition, Springer (1997).
[28] and , Input-to-state stability with respect to boundary disturbances for a class of semi-linear parabolic equations. Automatica 97 (2018) 271–277.
[29] and , A De Giorgi iteration-based approach for the establishment of ISS properties for Burgers’ equation with boundary and in-domain disturbances. IEEE Trans. Autom. Control 64 (2019) 3476–3483.
[30] and , A weak maximum principle-based approach for input-to-state stability analysis of nonlinear parabolic PDEs with boundary disturbances. Math. Control Signals Syst. 32 (2020) 157–176.
[31] and , ISS-like estimates for nonlinear parabolic PDEs with variable coefficients on higher dimensional domains. Syst. Control Lett. 146 (2020) 104808.
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