Optimal control in problems without initial conditions for
evolutionary variational inequalities

M. M. Bokalo; A. M. Tsebenko

doi:10.15330/ms.46.1.51-66

Optimal control in problems without initial conditions for the evolution variational inequal- ities is studied. Control functions are contained in a free member of the inequalities. The case of general cost function is considered, which includes either cases of distributed observations or final observations. The existence of an optimal control is proved.

1. David R. Adams, Suzanne Lenhart, Optimal control of the obstacle for a parabolic variational inequality, Journal of Mathematical Analysis and Applications, 268 (2002), 602–614.

2. J.-P. Aubin, Un theoreme de compacite, Comptes rendus hebdomadaires des seances de l’academie des sciences, 256 (2007), №24, 5042–5044.

3. V. Barbu, Optimal Control of Variational Inequalities, London: Pitman, 1983.

4. F. Bernis, Existence results for doubly nonlinear higher order parabolic equations on unbounded domains, Math. Ann., 279 (1988), 373–394.

5. M. Bokalo, Well-posedness of problems without initial conditions for nonlinear parabolic variational inequalities, Nonlinear boundary problem, 8 (1998), 58–63.

6. B. Mykola, L. Alfredo, Linear evolution first-order problems without initial conditions, A. Milan J. Math., 77 (2009), 437–494.

7. M. Bokalo, A. Tsebenko, Existence of optimal control in the coefficients for problem without initial condition for strongly nonlinear parabolic equations, Mat. Stud. 45 (2010), 40-56.

8. M. Bokalo, Optimal control of evolution systems without initial conditions, Visnyk of the Lviv University. Series Mechanics and Mathematics, 73 (2010), 85-113.

9. M. Bokalo, Problem without initial conditions for some classes of nonlinear parabolic equations, N.M. J. Math. Sci., 51 (1990), 2291-2322.

10. M. Boukrouche, D.A. Tarzia, Existence, uniqueness, and convergence of optimal control problems associated with parabolic variational inequalities of the second kind, arXiv:1309.4869v1 [math.AP], 2013.

11. H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, Dordrecht, Heidelberg, London, 2011.

12. H. BrЃLezis, OpЃLerateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, Amsterdam, London: North-Holland Publishing Comp., 1973.

13. O. Buhrii, Some parabolic variational inequalities without initial conditions, Visnyk of the Lviv University, Series Mechanics and Mathematics. 49 (1998), 113-121.

14. H.Gayevskyy, K. Greger, K. Zaharias, Nonlinear operator equations and operator differential equations. . M.: Mir, 1978.

15. S.D. Ivasishen, Parabolic boundary problems without initial conditions, Ukr. Mat. Zh., 34 (1982), no.5, 547-552.

16. I. Kazufumi, K. Kunisch, Optimal control of parabolic variational inequalities, J. Math. Pures Appl., 93 (2010), 329-360.

17. O. Kogut, On optimal control problem in coefficients for nonlinear elliptic variational inequalities, Visnik Dnipropetrovskogo Universitetu. Seria Modeluvanna, 19 (2011), no.8, 86.98.

18. S. Lavrenyuk, M. Ptashnyk, Problem without initial conditions for a nonlinear pseudoparabolic system, Differential Equations, 36 (2000), no.5, 739.748.

19. J.-L. Lions, Quelques methodes de resolution des problemes aux limites non lineaires, Paris (France): Dunod Gauthier-Villars, 1969.

20. O. Oleinik, G. Iosifjan, Analog of Saint-Venantfs principle and uniqueness of solutions of the boundary problems in unbounded domain for parabolic equations, Usp. Mat. Nauk., 31 (1976), no.6, 142-166.

21. A. Pankov, Bounded and almost periodic solutions of nonlinear operator differential equations, Kluwer, Dordrecht, 1990.

22. P. Pukach, On problem without initial conditions for some nonlinear degenarated parabolic system, Ukrainian Math. J., 46 (1994), no.4, 484-487.

23. R. Rockafellar, On the maximal monotonicity of subdifferential mappings, Pacific J. Math., 33 (1970), no.1, 209-216.

24. R. Showalter, Monotone operators in Banach space and nonlinear partial differential equations, volume 49 of Mathematical Surveys and Monographs, Providence: Amer. Math. Soc., 1997. . xiv+278 p.

25. R. Showalter, Singular nonlinear evolution equations, Rocky Mountain J. Math., 10 (1980), no.3, 499-507.

26. A. Tikhonov, A. Samarskii, Equations of mathematical physics, M.: Nauka, 1972.

27. A. Tychonoff Theor`emes dunicite pour lfequation de la chaleur, Mat. Sb., 42 (1935), no.2, 199-216. 28. K. Yoshida, Functional Analysis, Mir, Moscow, 1967.

