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March  2019, 24(3): 1229-1242. doi: 10.3934/dcdsb.2019013

Attractors of multivalued semi-flows generated by solutions of optimal control problems

1. 

Taras Shevchenko National University of Kyiv, Kyiv, Ukraine

2. 

Institute for Applied System Analysis, National Technical University "Igor Sikorsky Kyiv Polytechnic Institute", Kyiv, Ukraine

3. 

Centro de Investigación Operativa, Universidad Miguel Hernández de Elche, 03202-Elche, Alicante, Spain

4. 

National Technical University "Igor Sikorsky Kyiv Polytechnic Institute", Kyiv, Ukraine

To Professor Valery Melnik, in Memoriam

Received  February 2018 Revised  June 2018 Published  January 2019

Fund Project: The first two authors were partially supported by the State Fund for Fundamental Research of Ukraine under grants GP/F66/14921, GP/F78/187 and by the Grant of the National Academy of Sciences of Ukraine 2290/2018. The third author was partially supported by Spanish Ministry of Economy and Competitiveness and FEDER, projects MTM2015-63723-P and MTM2016-74921-P, and by Junta de Andalucía(Spain), project P12-FQM-1492.

In this paper we study the dynamical system generated by the solutions of optimal control problems. We obtain suitable conditions under which such systems generate multivalued semiprocesses. We prove the existence of uniform attractors for the multivalued semiprocess generated by the solutions of controlled reaction-diffusion equations and study its properties.

Citation: Olexiy V. Kapustyan, Pavlo O. Kasyanov, José Valero, Mikhail Z. Zgurovsky. Attractors of multivalued semi-flows generated by solutions of optimal control problems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1229-1242. doi: 10.3934/dcdsb.2019013
References:
[1]

A. V. Babin and M. I. Vishik, Maximal attractors of semigroups corresponding to evolutionary differential equations, Mat. Sb., 126 (1985), 397-419.   Google Scholar

[2]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Nauka, Moscow, 1989.  Google Scholar

[3]

J. M. Ball, Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations, in Mechanics: From Theory to Computation, Springer, New York, 2000,447–474.  Google Scholar

[4]

J. M. Ball, Global attractors for damped semilinear wave equations, Discrete Contin. Dyn. Syst., 10 (2004), 31-52.  doi: 10.3934/dcds.2004.10.31.  Google Scholar

[5]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Editura Academiei, Bucuresti, 1976.  Google Scholar

[6]

R. E. Bellman, I. Glicksberg and O. A. Gross, Some Aspects of the Mathematical Theory of Control Processes, Rand Corp., Santa-Monica, 1958.  Google Scholar

[7]

T. CaraballoJ. A. LangaV. S. Melnik and J. Valero, Pullback attractors of nonautonomous and stochastic multivalued dynamical systems, Set-Valued Anal., 11 (2003), 153-201.  doi: 10.1023/A:1022902802385.  Google Scholar

[8]

A. N. Carvalho, J. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Applied Mathematical Sciences, 182. Springer, New York, 2013. doi: 10.1007/978-1-4614-4581-4.  Google Scholar

[9]

D. N. Cheban, Global Attractors of Non-Autonomous Dissipative Dynamical Systems, Interdisciplinary Mathematical Sciences, Vol.1, World Scientific, New York, 2004. doi: 10.1142/9789812563088.  Google Scholar

[10]

D. N. Cheban, Global Attractors of Set-Valued Dynamical and Control Systems, Nova Science Publishers Inc, New York, 2010. Google Scholar

[11]

D. N. Cheban, Compact global attractors of control systems, Dyn. Control Syst., 16 (2010), 23-44.  doi: 10.1007/s10883-010-9086-8.  Google Scholar

[12]

D. N. Cheban and D. S. Fakhikh, Global Attractors of Dispersible Dynamical Systems, Sigma, Chisinau, 1994 (in Russian). Google Scholar

[13]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, AMS, Providence, Rhode Island, 2002.  Google Scholar

[14]

