September  2019, 8(3): 473-488. doi: 10.3934/eect.2019023

Optimal control of solutions of a multipoint initial-final problem for non-autonomous evolutionary Sobolev type equation

South Ural State University, Institute of Natural Sciences and Mathematics, 454080, Chelyabinsk, Lenin av, 76, Russian Federation

* Corresponding author: sagadeevama@susu.ru

Received  January 2018 Revised  November 2018 Published  May 2019

Fund Project: The work was supported by Act 211 Government of the Russian Federation, contract 02.A03.21.0011.

The paper presents sufficient conditions for existence of an optimal control of solutions to a non-autonomous degenerate operator-differential evolution equation. We construct families of operators that solve this equation, as well as classical and strong solutions of the multipoint initial-final problem for the equation. We show that there exists a solution of an optimal control problem for a given operator-differential equation with a multipoint initial-final condition. The paper, in addition to the introduction and the bibliography, contains five sections. The first three parts contain information about the solvability of the multipoint initial-final problem for a non-autonomous equation. The fourth section presents the main result of the article; that is, a theorem on existence of optimal control of solutions to a multipoint initial-final problem. In the fifth part, the optimal control problem for the non-autonomous modified Chen – Gurtin model with the multipoint initial-final condition is investigated on the basis of the obtained abstract results.

Citation: Minzilia A. Sagadeeva, Sophiya A. Zagrebina, Natalia A. Manakova. Optimal control of solutions of a multipoint initial-final problem for non-autonomous evolutionary Sobolev type equation. Evolution Equations & Control Theory, 2019, 8 (3) : 473-488. doi: 10.3934/eect.2019023
References:
[1]

A. B. Alshin, M. O. Korpusov and A. G. Sveshnikov, Blow Up in Nonlinear Sobolev Type Equations, Walter de Gruyter, 2011. doi: 10.1515/9783110255294.  Google Scholar

[2]

I. S. Aranson and L. Kramer, The world of the complex Ginzburg – Landau equation, Reviews of Modern Physics, 74 (2002), 99-143.  doi: 10.1103/RevModPhys.74.99.  Google Scholar

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J. Banasiak, Mathematical properties of inelastic scattering models in linear kinetic theory, Mathematical Models & Methods in Applied Sciences, 10 (2000), 163-186.  doi: 10.1142/S0218202500000112.  Google Scholar

[5]

G. I. BarenblattIu. P. Zheltov and I. N. Kochina, Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks [Strata], Journal of Applied Mathematics and Mechanics, 24 (1960), 1286-1303.  doi: 10.1016/0021-8928(60)90107-6.  Google Scholar

[6]

P. J. Chen and M. E. Gurtin, On a theory of heat conduction involving two temperatures, Journal of Applied Mathematics and Physics (ZAMP), 19 (1968), 614-627.  doi: 10.1007/BF01594969.  Google Scholar

[7]

G. V. Demidenko and S. V. Upsenskii, Partial Differential Equations and Systems not Solvable with Respect to the Highest-Order Derivative, Marcel Dekker, New York, Basel, 2003. doi: 10.1201/9780203911433.  Google Scholar

[8]

A. Favini and A. Yagi, Degenerate Differential Equations in Banach Spaces, Marcel Dekker Inc, New York, Basel, Hong Kong, 1999.  Google Scholar

[9]

A. FaviniG. A. Sviridyuk and A. A. Zamyshlyaeva, One class of Sobolev type equations of higher order with additive "white noise", Communications on Pure and Applied Analysis, 15 (2016), 185-196.  doi: 10.3934/cpaa.2016.15.185.  Google Scholar

[10]

A. FaviniG. A. Sviridyuk and M. A. Sagadeeva, Linear Sobolev type equations with relatively p-radial operators in space of "noises", Mediterranian Journal of Mathematics, 13 (2016), 4607-4621.  doi: 10.1007/s00009-016-0765-x.  Google Scholar

[11]

V. E. Fedorov, Degenerate strongly continuous semigroups of operators, St. Peterburg Math. J., 12 (2001), 471-489.   Google Scholar

[12]

M. Hallaire, Soil water movement in the film and vapor phase under the influence of evapotranspiration. Water and its conduction insoils, Proceedings of XXXVII Annual Meeting of the Highway Research Board, Highway Research Board Special Report, 40 (1958), 88-105.   Google Scholar

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E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups, American Mathematical Society, Providence, R.I., 1974.  Google Scholar

[14]

A. V. Keller and S. A. Zagrebina, Some generalizations of the Showalter – Sidorov problem for Sobolev-type models, Bulletin of the South Ural State University, Series: Mathematical Modelling, Programming and Computer Software, 8, issue 2 (2015), 5–23. (Russian) Google Scholar

