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Distributed optimal control of a nonstandard nonlocal phase field system

1 Dipartimento di Matematica “F. Casorati”, Università di Pavia, Via Ferrata 5, 27100 Pavia, Italy
2 Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstraße 39, 10117 Berlin andDepartment of Mathematics, Humboldt-Universit¨at zu Berlin, Unter den Linden 6, 10099 Berlin,Germany

Special Issues: Nonlinear Evolution PDEs, Interfaces and Applications

We investigate a distributed optimal control problem for a nonlocal phase field modelof viscous Cahn-Hilliard type. The model constitutes a nonlocal version of a model for two-speciesphase segregation on an atomic lattice under the presence of di usion that has been studied in a seriesof papers by P. Podio-Guidugli and the present authors. The model consists of a highly nonlinearparabolic equation coupled to an ordinary di erential equation. The latter equation contains bothnonlocal and singular terms that render the analysis di cult. Standard arguments of optimal controltheory do not apply directly, although the control constraints and the cost functional are of standardtype. We show that the problem admits a solution, and we derive the first-order necessary conditionsof optimality.
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1. C. Baiocchi, Sulle equazioni di erenziali astratte lineari del primo e del secondo ordine negli spazi di Hilbert, Ann. Mat Pura Appl. (4) 76 (1967), 233-304.
2. P. Colli, M. H. Farshbaf-Shaker, G. Gilardi, J. Sprekels, Optimal boundary control of a viscous Cahn–Hilliard system with dynamic boundary condition and double obstacle potentials, SIAM J. Control Optim. 53 (2015), 2696-2721.
3. P. Colli, M. H. Farshbaf-Shaker, G. Gilardi, J. Sprekels, Second-order analysis of a boundary control problem for the viscous Cahn–Hilliard equation with dynamic boundary conditions, Ann. Acad. Rom. Sci. Math. Appl. 7 (2015), 41-66.
4. P. Colli, G. Gilardi, P. Krejčí, P. Podio-Guidugli, J. Sprekels, Analysis of a time discretization scheme for a nonstandard viscous Cahn–Hilliard system, ESAIM Math. Model. Numer. Anal. 48 (2014), 1061-1087.
5. P. Colli, G. Gilardi, P. Krejčí, J. Sprekels, A continuous dependence result for a nonstandard system of phase field equations, Math. Methods Appl. Sci. 37 (2014), 1318-1324.
6. P. Colli, G. Gilardi, P. Krejčí, J. Sprekels, A vanishing di usion limit in a nonstandard system of phase field equations, Evol. Equ. Control Theory 3 (2014), 257-275.
7. P. Colli, G. Gilardi, P. Podio-Guidugli, J. Sprekels, Well-posedness and long-time behavior for a nonstandard viscous Cahn–Hilliard system, SIAM J. Appl. Math. 71 (2011), 1849-1870.
8. P. Colli, G. Gilardi, P. Podio-Guidugli, J. Sprekels, Distributed optimal control of a nonstandard system of phase field equations, Contin. Mech. Thermodyn. 24 (2012), 437-459.
9. P. Colli, G. Gilardi, P. Podio-Guidugli, J. Sprekels, Continuous dependence for a nonstandard Cahn–Hilliard system with nonlinear atom mobility, Rend. Sem. Mat. Univ. Politec. Torino 70 (2012), 27-52.
10. P. Colli, G. Gilardi, P. Podio-Guidugli, J. Sprekels, Global existence for a strongly coupled Cahn– Hilliard system with viscosity, Boll. Unione Mat. Ital. (9) 5 (2012), 495-513.
11. P. Colli, G. Gilardi, P. Podio-Guidugli, J. Sprekels, An asymptotic analysis for a nonstandard Cahn–Hilliard system with viscosity, Discrete Contin. Dyn. Syst. Ser. S 6 (2013), 353-368.
