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December  2021, 11(4): 905-934. doi: 10.3934/mcrf.2020051

Solvability and sliding mode control for the viscous Cahn–Hilliard system with a possibly singular potential

Pierluigi Colli 1, , Gianni Gilardi 1, and  Gabriela Marinoschi 2,

1. 

Dipartimento di Matematica ``F. Casorati'', Università di Pavia, via Ferrata 5, 27100 Pavia, Italy
2. 

``Gheorghe Mihoc-Caius Iacob'', Institute of Mathematical Statistics and Applied Mathematics, of the Romanian Academy (ISMMA), Calea 13 Septembrie 13, 050711 Bucharest, Romania

Received  April 2020 Revised  September 2020 Published  December 2021 Early access  December 2020

In the present contribution we study a viscous Cahn–Hilliard system where a further leading term in the expression for the chemical potential $ \mu $ is present. This term consists of a subdifferential operator $ S $ in $ L^2(\Omega) $ (where $ \Omega $ is the domain where the evolution takes place) acting on the difference of the phase variable $ \varphi $ and a given state $ {\varphi^*} $, which is prescribed and may depend on space and time. We prove existence and continuous dependence results in case of both homogeneous Neumann and Dirichlet boundary conditions for the chemical potential $ \mu $. Next, by assuming that $ S = \rho\;{\rm{sign}} $, a multiple of the $ \;{\rm{sign}} $ operator, and for smoother data, we first show regularity results. Then, in the case of Dirichlet boundary conditions for $ \mu $ and under suitable conditions on $ \rho $ and $ \Omega $, we also prove the sliding mode property, that is, that $ \varphi $ is forced to join the evolution of $ {\varphi^*} $ in some time $ T^* $ lower than the given final time $ T $. We point out that all our results hold true for a very general and possibly singular multi-well potential acting on $ \varphi $.

Keywords: Viscous Cahn-Hilliard equation, state-feedback control law, initial-boundary value problem, well-posedness, regularity, sliding mode property.
Mathematics Subject Classification: Primary: 35K52; Secondary: 58J35, 80A22, 93B52, 93C20.
Citation: Pierluigi Colli, Gianni Gilardi, Gabriela Marinoschi. Solvability and sliding mode control for the viscous Cahn–Hilliard system with a possibly singular potential. Mathematical Control and Related Fields, 2021, 11 (4) : 905-934. doi: 10.3934/mcrf.2020051
References:
[1]

V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer Monographs in Mathematics, Springer, New York, 2010. doi: 10.1007/978-1-4419-5542-5.

[2]

V. Barbu, P. Colli, G. Gilardi, G. Marinoschi and E. Rocca, Sliding mode control for a nonlinear phase-field system, SIAM J. Control Optim., 55 (2017), 2108-2133.  doi: 10.1137/15M102424X.

[3]

G. Bartolini, L. Fridman, A. Pisano and E. Usai, Modern Sliding Mode Control Theory. New Perspectives and Applications, Lecture Notes in Control and Information Sciences, 375, Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-79016-7.

[4]

A. L. Bertozzi, S. Esedoḡlu and A. Gillette, Inpainting of binary images using the Cahn–Hilliard equation, IEEE Trans. Image Process., 16 (2007), 285-291.  doi: 10.1109/TIP.2006.887728.

[5]

E. Bonetti, P. Colli, L. Scarpa and G. Tomassetti, Bounded solutions and their asymptotics for a doubly nonlinear Cahn–Hilliard system, Calc. Var. Partial Differential Equations, 59 (2020), 25pp. doi: 10.1007/s00526-020-1715-9.

[6]

J. W. Cahn, On spinodal decomposition, Acta Metall., 9 (1961), 795-801.  doi: 10.1016/0001-6160(61)90182-1.

[7]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. Ⅰ. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267.  doi: 10.1063/1.1744102.

[8]

M.-B. Cheng, V. Radisavljevic and W.-C. Su, Sliding mode boundary control of a parabolic PDE system with parameter variations and boundary uncertainties, Automatica J. IFAC, 47 (2011), 381-387.  doi: 10.1016/j.automatica.2010.10.045.

[9]

P. Colli, G. Gilardi, G. Marinoschi and E. Rocca, Sliding mode control for a phase field system related to tumor growth, Appl. Math. Optim., 79 (2019), 647-670.  doi: 10.1007/s00245-017-9451-z.