D. S. Fakhikh, The Levinson center of dispersible dissipative dynamical systems, Izv. Akad. Nauk Moldav. SSR Mat., (1990), 55–59, 78 (in Russian). Google Scholar

[15]

D. S. Fakhikh, The structure of the Levinson center of dispersible dissipative dynamical systems, Izv. Akad. Nauk Moldav. SSR Mat, (1991), 62–67, 92 (in Russian). Google Scholar

[16]

J. K. HaleJ. P. LaSalle and M. Slemrod, Theory of general class of dissipative processes, J. Math. Anal. Appl., 39 (1972), 177-191.  doi: 10.1016/0022-247X(72)90233-8.  Google Scholar

[17]

A. Haraux, Systemes Dynamiques Dissipatives et Applications, Masson, Paris, 1991.  Google Scholar

[18]

A. V. Kapustyan, Global attractors of a nonautonomous reaction-diffusion equation, Differential Equations, 38 (2002), 1467-1471.  doi: 10.1023/A:1022378831393.  Google Scholar

[19]

O. V. KapustyanO. A. Kapustian and A. V. Sukretna, Approximate stabilization for a nonlinear parabolic boundary-value problem, Ukrainian Math. J., 63 (2011), 759-767.  doi: 10.1007/s11253-011-0540-x.  Google Scholar

[20]

O. V. KapustyanP. O. Kasyanov and J. Valero, Pullback attractors for a class of extremal solutions of the 3D Navier-Stokes system, J. Math. Anal. Appl., 373 (2011), 535-547.  doi: 10.1016/j.jmaa.2010.07.040.  Google Scholar

[21]

O. V. KapustyanP. O. Kasyanov and J. Valero, Regular solutions and global attractors for reaction-diffusion systems without uniqueness, Commun. Pure Appl. Anal., 13 (2014), 1891-1906.  doi: 10.3934/cpaa.2014.13.1891.  Google Scholar

[22]

O. V. KapustyanV. S. Melnik and J. Valero, A weak attractor and properties of solutions for the three-dimensional Benard problem, Discrete Contin. Dyn. Syst., 18 (2007), 449-481.  doi: 10.3934/dcds.2007.18.449.  Google Scholar

[23]

O. V. Kapustyan, V. S. Melnik, J. Valero and V. V. Yasinsky, Global Attractors of Multi-Valued Dynamical Systems and Evolution Equations without Uniqueness, Naukova Dumka, Kyiv, 2008. Google Scholar

[24]

O. V. Kapustyan and D. V. Shkundin, Global attractor of one nonlinear parabolic equation, Ukrainian Math. J., 55 (2003), 446-455.  doi: 10.1023/B:UKMA.0000010155.48722.f2.  Google Scholar

[25]

O. V. Kapustyan and J. Valero, Attractors of differential inclusions and their approximations, Ukrainian Math. J., 52 (2000), 1118-1123.  doi: 10.1023/A:1005237902620.  Google Scholar

[26]

P. O. Kasyanov, Multivalued dynamics of solutions of autonomous operator differential equations with pseudomonotone nonlinearity, Math. Notes, 92 (2012), 205-218.  doi: 10.1134/S0001434612070231.  Google Scholar

[27]

O. A. Ladyzhenskaya, On dynamical system generated by Navier-Stokes equations, Zap. Nauch. Sem. LOMI, 27 (1972), 91-115.   Google Scholar

[28]

J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer, Berlin, 1971.  Google Scholar

[29]

P. Marín-RubioG. Planas and J. Real, Asymptotic behaviour of a phase-field model with three coupled equations without uniqueness, J. Differential Equations, 246 (2009), 4632-4652.  doi: 10.1016/j.jde.2009.01.021.  Google Scholar

[30]

V. S. Melnik, Multivalued Dynamics of Nonlinear Infinite-Dimensional Systems, Preprint NAS of Ukraine, 94-17, Kyiv, 1994.  Google Scholar

[31]