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S. G. Krein, Linear Differential Equations in Banach Space, American Mathematical Society, Providence, R.I., 1971.  Google Scholar

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S. G. Krein and S. Ja. L'vin, A general initial problem for a differential equation in a Banach space, Dokl. Akad. Nauk SSSR, 211 (1973), 530–533. (Russian)  Google Scholar

[17]

J.-L. Lions, Contrôle Optimal de Systémes Gouvernés par des Équations aux Dérivées Partielles, Dunod, Paris, 1968. (French)  Google Scholar

[18]

N. A. Manakova, Mathematical models and optimal control of the filtration and deformation processes, Bulletin of the South Ural State University, Series: Mathematical Modelling, Programming and Computer Software, 8, issue 3 (2015), 5–24. (Russian) Google Scholar

[19]

N. A. Manakova and A. G. Dylkov, Optimal control of the solutions of the initial-finish problem for the linear Hoff model, Mathematical Notes, 94 (2013), 220-230.  doi: 10.1134/S0001434613070225.  Google Scholar

[20]

N. A. Manakova and G. A. Sviridyuk, An optimal control of the solutions of the initial-final problem for linear Sobolev type equations with strongly relatively p-radial operator, in Semigroups of Operators – Theory and Applications, Springer Proc. Math. Stat. (eds. J. Banasiak, A. Bobrowski and M. Lachowicz), 113, Springer, Cham (2015), 213–224. doi: 10.1007/978-3-319-12145-1_13.  Google Scholar

[21]

A. P. Oskolkov, Nonlocal problems for some class nonlinear operator equations arising in the theory Sobolev type equations, Journal of Soviet Mathematics, 64 (1993), 724-736.  doi: 10.1007/BF02988478.  Google Scholar

[22]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[23]

M. A. Sagadeeva and G. A. Sviridyuk, The nonautonomous linear Oskolkov model on a geometrical graph: The stability of solutions and the optimal control problem, in Semigroups of Operators – Theory and Applications, Springer Proc. Math. Stat. (eds. J. Banasiak, A. Bobrowski and M. Lachowicz), 113, Springer, Cham, (2015), 257–271. doi: 10.1007/978-3-319-12145-1_16.  Google Scholar

[24]

M. A. Sagadeeva, Degenerate flows of solving operators for nonstationary Sobolev type equations, Bulletin of the South Ural State University, Series: Mathematics. Mechanics. Physics, 9, issue 1 (2017), 22–30. (Russian) Google Scholar

[25]

R. E. Showalter, The Sobolev type equations. Ⅰ, Applicable Analysis, 5 (1975), 15-22.  doi: 10.1080/00036817508839103.  Google Scholar

[26]

R. E. Showalter, The Sobolev type equations. Ⅱ, Applicable Analysis, 5 (1975), 81-99.  doi: 10.1080/00036817508839111.  Google Scholar

[27]

T. G. Sukacheva and A. O. Kondyukov, On a class of Sobolev-type equations, Bulletin of the South Ural State University, Series: Mathematical Modelling, Programming and Computer Software, 7, issue 4 (2014), 5–21. doi: 10.14529/mmp140401.  Google Scholar

[28]

G. A. Sviridyuk and A. S. Shipilov, On the stability of solutions of the Oskolkov equations on a graph, Differential Equations, 46 (2010), 1157-1163.  doi: 10.1134/S0012266110080094.  Google Scholar

[29]

G. A. Sviridyuk and V. E. Fedorov, Linear Sobolev Type Equations and Degenerate Semigroups of Operators, VSP, Utrecht – Boston, 2003. doi: 10.1515/9783110915501.  Google Scholar

[30]

G. A. Sviridyuk, Sobolev-type linear equations and strongly continuous semigroups of resolving operators with kernels, Russian Acad. Sci. Dokl. Math., 50 (1995), 137-142.   Google Scholar

[31]

G. A. Sviridyuk, A problem of Showalter, Differential Equations, 25 (1989), 338-339.   Google Scholar

[32]

G. A. Sviridyuk and A. A. Efremov, Optimal control of Sobolev-type linear equations with relatively p-sectorial operators, Differential Equations, 31 (1995), 1882-1890.   Google Scholar

[33]

S. A. Zagrebina, A multipoint initial-final value problem for a linear model of plane-parallel thermal convection in viscoelastic incompressible fluid, Bulletin of the South Ural State University, Series: Mathematical Modelling, Programming and Computer Software, 7, issue 3 (2014), 5–22. doi: 10.14529/mmp140301.  Google Scholar

[34]

S. A. Zagrebina and M. A. Sagadeeva, The generalized splitting theorem for linear Sobolev type equations in relatively radial case, The Bulletin of Irkutsk State University, Series: Mathematics, 7 (2014), 19-33.   Google Scholar

