Research Article
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Year 2024, , 15 - 24, 31.01.2024
https://doi.org/10.54974/fcmathsci.1243111

Abstract

References

  • [1] Adigüzel R.S., Aksoy U., Karapinar E., Erhan I.M., On the solutions of fractional differential equations via Geraghty type hybrid contractions, Applied and Computational Mathematics, 20(2), 313-333, 2021.
  • [2] Astashova I., Filinovskiy A., Lashin D., On properties of the control function in a control problem with a point observation for a parabolic equation, Functional Differential Equations, 28(3-4), 99-102, 2021.
  • [3] Bollo C.M., Gariboldi C.M., Tarzia D.A., Neumann boundary optimal control problems governed by parabolic variational equalities, Control and Cybernetics, 50(2), 227-252, 2021.
  • [4] Clarkson J.A., Uniformly convex spaces, Transactions of the American Mathematical Society, 40(3), 396-414, 1936.
  • [5] Dhamo V., Tröltzsch F., Some aspects of reachability for parabolic boundary control problems with control constraints, Computational Optimization and Applications, 50, 75-110, 2011.
  • [6] Ergün A., A half inverse problem for the singular diffusion operator with jump condition, Miskolch Mathematical Notes, 21(2), 805-821, 2020.
  • [7] Ergün A., The multiplicity of eigenvalues of a vectorial diffusion equations with discontinuous function inside a finite interval, Turkish Journal of Science, 5(2), 73-85, 2020.
  • [8] Ergün A., Amirov R.K., Half inverse problem for diffusion operators with jump conditions dependent on the spectral parameter, Numerical Methods for Partial Differential Equations, 38, 577-590, 2022.
  • [9] Fardigola L., Khalina K., Controllability problems for the heat equation with variable coefficients on a half-axis, ESAIM: Control, Optimisation and Calculus of Variations, 28, 1-21, 2022.
  • [10] Flandoli F., Boundary control approach to the regularization of a Cauchy problem for the heat equation, IFAC Proceedings Volumes, 22(4), 271-275, 1989.
  • [11] Goebel M., On existence of optimal control, Mathematische Nachrichten, 93, 67-73, 1979.
  • [12] Hasanoğlu A., Simultaneous determination of the source terms in a linear parabolic problem from the final overdetermination: Weak solution approach, Journal of Mathematical Analysis and Applications, 330, 766-779, 2007.
  • [13] Iskenderov A.D., Tagiyev R.Q., Yagubov Q.Y., Optimization Methods, Çaşıoğlu, Baku, 2002.
  • [14] Ji G., Martin C., Optimal boundary control of the heat equation with target function at terminal time, Applied Mathematics and Computation, 127, 335-345, 2002.
  • [15] Kumpf M., Nickel G., Dymanic boundary conditions and boundary control for the one-dimensional heat equation, Journal of Dynamical and Control Systmes, 10(2), 213-225, 2004.
  • [16] Ladyzhenskaya O.A., The Boundary Value Problems of Mathematical Physics, Applied Mathematical Sciences, 49, Springer, 1985.
  • [17] Lions J.L., Optimal Control of Systems Governed by Partial Differential Equations, Springer, 1971.
  • [18] Lions J.L., Magenes E., Non-Homogeneous Boundary Value Problems and Applications, Springer, 1972.
  • [19] Martin P., Rosier L., Rouchon P., On the reachable states for the boundary control of the heat equation, Applied Mathematics Research Express, 2016(2), 181-216, 2016.
  • [20] Micu S., Roventa I., Tucsnak M., Time optimal boundary controls for the heat equation, Journal of Functional Analysis, 263, 25-49, 2012.
  • [21] Musaev H.K., The Cauchy problem for degenerate parabolic convolution equation, TWMS Journal of Pure and Applied Mathematics, 12(2), 278-288, 2021.
  • [22] Pankov P.S., Zheentaeva Z.K., Shirinov T., Asymptotic reduction of solution space dimension for dynamical systems, TWMS Journal of Pure and Applied Mathematics, 12(2), 243-253, 2021.
  • [23] Sadek I.S., Bokhari M.A., Optimal boundary control of heat conduction problems on an infinite time domain by control parameterization, Journal of the Franklin Instute, 348, 1656-1667, 2011.
  • [24] Shokri A., The multistep multiderivative methods for the numerical solution of first order initial value problems, TWMS Journal of Pure and Applied Mathematics, 7(1), 88-97, 2016.
  • [25] Shokri A., Saadat H., P-stability, TF and VSDPL technique in Obrechkoff methods for the numerical solution of the Schrödinger equation, Bulletin of the Iranian Mathematical Society, 42(3), 687-706, 2016.
  • [26] Shokri A., Saadat H., Khodadadi A., A new high order closed Newton-Cotes trigonometrically-fitted formulae for the numerical solution of the Schrödinger equation, Iranian Journal of Mathematical Sciences and Informatics, 13(1), 111-129, 2018.
  • [27] Şener Ş.S., Subaşi M., On a Neumann boundary control in a parabolic system, Boundary Value Problems, 2015, 1-12, 2015.
  • [28] Vasilyev F.P., Methods for Solving Extremal Problems„ Nauka, 1981.

On Optimal Control of the Heat Flux at the Left-Hand Side in a Heat Conductivity System

Year 2024, , 15 - 24, 31.01.2024
https://doi.org/10.54974/fcmathsci.1243111

Abstract

We deal with an optimal boundary control problem in a 1-d heat equation with Neumann boundary conditions. We search for a Neumann boundary function which is the minimum element of a quadratic cost functional involving the $H^1$-norm of boundary controls. We prove that the cost functional has a unique minimum element and is Frechet differentiable. We give a necessary condition for the optimal solution and construct a minimizing sequence using the gradient of the cost functional.