V. S. Melnik and O. V. Kapustyan, On global attractors of multivalued semidynamic systems and their approximations, Dokl. Akad. Nauk, 366 (1999), 445-448.   Google Scholar

[32]

V. S. Melnik and J. Valero, On attractors of multi-valued semi-flows and differential inclusions, Set-Valued Anal., 6 (1998), 83-111.  doi: 10.1023/A:1008608431399.  Google Scholar

[33]

A. Segatti, Global attractor for a class of doubly nonlinear abstract evolution equations, Discrete Contin. Dyn. Syst., 14 (2006), 801-820.  doi: 10.3934/dcds.2006.14.801.  Google Scholar

[34]

G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Springer, New-York, 2002. doi: 10.1007/978-1-4757-5037-9.  Google Scholar

[35]

J. Simsen and E. N. Neres Junior, Existence and upper semicontinuity of global attractors for a p-Laplacian differential inclusion, Bol. Soc. Paran. Mat, 33 (2015), 235-245.  doi: 10.5269/bspm.v33i1.21767.  Google Scholar

[36]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[37]

J. Valero, Attractors of parabolic equations without uniqueness, J. Dynamics Differential Equations, 13 (2001), 711-744.  doi: 10.1023/A:1016642525800.  Google Scholar

[38]

J. Valero and A. V. Kapustyan, On the connectedness and asymptotic behaviour of solutions of reaction-diffusion systems, J. Math. Anal. Appl., 323 (2006), 614-633.  doi: 10.1016/j.jmaa.2005.10.042.  Google Scholar

[39]

M. Z. Zgurovsky and P. O. Kasyanov, Evolution inclusions in nonsmooth systems with applications for earth data processing : uniform trajectory attractors for nonautonomous evolution inclusions solutions with pointwise pseudomonotone mappings, in Advances in Global Optimization, Springer Proceedings in Mathematics and Statistics, Cham, 95 (2015), 283-294. doi: 10.1007/978-3-319-08377-3_28 .  Google Scholar

[40]

M. Z. Zgurovsky and P. O. Kasyanov, Qualitative and Quantitative Analysis of Nonlinear Systems : Theory and Applications, Springer, Cham, 2018 doi: 10.1007/978-3-319-59840-6.  Google Scholar

[41]

M. Z. Zgurovsky, P. O. Kasyanov, O. V. Kapustyan, J. Valero and N. V. Zadoianchuk, Evolution Inclusions and Variation Inequalities for Earth Data Processing Ⅲ. Long-time Behavior of Evolution Inclusions Solutions in Earth Data Analysis, Springer, Berlin, 2012. doi: 10.1007/978-3-642-28512-7.  Google Scholar

[42]

M. Z. Zgurovsky and V. S. Melnik, Nonlinear Analysis and Control of Physical Processes and Fields, Springer, Berlin, 2004. doi: 10.1007/978-3-642-18770-4.  Google Scholar

show all references

References:
[1]

A. V. Babin and M. I. Vishik, Maximal attractors of semigroups corresponding to evolutionary differential equations, Mat. Sb., 126 (1985), 397-419.   Google Scholar

[2]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Nauka, Moscow, 1989.  Google Scholar

[3]

J. M. Ball, Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations, in Mechanics: From Theory to Computation, Springer, New York, 2000,447–474.  Google Scholar

[4]

J. M. Ball, Global attractors for damped semilinear wave equations, Discrete Contin. Dyn. Syst., 10 (2004), 31-52.  doi: 10.3934/dcds.2004.10.31.  Google Scholar

[5]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Editura Academiei, Bucuresti, 1976.  Google Scholar

[6]

R. E. Bellman, I. Glicksberg and O. A. Gross, Some Aspects of the Mathematical Theory of Control Processes, Rand Corp., Santa-Monica, 1958.  Google Scholar

[7]

T. CaraballoJ. A. LangaV. S. Melnik and J. Valero, Pullback attractors of nonautonomous and stochastic multivalued dynamical systems, Set-Valued Anal., 11 (2003), 153-201.  doi: 10.1023/A:1022902802385.  Google Scholar