[10]

P. Colli, G. Gilardi, P. Podio-Guidugli and J. Sprekels, Global existence and uniqueness for a singular/degenerate Cahn–Hilliard system with viscosity, J. Differential Equations, 254 (2013), 4217-4244.  doi: 10.1016/j.jde.2013.02.014.

[11]

P. Colli, G. Gilardi, P. Podio-Guidugli and J. Sprekels, Well-posedness and long-time behavior for a nonstandard viscous Cahn–Hilliard system, SIAM J. Appl. Math., 71 (2011), 1849-1870.  doi: 10.1137/110828526.

[12]

P. Colli and D. Manini, Sliding mode control for a generalization of the Caginalp phase-field system, Appl. Math. Optim., (2020). doi: 10.1007/s00245-020-09682-3.

[13]

M. Colturato, On a class of conserved phase field systems with a maximal monotone perturbation, Appl. Math. Optim., 78 (2018), 545-585.  doi: 10.1007/s00245-017-9415-3.

[14]

M. Colturato, Solvability of a class of phase field systems related to a sliding mode control problem, Appl. Math., 6 (2016), 623-650.  doi: 10.1007/s10492-016-0150-x.

[15]

C. Edwards, E. Fossas Colet and L. Fridman, Advances in Variable Structure and Sliding Mode Control, Lecture Notes in Control and Information Sciences, 334, Springer-Verlag, Berlin, 2006. doi: 10.1007/11612735.

[16] C. Edwards and S. Spurgeon, Sliding Mode Control: Theory and Applications, CRC Press, London, 1998.  doi: 10.1201/9781498701822.
[17]

L. Fridman, J. Moreno, R. Iriarte, Sliding Modes After the First Decade of the 21st Century. State of the Art, Lecture Notes in Control and Information Sciences, 412, Springer-Verlag, Berlin, 2011. doi: 10.1007/978-3-642-22164-4.

[18]

C. G. Gal, M. Grasselli and H. Wu, Global weak solutions to a diffuse interface model for incompressible two-phase flows with moving contact lines and different densities, Arch. Ration. Mech. Anal., 234 (2019), 1-56.  doi: 10.1007/s00205-019-01383-8.

[19]

G. Gilardi, A. Miranville and G. Schimperna, On the Cahn–Hilliard equation with irregular potentials and dynamic boundary conditions, Commun. Pure Appl. Anal., 8 (2009), 881-912.  doi: 10.3934/cpaa.2009.8.881.

[20]

J.-J. Gu and J.-M. Wang, Sliding mode control for $N$-coupled reaction-diffusion PDEs with boundary input disturbances, Internat. J. Robust Nonlinear Control, 29 (2019), 1437-1461.  doi: 10.1002/rnc.4448.

[21]

M. E. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Phys. D, 92 (1996), 178-192.  doi: 10.1016/0167-2789(95)00173-5.

[22]

U. Itkis, Control Systems of Variable Structure, Wiley, 1976.

[23]

L. Levaggi, Existence of sliding motions for nonlinear evolution equations in Banach spaces, Discrete Contin. Dyn. Syst., (2013), 477–487. doi: 10.3934/proc.2013.2013.477.

[24]

L. Levaggi, Infinite dimensional systems' sliding motions, Eur. J. Control, 8 (2002), 508-516.  doi: 10.3166/ejc.8.508-516.

[25]

J.-L. Lions, Quelques Méthodes de Résolution des Probl\`emes aux Limites non Linéaires, Dunod; Gauthier-Villars, Paris, 1969.

[26]

Q.-X. Liu, M. Rietkerk, P. M. J. Herman, T. Piersma, J. M. Fryxell and J. van de Koppel, Phase separation driven by density-dependent movement: A novel mechanism for ecological patterns, Phys. Life Rev., 19 (2016), 107-121.  doi: 10.1016/j.plrev.2016.07.009.

[27]

A. Miranville, Some generalizations of the Cahn–Hilliard equation, Asymptot. Anal., 22 (2000), 235-259. 

[28]

A. Miranville, The Cahn–Hilliard equation and some of its variants, AIMS Math., 2 (2017), 479-544.  doi: 10.3934/Math.2017.2.479.