References

  • [1] Adigüzel R.S., Aksoy U., Karapinar E., Erhan I.M., On the solutions of fractional differential equations via Geraghty type hybrid contractions, Applied and Computational Mathematics, 20(2), 313-333, 2021.
  • [2] Astashova I., Filinovskiy A., Lashin D., On properties of the control function in a control problem with a point observation for a parabolic equation, Functional Differential Equations, 28(3-4), 99-102, 2021.
  • [3] Bollo C.M., Gariboldi C.M., Tarzia D.A., Neumann boundary optimal control problems governed by parabolic variational equalities, Control and Cybernetics, 50(2), 227-252, 2021.
  • [4] Clarkson J.A., Uniformly convex spaces, Transactions of the American Mathematical Society, 40(3), 396-414, 1936.
  • [5] Dhamo V., Tröltzsch F., Some aspects of reachability for parabolic boundary control problems with control constraints, Computational Optimization and Applications, 50, 75-110, 2011.
  • [6] Ergün A., A half inverse problem for the singular diffusion operator with jump condition, Miskolch Mathematical Notes, 21(2), 805-821, 2020.
  • [7] Ergün A., The multiplicity of eigenvalues of a vectorial diffusion equations with discontinuous function inside a finite interval, Turkish Journal of Science, 5(2), 73-85, 2020.
  • [8] Ergün A., Amirov R.K., Half inverse problem for diffusion operators with jump conditions dependent on the spectral parameter, Numerical Methods for Partial Differential Equations, 38, 577-590, 2022.
  • [9] Fardigola L., Khalina K., Controllability problems for the heat equation with variable coefficients on a half-axis, ESAIM: Control, Optimisation and Calculus of Variations, 28, 1-21, 2022.
  • [10] Flandoli F., Boundary control approach to the regularization of a Cauchy problem for the heat equation, IFAC Proceedings Volumes, 22(4), 271-275, 1989.
  • [11] Goebel M., On existence of optimal control, Mathematische Nachrichten, 93, 67-73, 1979.
  • [12] Hasanoğlu A., Simultaneous determination of the source terms in a linear parabolic problem from the final overdetermination: Weak solution approach, Journal of Mathematical Analysis and Applications, 330, 766-779, 2007.
  • [13] Iskenderov A.D., Tagiyev R.Q., Yagubov Q.Y., Optimization Methods, Çaşıoğlu, Baku, 2002.
  • [14] Ji G., Martin C., Optimal boundary control of the heat equation with target function at terminal time, Applied Mathematics and Computation, 127, 335-345, 2002.
  • [15] Kumpf M., Nickel G., Dymanic boundary conditions and boundary control for the one-dimensional heat equation, Journal of Dynamical and Control Systmes, 10(2), 213-225, 2004.
  • [16] Ladyzhenskaya O.A., The Boundary Value Problems of Mathematical Physics, Applied Mathematical Sciences, 49, Springer, 1985.
  • [17] Lions J.L., Optimal Control of Systems Governed by Partial Differential Equations, Springer, 1971.
  • [18] Lions J.L., Magenes E., Non-Homogeneous Boundary Value Problems and Applications, Springer, 1972.
  • [19] Martin P., Rosier L., Rouchon P., On the reachable states for the boundary control of the heat equation, Applied Mathematics Research Express, 2016(2), 181-216, 2016.
  • [20] Micu S., Roventa I., Tucsnak M., Time optimal boundary controls for the heat equation, Journal of Functional Analysis, 263, 25-49, 2012.
  • [21] Musaev H.K., The Cauchy problem for degenerate parabolic convolution equation, TWMS Journal of Pure and Applied Mathematics, 12(2), 278-288, 2021.
  • [22] Pankov P.S., Zheentaeva Z.K., Shirinov T., Asymptotic reduction of solution space dimension for dynamical systems, TWMS Journal of Pure and Applied Mathematics, 12(2), 243-253, 2021.
  • [23] Sadek I.S., Bokhari M.A., Optimal boundary control of heat conduction problems on an infinite time domain by control parameterization, Journal of the Franklin Instute, 348, 1656-1667, 2011.
  • [24] Shokri A., The multistep multiderivative methods for the numerical solution of first order initial value problems, TWMS Journal of Pure and Applied Mathematics, 7(1), 88-97, 2016.
  • [25] Shokri A., Saadat H., P-stability, TF and VSDPL technique in Obrechkoff methods for the numerical solution of the Schrödinger equation, Bulletin of the Iranian Mathematical Society, 42(3), 687-706, 2016.
  • [26] Shokri A., Saadat H., Khodadadi A., A new high order closed Newton-Cotes trigonometrically-fitted formulae for the numerical solution of the Schrödinger equation, Iranian Journal of Mathematical Sciences and Informatics, 13(1), 111-129, 2018.
  • [27] Şener Ş.S., Subaşi M., On a Neumann boundary control in a parabolic system, Boundary Value Problems, 2015, 1-12, 2015.
  • [28] Vasilyev F.P., Methods for Solving Extremal Problems„ Nauka, 1981.
There are 28 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Articles
Authors

Taha Koç 0000-0002-5087-4852

Yeşim Akbulut 0000-0003-2782-8900

Seher Aslancı 0000-0002-5749-0958

Publication Date January 31, 2024
Published in Issue Year 2024

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19113 FCMS is licensed under the Creative Commons Attribution 4.0 International Public License.