[35]

A. A. Zamyshlyaeva, The higher-order Sobolev-type models, Bulletin of the South Ural State University, Series: Mathematical Modelling, Programming and Computer Software, 7, issue 2 (2014), 5–28. (Russian) Google Scholar

[36]

A. A. Zamyshlyaeva, O. N. Tsyplenkova and E. V. Bychkov, Optimal control of solutions to the initial-final problem for the Sobolev type equation of higher order, Journal of Computational and Engineering Mathematics, 3, issue 2 (2016), 57–67. doi: 10.14529/jcem1602007.  Google Scholar

show all references

References:
[1]

A. B. Alshin, M. O. Korpusov and A. G. Sveshnikov, Blow Up in Nonlinear Sobolev Type Equations, Walter de Gruyter, 2011. doi: 10.1515/9783110255294.  Google Scholar

[2]

I. S. Aranson and L. Kramer, The world of the complex Ginzburg – Landau equation, Reviews of Modern Physics, 74 (2002), 99-143.  doi: 10.1103/RevModPhys.74.99.  Google Scholar

[3]

L. Arlotti and J. Banasiak, Nonautonomous fragmentation equation via evolution semigroups, Mathematical Methods in the Applied Sciences, 33 (2010), 1201-1210.  doi: 10.1002/mma.1282.  Google Scholar

[4]

J. Banasiak, Mathematical properties of inelastic scattering models in linear kinetic theory, Mathematical Models & Methods in Applied Sciences, 10 (2000), 163-186.  doi: 10.1142/S0218202500000112.  Google Scholar

[5]

G. I. BarenblattIu. P. Zheltov and I. N. Kochina, Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks [Strata], Journal of Applied Mathematics and Mechanics, 24 (1960), 1286-1303.  doi: 10.1016/0021-8928(60)90107-6.  Google Scholar

[6]

P. J. Chen and M. E. Gurtin, On a theory of heat conduction involving two temperatures, Journal of Applied Mathematics and Physics (ZAMP), 19 (1968), 614-627.  doi: 10.1007/BF01594969.  Google Scholar

[7]

G. V. Demidenko and S. V. Upsenskii, Partial Differential Equations and Systems not Solvable with Respect to the Highest-Order Derivative, Marcel Dekker, New York, Basel, 2003. doi: 10.1201/9780203911433.  Google Scholar

[8]

A. Favini and A. Yagi, Degenerate Differential Equations in Banach Spaces, Marcel Dekker Inc, New York, Basel, Hong Kong, 1999.  Google Scholar

[9]

A. FaviniG. A. Sviridyuk and A. A. Zamyshlyaeva, One class of Sobolev type equations of higher order with additive "white noise", Communications on Pure and Applied Analysis, 15 (2016), 185-196.  doi: 10.3934/cpaa.2016.15.185.  Google Scholar

[10]

A. FaviniG. A. Sviridyuk and M. A. Sagadeeva, Linear Sobolev type equations with relatively p-radial operators in space of "noises", Mediterranian Journal of Mathematics, 13 (2016), 4607-4621.  doi: 10.1007/s00009-016-0765-x.  Google Scholar

[11]

V. E. Fedorov, Degenerate strongly continuous semigroups of operators, St. Peterburg Math. J., 12 (2001), 471-489.   Google Scholar

[12]

M. Hallaire, Soil water movement in the film and vapor phase under the influence of evapotranspiration. Water and its conduction insoils, Proceedings of XXXVII Annual Meeting of the Highway Research Board, Highway Research Board Special Report, 40 (1958), 88-105.   Google Scholar

[13]

E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups, American Mathematical Society, Providence, R.I., 1974.  Google Scholar

[14]

A. V. Keller and S. A. Zagrebina, Some generalizations of the Showalter – Sidorov problem for Sobolev-type models, Bulletin of the South Ural State University, Series: Mathematical Modelling, Programming and Computer Software, 8, issue 2 (2015), 5–23. (Russian) Google Scholar

[15]

S. G. Krein, Linear Differential Equations in Banach Space, American Mathematical Society, Providence, R.I., 1971.  Google Scholar

[16]

S. G. Krein and S. Ja. L'vin, A general initial problem for a differential equation in a Banach space, Dokl. Akad. Nauk SSSR, 211 (1973), 530–533. (Russian)  Google Scholar

[17]

J.-L. Lions, Contrôle Optimal de Systémes Gouvernés par des Équations aux Dérivées Partielles, Dunod, Paris, 1968. (French)  Google Scholar

[18]

