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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  March 2019 Early access  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 and 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. 

[2]

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

[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.

[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.

[5]

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

[6]

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

[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.

[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.

[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.

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[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).

[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.

[17]

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

[18]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[27]

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

[28]

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

[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.

[30]

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

[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. 

[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.

[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.

[34]

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

[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.

[36]

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

[37]

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

[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.

[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 .

[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.

[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.

[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.

show all references

To Professor Valery Melnik, in Memoriam

References:
[1]

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

[2]

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

[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.

[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.

[5]

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

[6]

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

[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.

[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.

[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.

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[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).

[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.

[17]

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

[18]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[27]

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

[28]

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

[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.

[30]

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

[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. 

[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.

[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.

[34]

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

[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.

[36]

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

[37]

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

[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.

[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 .

[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.

[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.

[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.

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