12. P. Colli, G. Gilardi, P. Podio-Guidugli, J. Sprekels, Global existence and uniqueness for a singular /degenerate Cahn–Hilliard system with viscosity, J. Di erential Equations 254 (2013), 4217- 4244.
13. P. Colli, G. Gilardi, E. Rocca, J. Sprekels, Optimal distributed control of a di use interface model of tumor growth, preprint arXiv:1601.04567 [math.AP] (2016), 1-32.
14. P. Colli, G. Gilardi, J. Sprekels, Analysis and optimal boundary control of a nonstandard system of phase field equations, Milan J. Math. 80 (2012), 119-149.
15. P. Colli, G. Gilardi, J. Sprekels, Regularity of the solution to a nonstandard system of phase field equations, Rend. Cl. Sci. Mat. Nat. 147 (2013), 3-19.
16. P. Colli, G. Gilardi, J. Sprekels, A boundary control problem for the pure Cahn–Hilliard equation with dynamic boundary conditions, Adv. Nonlinear Anal. 4 (2015), 311-325.
17. P. Colli, G. Gilardi, J. Sprekels, A boundary control problem for the viscous Cahn–Hilliard equation with dynamic boundary conditions, Appl. Math. Optim. 73 (2016), 195-225.
18. P. Colli, G. Gilardi, J. Sprekels, On an application of Tikhonov’s fixed point theorem to a nonlocal Cahn–Hilliard type system modeling phase separation, J. Di erential Equations 260 (2016), 7940- 7964.
19. S. Frigeri, E. Rocca, J. Sprekels, Optimal distributed control of a nonlocal Cahn–Hilliard/Navier– Stokes system in 2D, SIAM J. Control Optim. 54 (2016), 221-250.
20. M. Hinterm¨uller, T. Keil, D. Wegner, Optimal control of a semidiscrete Cahn–Hilliard–Navier– Stokes system with non-matched fluid densities, preprint arXiv:1506.03591 [math.AP] (2015), 1- 35.
21. M. Hinterm¨uller, D.Wegner, Distributed optimal control of the Cahn–Hilliard system including the case of a double-obstacle homogeneous free energy density, SIAM J. Control Optim. 50 (2012), 388-418.
22. M. Hinterm¨uller, D. Wegner, Optimal control of a semidiscrete Cahn–Hilliard–Navier–Stokes system, SIAM J. Control Optim. 52 (2014), 747-772.
23. M. Hinterm¨uller, D. Wegner, Distributed and boundary control problems for the semidiscrete Cahn–Hilliard/Navier–Stokes system with nonsmooth Ginzburg–Landau energies, Isaac Newton Institute Preprint Series No. NI14042-FRB (2014), 1-29.
24. J. L. Lions, “Quelques méthodes de résolution des problèmes aux limites non linéaires”, Dunod Gauthier-Villars, Paris, 1969.
25. P. Podio-Guidugli, Models of phase segregation and diffusion of atomic species on a lattice, Ric. Mat. 55 (2006), pp. 105-118.
26. E. Rocca, J. Sprekels, Optimal distributed control of a nonlocal convective Cahn–Hilliard equation by the velocity in three dimensions, SIAM J. Control Optim. 53 (2015), 1654-1680.
27. J. Simon, Compact sets in the space Lp(0; T; B) , Ann. Mat. Pura Appl. (4) 146 (1987), pp. 65-96.
28. Q.-F. Wang, S.-i. Nakagiri, Weak solutions of Cahn–Hilliard equations having forcing terms and optimal control problems, Mathematical models in functional equations (Japanese) (Kyoto, 1999), Sῡrikaisekikenkyῡsho Kōkyῡroku No. 1128 (2000), 172-180.
29. X. Zhao, C. Liu, Optimal control of the convective Cahn–Hilliard equation, Appl. Anal. 92 (2013), 1028-1045.
30. X. Zhao, C. Liu, Optimal control for the convective Cahn–Hilliard equation in 2D case, Appl. Math. Optim. 70 (2014), 61-82.

Copyright Info: © 2016, Gianni Gilardi, et al., licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

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