N. A. Manakova, Mathematical models and optimal control of the filtration and deformation processes, Bulletin of the South Ural State University, Series: Mathematical Modelling, Programming and Computer Software, 8, issue 3 (2015), 5–24. (Russian) Google Scholar

[19]

N. A. Manakova and A. G. Dylkov, Optimal control of the solutions of the initial-finish problem for the linear Hoff model, Mathematical Notes, 94 (2013), 220-230.  doi: 10.1134/S0001434613070225.  Google Scholar

[20]

N. A. Manakova and G. A. Sviridyuk, An optimal control of the solutions of the initial-final problem for linear Sobolev type equations with strongly relatively p-radial operator, in Semigroups of Operators – Theory and Applications, Springer Proc. Math. Stat. (eds. J. Banasiak, A. Bobrowski and M. Lachowicz), 113, Springer, Cham (2015), 213–224. doi: 10.1007/978-3-319-12145-1_13.  Google Scholar

[21]

A. P. Oskolkov, Nonlocal problems for some class nonlinear operator equations arising in the theory Sobolev type equations, Journal of Soviet Mathematics, 64 (1993), 724-736.  doi: 10.1007/BF02988478.  Google Scholar

[22]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[23]

M. A. Sagadeeva and G. A. Sviridyuk, The nonautonomous linear Oskolkov model on a geometrical graph: The stability of solutions and the optimal control problem, in Semigroups of Operators – Theory and Applications, Springer Proc. Math. Stat. (eds. J. Banasiak, A. Bobrowski and M. Lachowicz), 113, Springer, Cham, (2015), 257–271. doi: 10.1007/978-3-319-12145-1_16.  Google Scholar

[24]

M. A. Sagadeeva, Degenerate flows of solving operators for nonstationary Sobolev type equations, Bulletin of the South Ural State University, Series: Mathematics. Mechanics. Physics, 9, issue 1 (2017), 22–30. (Russian) Google Scholar

[25]

R. E. Showalter, The Sobolev type equations. Ⅰ, Applicable Analysis, 5 (1975), 15-22.  doi: 10.1080/00036817508839103.  Google Scholar

[26]

R. E. Showalter, The Sobolev type equations. Ⅱ, Applicable Analysis, 5 (1975), 81-99.  doi: 10.1080/00036817508839111.  Google Scholar

[27]

T. G. Sukacheva and A. O. Kondyukov, On a class of Sobolev-type equations, Bulletin of the South Ural State University, Series: Mathematical Modelling, Programming and Computer Software, 7, issue 4 (2014), 5–21. doi: 10.14529/mmp140401.  Google Scholar

[28]

G. A. Sviridyuk and A. S. Shipilov, On the stability of solutions of the Oskolkov equations on a graph, Differential Equations, 46 (2010), 1157-1163.  doi: 10.1134/S0012266110080094.  Google Scholar

[29]

G. A. Sviridyuk and V. E. Fedorov, Linear Sobolev Type Equations and Degenerate Semigroups of Operators, VSP, Utrecht – Boston, 2003. doi: 10.1515/9783110915501.  Google Scholar

[30]

G. A. Sviridyuk, Sobolev-type linear equations and strongly continuous semigroups of resolving operators with kernels, Russian Acad. Sci. Dokl. Math., 50 (1995), 137-142.   Google Scholar

[31]

G. A. Sviridyuk, A problem of Showalter, Differential Equations, 25 (1989), 338-339.   Google Scholar

[32]

G. A. Sviridyuk and A. A. Efremov, Optimal control of Sobolev-type linear equations with relatively p-sectorial operators, Differential Equations, 31 (1995), 1882-1890.   Google Scholar

[33]

S. A. Zagrebina, A multipoint initial-final value problem for a linear model of plane-parallel thermal convection in viscoelastic incompressible fluid, Bulletin of the South Ural State University, Series: Mathematical Modelling, Programming and Computer Software, 7, issue 3 (2014), 5–22. doi: 10.14529/mmp140301.  Google Scholar

[34]

S. A. Zagrebina and M. A. Sagadeeva, The generalized splitting theorem for linear Sobolev type equations in relatively radial case, The Bulletin of Irkutsk State University, Series: Mathematics, 7 (2014), 19-33.   Google Scholar

[35]

A. A. Zamyshlyaeva, The higher-order Sobolev-type models, Bulletin of the South Ural State University, Series: Mathematical Modelling, Programming and Computer Software, 7, issue 2 (2014), 5–28. (Russian) Google Scholar

[36]

A. A. Zamyshlyaeva, O. N. Tsyplenkova and E. V. Bychkov, Optimal control of solutions to the initial-final problem for the Sobolev type equation of higher order, Journal of Computational and Engineering Mathematics, 3, issue 2 (2016), 57–67. doi: 10.14529/jcem1602007.  Google Scholar

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