[8]

A. N. Carvalho, J. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Applied Mathematical Sciences, 182. Springer, New York, 2013. doi: 10.1007/978-1-4614-4581-4.  Google Scholar

[9]

D. N. Cheban, Global Attractors of Non-Autonomous Dissipative Dynamical Systems, Interdisciplinary Mathematical Sciences, Vol.1, World Scientific, New York, 2004. doi: 10.1142/9789812563088.  Google Scholar

[10]

D. N. Cheban, Global Attractors of Set-Valued Dynamical and Control Systems, Nova Science Publishers Inc, New York, 2010. Google Scholar

[11]

D. N. Cheban, Compact global attractors of control systems, Dyn. Control Syst., 16 (2010), 23-44.  doi: 10.1007/s10883-010-9086-8.  Google Scholar

[12]

D. N. Cheban and D. S. Fakhikh, Global Attractors of Dispersible Dynamical Systems, Sigma, Chisinau, 1994 (in Russian). Google Scholar

[13]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, AMS, Providence, Rhode Island, 2002.  Google Scholar

[14]

D. S. Fakhikh, The Levinson center of dispersible dissipative dynamical systems, Izv. Akad. Nauk Moldav. SSR Mat., (1990), 55–59, 78 (in Russian). Google Scholar

[15]

D. S. Fakhikh, The structure of the Levinson center of dispersible dissipative dynamical systems, Izv. Akad. Nauk Moldav. SSR Mat, (1991), 62–67, 92 (in Russian). Google Scholar

[16]

J. K. HaleJ. P. LaSalle and M. Slemrod, Theory of general class of dissipative processes, J. Math. Anal. Appl., 39 (1972), 177-191.  doi: 10.1016/0022-247X(72)90233-8.  Google Scholar

[17]

A. Haraux, Systemes Dynamiques Dissipatives et Applications, Masson, Paris, 1991.  Google Scholar

[18]

A. V. Kapustyan, Global attractors of a nonautonomous reaction-diffusion equation, Differential Equations, 38 (2002), 1467-1471.  doi: 10.1023/A:1022378831393.  Google Scholar

[19]

O. V. KapustyanO. A. Kapustian and A. V. Sukretna, Approximate stabilization for a nonlinear parabolic boundary-value problem, Ukrainian Math. J., 63 (2011), 759-767.  doi: 10.1007/s11253-011-0540-x.  Google Scholar

[20]

O. V. KapustyanP. O. Kasyanov and J. Valero, Pullback attractors for a class of extremal solutions of the 3D Navier-Stokes system, J. Math. Anal. Appl., 373 (2011), 535-547.  doi: 10.1016/j.jmaa.2010.07.040.  Google Scholar

[21]

O. V. KapustyanP. O. Kasyanov and J. Valero, Regular solutions and global attractors for reaction-diffusion systems without uniqueness, Commun. Pure Appl. Anal., 13 (2014), 1891-1906.  doi: 10.3934/cpaa.2014.13.1891.  Google Scholar

[22]

O. V. KapustyanV. S. Melnik and J. Valero, A weak attractor and properties of solutions for the three-dimensional Benard problem, Discrete Contin. Dyn. Syst., 18 (2007), 449-481.  doi: 10.3934/dcds.2007.18.449.  Google Scholar

[23]

O. V. Kapustyan, V. S. Melnik, J. Valero and V. V. Yasinsky, Global Attractors of Multi-Valued Dynamical Systems and Evolution Equations without Uniqueness, Naukova Dumka, Kyiv, 2008. Google Scholar

[24]

O. V. Kapustyan and D. V. Shkundin, Global attractor of one nonlinear parabolic equation, Ukrainian Math. J., 55 (2003), 446-455.  doi: 10.1023/B:UKMA.0000010155.48722.f2.  Google Scholar

[25]

O. V. Kapustyan and J. Valero, Attractors of differential inclusions and their approximations, Ukrainian Math. J., 52 (2000), 1118-1123.  doi: 10.1023/A:1005237902620.  Google Scholar