[29]

A. Miranville and S. Zelik, Robust exponential attractors for Cahn-Hilliard type equations with singular potentials, Math. Methods Appl. Sci., 27 (2004), 545-582.  doi: 10.1002/mma.464.

[30]

A. Novick-Cohen, On the viscous Cahn-Hilliard equation, in Material Instabilities in Continuum Mechanics, Oxford Sci. Publ., Oxford Univ. Press, New York, 1988, 329-342.

[31]

A. Novick-Cohen and R. L. Pego, Stable patterns in a viscous diffusion equation, Trans. Amer. Math. Soc., 324 (1991), 331-351.  doi: 10.1090/S0002-9947-1991-1015926-7.

[32]

Y. Orlov and V. I. Utkin, Unit sliding mode control in infinite-dimensional systems. Adaptive learning and control using sliding modes, Appl. Math. Comput. Sci., 8 (1998), 7-20. 

[33]

Y. V. Orlov, Application of Lyapunov method in distributed systems, Autom. Remote Control, 44 (1983), 426-430. 

[34]

Y. V. Orlov, Discontinuous unit feedback control of uncertain infinite dimensional systems, IEEE Trans. Automatic Control, 45 (2000), 834-843.  doi: 10.1109/9.855545.

[35]

Y. V. Orlov and V. I. Utkin, Sliding mode control in indefinite-dimensional systems, Automatica J. IFAC, 23 (1987), 753-757.  doi: 10.1016/0005-1098(87)90032-X.

[36]

Y. V. Orlov and V. I. Utkin, Use of sliding modes in distributed system control problems, Automat. Remote Control, 43 (1982), 1127-1135. 

[37]

A. Pilloni, A. Pisano, Y. Orlov and E. Usai, Consensus-based control for a network of diffusion PDEs with boundary local interaction, IEEE Trans. Automat. Control, 61 (2016), 2708-2713.  doi: 10.1109/TAC.2015.2506990.

[38]

P. Podio-Guidugli, Models of phase segregation and diffusion of atomic species on a lattice, Ric. Mat., 55 (2006), 105-118.  doi: 10.1007/s11587-006-0008-8.

[39]

J. Simon, Compact sets in the space $L^p(0,T; B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.

[40]

V. I. Utkin, Sliding Modes in Control and Optimization, Communications and Control Engineering Series, Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-642-84379-2.

[41] V. Utkin, J. Guldner and J. Shi, Sliding Mode Control in Electro-Mechanical Systems, CRC Press, Boca Raton, 2009.  doi: 10.1201/9781420065619.
[42]

S. M. Wise, J. S. Lowengrub, H. B. Frieboes and V. Cristini, Three-dimensional multispecies nonlinear tumor growth–I: Model and numerical method, J. Theoret. Biol., 253 (2008), 524-543.  doi: 10.1016/j.jtbi.2008.03.027.

[43]

H. Xing, D. Li, C. Gao and Y. Kao, Delay-independent sliding mode control for a class of quasi-linear parabolic distributed parameter systems with time-varying delay, J. Franklin Inst., 350 (2013), 397-418.  doi: 10.1016/j.jfranklin.2012.12.007.

[44]

K. D. Young and Ü. Özgüner, Variable Structure Systems, Sliding Mode and Nonlinear Control, Lecture Notes in Control and Information Sciences, 247, Springer-Verlag, Ltd., London, 1999. doi: 10.1007/BFb0109967.

show all references

References:
[1]

V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer Monographs in Mathematics, Springer, New York, 2010. doi: 10.1007/978-1-4419-5542-5.

[2]

V. Barbu, P. Colli, G. Gilardi, G. Marinoschi and E. Rocca, Sliding mode control for a nonlinear phase-field system, SIAM J. Control Optim., 55 (2017), 2108-2133.  doi: 10.1137/15M102424X.

[3]

G. Bartolini, L. Fridman, A. Pisano and E. Usai, Modern Sliding Mode Control Theory. New Perspectives and Applications, Lecture Notes in Control and Information Sciences, 375, Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-79016-7.

[4]

A. L. Bertozzi, S. Esedoḡlu and A. Gillette, Inpainting of binary images using the Cahn–Hilliard equation, IEEE Trans. Image Process., 16 (2007), 285-291.  doi: 10.1109/TIP.2006.887728.

