doi: 10.3934/mcrf.2020051
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Solvability and sliding mode control for the viscous Cahn–Hilliard system with a possibly singular potential

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 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 $.

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 & Related Fields, 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.  Google Scholar

[2]

V. BarbuP. ColliG. GilardiG. 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.  Google Scholar

[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.  Google Scholar

[4]

A. L. BertozziS. 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.  Google Scholar

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[8]

M.-B. ChengV. 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.  Google Scholar

[9]

P. ColliG. GilardiG. 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.  Google Scholar

[10]

P. ColliG. GilardiP. 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.  Google Scholar

[11]

P. ColliG. GilardiP. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[18]

C. G. GalM. 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.  Google Scholar

[19]

G. GilardiA. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[24]

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

[25]

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

[26]

Q.-X. LiuM. RietkerkP. M. J. HermanT. PiersmaJ. 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.  Google Scholar

[27]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[33]

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

[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.  Google Scholar

[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.  Google Scholar

[36]

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

[37]

A. PilloniA. PisanoY. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

S. M. WiseJ. S. LowengrubH. 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.  Google Scholar

[43]

H. XingD. LiC. 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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

V. BarbuP. ColliG. GilardiG. 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.  Google Scholar

[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.  Google Scholar

[4]

A. L. BertozziS. 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.  Google Scholar

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[8]

M.-B. ChengV. 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.  Google Scholar

[9]

P. ColliG. GilardiG. 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.  Google Scholar

[10]

P. ColliG. GilardiP. 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.  Google Scholar

[11]

P. ColliG. GilardiP. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[18]

C. G. GalM. 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.  Google Scholar

[19]

G. GilardiA. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[24]

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

[25]

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

[26]

Q.-X. LiuM. RietkerkP. M. J. HermanT. PiersmaJ. 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.  Google Scholar

[27]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[33]

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

[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.  Google Scholar

[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.  Google Scholar

[36]

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

[37]

A. PilloniA. PisanoY. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

S. M. WiseJ. S. LowengrubH. 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.  Google Scholar

[43]

H. XingD. LiC. 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.  Google Scholar

[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.  Google Scholar

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Linglong Du, Caixuan Ren. Pointwise wave behavior of the initial-boundary value problem for the nonlinear damped wave equation in $\mathbb{R}_{+}^{n} $. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3265-3280. doi: 10.3934/dcdsb.2018319

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