	Optimal control in problems without initial conditions for evolutionary variational inequalities (in Ukrainian)
Author	M. M. Bokalo, A. M. Tsebenko mm.bokalo@gmail.com Ivan Franko National University of Lviv
Abstract	Optimal control in problems without initial conditions for the evolution variational inequal- ities is studied. Control functions are contained in a free member of the inequalities. The case of general cost function is considered, which includes either cases of distributed observations or final observations. The existence of an optimal control is proved.
Keywords	optimal control; parabolic variational inequality; variational inclusion
DOI	doi:10.15330/ms.46.1.51-66
Reference	1. David R. Adams, Suzanne Lenhart, Optimal control of the obstacle for a parabolic variational inequality, Journal of Mathematical Analysis and Applications, 268 (2002), 602–614. 2. J.-P. Aubin, Un theoreme de compacite, Comptes rendus hebdomadaires des seances de l’academie des sciences, 256 (2007), №24, 5042–5044. 3. V. Barbu, Optimal Control of Variational Inequalities, London: Pitman, 1983. 4. F. Bernis, Existence results for doubly nonlinear higher order parabolic equations on unbounded domains, Math. Ann., 279 (1988), 373–394. 5. M. Bokalo, Well-posedness of problems without initial conditions for nonlinear parabolic variational inequalities, Nonlinear boundary problem, 8 (1998), 58–63. 6. B. Mykola, L. Alfredo, Linear evolution first-order problems without initial conditions, A. Milan J. Math., 77 (2009), 437–494. 7. M. Bokalo, A. Tsebenko, Existence of optimal control in the coefficients for problem without initial condition for strongly nonlinear parabolic equations, Mat. Stud. 45 (2010), 40-56. 8. M. Bokalo, Optimal control of evolution systems without initial conditions, Visnyk of the Lviv University. Series Mechanics and Mathematics, 73 (2010), 85-113. 9. M. Bokalo, Problem without initial conditions for some classes of nonlinear parabolic equations, N.M. J. Math. Sci., 51 (1990), 2291-2322. 10. M. Boukrouche, D.A. Tarzia, Existence, uniqueness, and convergence of optimal control problems associated with parabolic variational inequalities of the second kind, arXiv:1309.4869v1 [math.AP], 2013. 11. H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, Dordrecht, Heidelberg, London, 2011. 12. H. BrЃLezis, OpЃLerateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, Amsterdam, London: North-Holland Publishing Comp., 1973. 13. O. Buhrii, Some parabolic variational inequalities without initial conditions, Visnyk of the Lviv University, Series Mechanics and Mathematics. 49 (1998), 113-121. 14. H.Gayevskyy, K. Greger, K. Zaharias, Nonlinear operator equations and operator differential equations. . M.: Mir, 1978. 15. S.D. Ivasishen, Parabolic boundary problems without initial conditions, Ukr. Mat. Zh., 34 (1982), no.5, 547-552. 16. I. Kazufumi, K. Kunisch, Optimal control of parabolic variational inequalities, J. Math. Pures Appl., 93 (2010), 329-360. 17. O. Kogut, On optimal control problem in coefficients for nonlinear elliptic variational inequalities, Visnik Dnipropetrovskogo Universitetu. Seria Modeluvanna, 19 (2011), no.8, 86.98. 18. S. Lavrenyuk, M. Ptashnyk, Problem without initial conditions for a nonlinear pseudoparabolic system, Differential Equations, 36 (2000), no.5, 739.748. 19. J.-L. Lions, Quelques methodes de resolution des problemes aux limites non lineaires, Paris (France): Dunod Gauthier-Villars, 1969. 20. O. Oleinik, G. Iosifjan, Analog of Saint-Venantfs principle and uniqueness of solutions of the boundary problems in unbounded domain for parabolic equations, Usp. Mat. Nauk., 31 (1976), no.6, 142-166. 21. A. Pankov, Bounded and almost periodic solutions of nonlinear operator differential equations, Kluwer, Dordrecht, 1990. 22. P. Pukach, On problem without initial conditions for some nonlinear degenarated parabolic system, Ukrainian Math. J., 46 (1994), no.4, 484-487. 23. R. Rockafellar, On the maximal monotonicity of subdifferential mappings, Pacific J. Math., 33 (1970), no.1, 209-216. 24. R. Showalter, Monotone operators in Banach space and nonlinear partial differential equations, volume 49 of Mathematical Surveys and Monographs, Providence: Amer. Math. Soc., 1997. . xiv+278 p. 25. R. Showalter, Singular nonlinear evolution equations, Rocky Mountain J. Math., 10 (1980), no.3, 499-507. 26. A. Tikhonov, A. Samarskii, Equations of mathematical physics, M.: Nauka, 1972. 27. A. Tychonoff Theor`emes dunicite pour lfequation de la chaleur, Mat. Sb., 42 (1935), no.2, 199-216. 28. K. Yoshida, Functional Analysis, Mir, Moscow, 1967.
Pages	51-66
Volume	46
Issue	1
Year	2016
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

Optimal control in problems without initial conditions for evolutionary variational inequalities (in Ukrainian)