On properties of minimizers of a control problem with time-distributed functional related to parabolic equations

• Autorzy: Astashova I. V. , Filinovskiy A. V.

Identyfikatory

DOI

10.7494/OpMath.2019.39.5.595

• Języki publikacji: EN

Abstrakty

EN

We consider a control problem given by a mathematical model of the temperature control in industrial hothouses. The model is based on one-dimensional parabolic equations with variable coefficients. The optimal control is defined as a minimizer of a quadratic cost functional. We describe qualitative properties of this minimizer, study the structure of the set of accessible temperature functions, and prove the dense controllability for some set of control functions.

Słowa kluczowe

EN

parabolic equation; extremal problem; quadratic cost functional; minimizer; exact controllability; dense controllability

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2019
• Tom: Vol. 39, no. 5
• Strony: 595--609
• Opis fizyczny: Bibliogr. 23 poz.

Twórcy

autor

Astashova I. V.

• Lomonosov Moscow State University Moscow, Russia
• Plekhanov Russian University of Economics Moscow, Russia

autor

Filinovskiy A. V.

• Bauman Moscow State Technical University Moscow, Russia
• Lomonosov Moscow State University Moscow, Russia

Bibliografia

• [1] I.V. Astashova, A.V. Filinovskiy, V.A. Kondratiev, L.A. Muravei, Some problems in the qualitative theory of differential equations, J. Nat. Geom. 23 (2003), 1-126.
• [2] I.V. Astashova (ed.), Qualitative Properties of Solutions to Differential Equations and Related Topics of Spectral Analysis, UNITY-DANA, Moscow, 2012 [in Russian].
• [3] I.V. Astashova, A.V. Filinovskiy, D.A. Lashin, On maintaining optimal temperatures in greenhouses, WSEAS Trans, on Circuits and Systems 15 (2016), 198-204.
• [4] I. Astashova, A. Filinovskiy, D. Lashin, On a model of maintaining the optimal temperature in greenhouse, Funct. Differ. Equ. 23 (2016) 3-4, 97-108.
• [5] I. Astashova, A. Filinovskiy, D. Lashin, On optimal temperature control in hothouses, [in:] Th. Simos, Ch. Tsitouras (eds.), Proc. Int. Conf. on Numerical Analysis and Applied Mathematics 2016, Proceedings of 14th International Conference on Numerical Analysis and Applied Mathematics (ICNAAM 2016) (19-25 September 2016, Rhodes, Greece), AIP Conf. Proc. V. 1863, 2017, pp. 4-8.
• [6] I.V. Astashova, A.V. Filinovskiy, On the dense controllability for the parabolic problem, with time-distributed functional, Tatr. Mt. Math. Publ. 71 (2018), 9-25.
• [7] A.G. Butkovsky, Optimal control in the systems with distributed parameters, Avtomat. i Telemekh. 22 (1961), 17-26 [in Russian].
• [8] A.G. Butkovsky, A.I. Egorov, K.A. Lurie, Optimal control of distributed systems, SIAM J. Control 6 (1968), 437-476.
• [9] A.I. Egorov, Optimal Control by Heat and Diffusion Processes, Nauka, Moscow, 1978 [in Russian].
• [10] Yu.V. Egorov, Some problems of theory of optimal control, Zh. Vychisl. Mat. Mat. Fiz. 3 (1963), 887-904 [in Russian].
• [11] L.C. Evans, Partial Differential Equations, AMS, Graduate Series in Mathematics, vol. 19, Providence, 1998.
• [12] M.H. Farag, T.A. Talaat, E.M. Kamal, Existence and uniqueness solution of a class of quasilinear parabolic boundary control problems, Cubo 15 (2013), 111-119.
• [13] A. Friedman, Optimal control for parabolic equations, J. Math. Anal. Appl. 6 (1968), 437-476.
• [14] O.A. Ladyzhenskaya, V.A. Solonnikov, N.N. Ural'seva, Linear and quasi-linear equations of parabolic type, Translations of Mathematical Monographs, vol. 23, American Mathematical Society, Providence, RI, 1968.
• [15] O.A. Ladyzhenskaya, Boundary Value Problems of Mathematical Physics, Fizmatlit, Moscow, 1973 [in Russian].
• [16] J.L. Lions, Optimal Control of Systems governed by Partial Differential Equations, Springer, Berlin, 1971.
• [17] K.A. Lurie, Applied Optimal Control Theory of Distributed Systems, Springer, Berlin, 2013.
• [18] L.A. Lusternik, V.I. Sobolev, Elements of Functional Analysis, Fizmatlit, Moscow, 1965 [in Russian].
• [19] S. Mazur, Uber convexe Mengen in linearen normierte Raurnen, Studia Math. 4 (1933), Issue 1, 70-84.
• [20] F. Riesz, B. Szokefalvi-Nagy, Functional Analysis, Dover Books on Advanced Mathematics, Dover Publications, New York, 1990.
• [21] J.-C. Saut, B. Scheurer, Unique continuation for some evolution equations, J. Diff. Equ. 66 (1987), 118-139.
• [22] E.C. Titchmarsh, The zeros of certain integral functions, Proc. Lond. Math. Soc. 25 (1926), Issue S2, 283-302.
• [23] F. Troltzsch, Optimal Control of Partial Differential Equations. Theory, Methods and Applications, Graduate Studies in Mathematics, vol. 112, AMS, Providence, 2010.