[5]

E. Bonetti, P. Colli, L. Scarpa and G. Tomassetti, Bounded solutions and their asymptotics for a doubly nonlinear Cahn–Hilliard system, Calc. Var. Partial Differential Equations, 59 (2020), 25pp. doi: 10.1007/s00526-020-1715-9.

[6]

J. W. Cahn, On spinodal decomposition, Acta Metall., 9 (1961), 795-801.  doi: 10.1016/0001-6160(61)90182-1.

[7]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. Ⅰ. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267.  doi: 10.1063/1.1744102.

[8]

M.-B. Cheng, V. Radisavljevic and W.-C. Su, Sliding mode boundary control of a parabolic PDE system with parameter variations and boundary uncertainties, Automatica J. IFAC, 47 (2011), 381-387.  doi: 10.1016/j.automatica.2010.10.045.

[9]

P. Colli, G. Gilardi, G. Marinoschi and E. Rocca, Sliding mode control for a phase field system related to tumor growth, Appl. Math. Optim., 79 (2019), 647-670.  doi: 10.1007/s00245-017-9451-z.

[10]

P. Colli, G. Gilardi, P. Podio-Guidugli and J. Sprekels, Global existence and uniqueness for a singular/degenerate Cahn–Hilliard system with viscosity, J. Differential Equations, 254 (2013), 4217-4244.  doi: 10.1016/j.jde.2013.02.014.

[11]

P. Colli, G. Gilardi, P. Podio-Guidugli and J. Sprekels, Well-posedness and long-time behavior for a nonstandard viscous Cahn–Hilliard system, SIAM J. Appl. Math., 71 (2011), 1849-1870.  doi: 10.1137/110828526.

[12]

P. Colli and D. Manini, Sliding mode control for a generalization of the Caginalp phase-field system, Appl. Math. Optim., (2020). doi: 10.1007/s00245-020-09682-3.

[13]

M. Colturato, On a class of conserved phase field systems with a maximal monotone perturbation, Appl. Math. Optim., 78 (2018), 545-585.  doi: 10.1007/s00245-017-9415-3.

[14]

M. Colturato, Solvability of a class of phase field systems related to a sliding mode control problem, Appl. Math., 6 (2016), 623-650.  doi: 10.1007/s10492-016-0150-x.

[15]

C. Edwards, E. Fossas Colet and L. Fridman, Advances in Variable Structure and Sliding Mode Control, Lecture Notes in Control and Information Sciences, 334, Springer-Verlag, Berlin, 2006. doi: 10.1007/11612735.

[16] C. Edwards and S. Spurgeon, Sliding Mode Control: Theory and Applications, CRC Press, London, 1998.  doi: 10.1201/9781498701822.
[17]

L. Fridman, J. Moreno, R. Iriarte, Sliding Modes After the First Decade of the 21st Century. State of the Art, Lecture Notes in Control and Information Sciences, 412, Springer-Verlag, Berlin, 2011. doi: 10.1007/978-3-642-22164-4.

[18]

C. G. Gal, M. Grasselli and H. Wu, Global weak solutions to a diffuse interface model for incompressible two-phase flows with moving contact lines and different densities, Arch. Ration. Mech. Anal., 234 (2019), 1-56.  doi: 10.1007/s00205-019-01383-8.

[19]

G. Gilardi, A. Miranville and G. Schimperna, On the Cahn–Hilliard equation with irregular potentials and dynamic boundary conditions, Commun. Pure Appl. Anal., 8 (2009), 881-912.  doi: 10.3934/cpaa.2009.8.881.

[20]

J.-J. Gu and J.-M. Wang, Sliding mode control for $N$-coupled reaction-diffusion PDEs with boundary input disturbances, Internat. J. Robust Nonlinear Control, 29 (2019), 1437-1461.  doi: 10.1002/rnc.4448.

[21]

M. E. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Phys. D, 92 (1996), 178-192.  doi: 10.1016/0167-2789(95)00173-5.

[22]

U. Itkis, Control Systems of Variable Structure, Wiley, 1976.

[23]

L. Levaggi, Existence of sliding motions for nonlinear evolution equations in Banach spaces, Discrete Contin. Dyn. Syst., (2013), 477–487. doi: 10.3934/proc.2013.2013.477.

[24]

L. Levaggi, Infinite dimensional systems' sliding motions, Eur. J. Control, 8 (2002), 508-516.  doi: 10.3166/ejc.8.508-516.

