# American Institute of Mathematical Sciences

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

Received  January 2018 Revised  November 2018 Published  September 2019 Early access  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 and 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. [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. [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. [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. [5] G. I. Barenblatt, Iu. 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. [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. [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. [8] A. Favini and A. Yagi, Degenerate Differential Equations in Banach Spaces, Marcel Dekker Inc, New York, Basel, Hong Kong, 1999. [9] A. Favini, G. 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. [10] A. Favini, G. 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. [11] V. E. Fedorov, Degenerate strongly continuous semigroups of operators, St. Peterburg Math. J., 12 (2001), 471-489. [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. [13] E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups, American Mathematical Society, Providence, R.I., 1974. [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) [15] S. G. Krein, Linear Differential Equations in Banach Space, American Mathematical Society, Providence, R.I., 1971. [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) [17] J.-L. Lions, Contrôle Optimal de Systémes Gouvernés par des Équations aux Dérivées Partielles, Dunod, Paris, 1968. (French) [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) [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. [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. [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. [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. [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. [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) [25] R. E. Showalter, The Sobolev type equations. Ⅰ, Applicable Analysis, 5 (1975), 15-22.  doi: 10.1080/00036817508839103. [26] R. E. Showalter, The Sobolev type equations. Ⅱ, Applicable Analysis, 5 (1975), 81-99.  doi: 10.1080/00036817508839111. [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. [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. [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. [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. [31] G. A. Sviridyuk, A problem of Showalter, Differential Equations, 25 (1989), 338-339. [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. [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. [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. [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) [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.

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##### 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. [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. [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. [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. [5] G. I. Barenblatt, Iu. 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. [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. [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. [8] A. Favini and A. Yagi, Degenerate Differential Equations in Banach Spaces, Marcel Dekker Inc, New York, Basel, Hong Kong, 1999. [9] A. Favini, G. 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. [10] A. Favini, G. 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. [11] V. E. Fedorov, Degenerate strongly continuous semigroups of operators, St. Peterburg Math. J., 12 (2001), 471-489. [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. [13] E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups, American Mathematical Society, Providence, R.I., 1974. [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) [15] S. G. Krein, Linear Differential Equations in Banach Space, American Mathematical Society, Providence, R.I., 1971. [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) [17] J.-L. Lions, Contrôle Optimal de Systémes Gouvernés par des Équations aux Dérivées Partielles, Dunod, Paris, 1968. (French) [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) [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. [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. [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. [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. [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. [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) [25] R. E. Showalter, The Sobolev type equations. Ⅰ, Applicable Analysis, 5 (1975), 15-22.  doi: 10.1080/00036817508839103. [26] R. E. Showalter, The Sobolev type equations. Ⅱ, Applicable Analysis, 5 (1975), 81-99.  doi: 10.1080/00036817508839111. [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. [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. [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. [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. [31] G. A. Sviridyuk, A problem of Showalter, Differential Equations, 25 (1989), 338-339. [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. [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. [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. [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) [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.
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