[26]

P. O. Kasyanov, Multivalued dynamics of solutions of autonomous operator differential equations with pseudomonotone nonlinearity, Math. Notes, 92 (2012), 205-218.  doi: 10.1134/S0001434612070231.  Google Scholar

[27]

O. A. Ladyzhenskaya, On dynamical system generated by Navier-Stokes equations, Zap. Nauch. Sem. LOMI, 27 (1972), 91-115.   Google Scholar

[28]

J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer, Berlin, 1971.  Google Scholar

[29]

P. Marín-RubioG. Planas and J. Real, Asymptotic behaviour of a phase-field model with three coupled equations without uniqueness, J. Differential Equations, 246 (2009), 4632-4652.  doi: 10.1016/j.jde.2009.01.021.  Google Scholar

[30]

V. S. Melnik, Multivalued Dynamics of Nonlinear Infinite-Dimensional Systems, Preprint NAS of Ukraine, 94-17, Kyiv, 1994.  Google Scholar

[31]

V. S. Melnik and O. V. Kapustyan, On global attractors of multivalued semidynamic systems and their approximations, Dokl. Akad. Nauk, 366 (1999), 445-448.   Google Scholar

[32]

V. S. Melnik and J. Valero, On attractors of multi-valued semi-flows and differential inclusions, Set-Valued Anal., 6 (1998), 83-111.  doi: 10.1023/A:1008608431399.  Google Scholar

[33]

A. Segatti, Global attractor for a class of doubly nonlinear abstract evolution equations, Discrete Contin. Dyn. Syst., 14 (2006), 801-820.  doi: 10.3934/dcds.2006.14.801.  Google Scholar

[34]

G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Springer, New-York, 2002. doi: 10.1007/978-1-4757-5037-9.  Google Scholar

[35]

J. Simsen and E. N. Neres Junior, Existence and upper semicontinuity of global attractors for a p-Laplacian differential inclusion, Bol. Soc. Paran. Mat, 33 (2015), 235-245.  doi: 10.5269/bspm.v33i1.21767.  Google Scholar

[36]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[37]

J. Valero, Attractors of parabolic equations without uniqueness, J. Dynamics Differential Equations, 13 (2001), 711-744.  doi: 10.1023/A:1016642525800.  Google Scholar

[38]

J. Valero and A. V. Kapustyan, On the connectedness and asymptotic behaviour of solutions of reaction-diffusion systems, J. Math. Anal. Appl., 323 (2006), 614-633.  doi: 10.1016/j.jmaa.2005.10.042.  Google Scholar

[39]

M. Z. Zgurovsky and P. O. Kasyanov, Evolution inclusions in nonsmooth systems with applications for earth data processing : uniform trajectory attractors for nonautonomous evolution inclusions solutions with pointwise pseudomonotone mappings, in Advances in Global Optimization, Springer Proceedings in Mathematics and Statistics, Cham, 95 (2015), 283-294. doi: 10.1007/978-3-319-08377-3_28 .  Google Scholar

[40]

M. Z. Zgurovsky and P. O. Kasyanov, Qualitative and Quantitative Analysis of Nonlinear Systems : Theory and Applications, Springer, Cham, 2018 doi: 10.1007/978-3-319-59840-6.  Google Scholar

[41]

M. Z. Zgurovsky, P. O. Kasyanov, O. V. Kapustyan, J. Valero and N. V. Zadoianchuk, Evolution Inclusions and Variation Inequalities for Earth Data Processing Ⅲ. Long-time Behavior of Evolution Inclusions Solutions in Earth Data Analysis, Springer, Berlin, 2012. doi: 10.1007/978-3-642-28512-7.  Google Scholar

[42]

M. Z. Zgurovsky and V. S. Melnik, Nonlinear Analysis and Control of Physical Processes and Fields, Springer, Berlin, 2004. doi: 10.1007/978-3-642-18770-4.  Google Scholar

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