[25]

J.-L. Lions, Quelques Méthodes de Résolution des Probl\`emes aux Limites non Linéaires, Dunod; Gauthier-Villars, Paris, 1969.

[26]

Q.-X. Liu, M. Rietkerk, P. M. J. Herman, T. Piersma, J. M. Fryxell and J. van de Koppel, Phase separation driven by density-dependent movement: A novel mechanism for ecological patterns, Phys. Life Rev., 19 (2016), 107-121.  doi: 10.1016/j.plrev.2016.07.009.

[27]

A. Miranville, Some generalizations of the Cahn–Hilliard equation, Asymptot. Anal., 22 (2000), 235-259. 

[28]

A. Miranville, The Cahn–Hilliard equation and some of its variants, AIMS Math., 2 (2017), 479-544.  doi: 10.3934/Math.2017.2.479.

[29]

A. Miranville and S. Zelik, Robust exponential attractors for Cahn-Hilliard type equations with singular potentials, Math. Methods Appl. Sci., 27 (2004), 545-582.  doi: 10.1002/mma.464.

[30]

A. Novick-Cohen, On the viscous Cahn-Hilliard equation, in Material Instabilities in Continuum Mechanics, Oxford Sci. Publ., Oxford Univ. Press, New York, 1988, 329-342.

[31]

A. Novick-Cohen and R. L. Pego, Stable patterns in a viscous diffusion equation, Trans. Amer. Math. Soc., 324 (1991), 331-351.  doi: 10.1090/S0002-9947-1991-1015926-7.

[32]

Y. Orlov and V. I. Utkin, Unit sliding mode control in infinite-dimensional systems. Adaptive learning and control using sliding modes, Appl. Math. Comput. Sci., 8 (1998), 7-20. 

[33]

Y. V. Orlov, Application of Lyapunov method in distributed systems, Autom. Remote Control, 44 (1983), 426-430. 

[34]

Y. V. Orlov, Discontinuous unit feedback control of uncertain infinite dimensional systems, IEEE Trans. Automatic Control, 45 (2000), 834-843.  doi: 10.1109/9.855545.

[35]

Y. V. Orlov and V. I. Utkin, Sliding mode control in indefinite-dimensional systems, Automatica J. IFAC, 23 (1987), 753-757.  doi: 10.1016/0005-1098(87)90032-X.

[36]

Y. V. Orlov and V. I. Utkin, Use of sliding modes in distributed system control problems, Automat. Remote Control, 43 (1982), 1127-1135. 

[37]

A. Pilloni, A. Pisano, Y. Orlov and E. Usai, Consensus-based control for a network of diffusion PDEs with boundary local interaction, IEEE Trans. Automat. Control, 61 (2016), 2708-2713.  doi: 10.1109/TAC.2015.2506990.

[38]

P. Podio-Guidugli, Models of phase segregation and diffusion of atomic species on a lattice, Ric. Mat., 55 (2006), 105-118.  doi: 10.1007/s11587-006-0008-8.

[39]

J. Simon, Compact sets in the space $L^p(0,T; B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.

[40]

V. I. Utkin, Sliding Modes in Control and Optimization, Communications and Control Engineering Series, Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-642-84379-2.

[41] V. Utkin, J. Guldner and J. Shi, Sliding Mode Control in Electro-Mechanical Systems, CRC Press, Boca Raton, 2009.  doi: 10.1201/9781420065619.
[42]

S. M. Wise, J. S. Lowengrub, H. B. Frieboes and V. Cristini, Three-dimensional multispecies nonlinear tumor growth–I: Model and numerical method, J. Theoret. Biol., 253 (2008), 524-543.  doi: 10.1016/j.jtbi.2008.03.027.

[43]

H. Xing, D. Li, C. Gao and Y. Kao, Delay-independent sliding mode control for a class of quasi-linear parabolic distributed parameter systems with time-varying delay, J. Franklin Inst., 350 (2013), 397-418.  doi: 10.1016/j.jfranklin.2012.12.007.

[44]

K. D. Young and Ü. Özgüner, Variable Structure Systems, Sliding Mode and Nonlinear Control, Lecture Notes in Control and Information Sciences, 247, Springer-Verlag, Ltd., London, 1999. doi: 10.1007/BFb0109967.

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