# American Institute of Mathematical Sciences

## On the optimal coefficient control in a heat equation

 1 Siirt University, Department of Mathematics Education, Siirt, Turkey 2 Ataturk University, Science Faculty, Department of Mathematics, Erzurum, Turkey

Received  March 2019 Revised  April 2019 Published  October 2019

In this study, optimal control of a senior coefficient in a heat conduction problem is discussed. Since such problems are severely ill-posed, it is necessary to regularize them. So, we present the regularization process which has a great importance for a optimal control problem in this study. After proving the existence and uniqueness of optimal solution, a necessary condition for optimality is stated. Then, an iterative algorithm is purposed for numerical approximations. Finally, optimal control has been approximately calculated using the obtained results. The outcomes have also been tested with numerical examples.

Citation: Seda İǧret Araz, Murat Subașı. On the optimal coefficient control in a heat equation. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020124
##### References:

show all references

##### References:
The effect of α param eter to the x-m inim um norm control with different starting elem ents
The effect of α param eter to the 0.1-minimum norm control with different starting elements
J(k) and ‖k-k+L2(0, 1)2 values for some α numbers
J(k) and ‖k-k+L2(0, 1)2 valuesfor some α numbers
Some α values and corresponding k*(x) controls
Some α values and corresponding k*(x) controls
 [1] Fulvia Confortola, Elisa Mastrogiacomo. Optimal control for stochastic heat equation with memory. Evolution Equations & Control Theory, 2014, 3 (1) : 35-58. doi: 10.3934/eect.2014.3.35 [2] Ulisse Stefanelli, Daniel Wachsmuth, Gerd Wachsmuth. Optimal control of a rate-independent evolution equation via viscous regularization. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1467-1485. doi: 10.3934/dcdss.2017076 [3] Arturo de Pablo, Guillermo Reyes, Ariel Sánchez. The Cauchy problem for a nonhomogeneous heat equation with reaction. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 643-662. doi: 10.3934/dcds.2013.33.643 [4] Xiangfeng Yang, Yaodong Ni. Extreme values problem of uncertain heat equation. Journal of Industrial & Management Optimization, 2019, 15 (4) : 1995-2008. doi: 10.3934/jimo.2018133 [5] Eduardo Casas, Fredi Tröltzsch. Sparse optimal control for the heat equation with mixed control-state constraints. Mathematical Control & Related Fields, 2019, 0 (0) : 0-0. doi: 10.3934/mcrf.2020007 [6] Luz de Teresa, Enrique Zuazua. Identification of the class of initial data for the insensitizing control of the heat equation. Communications on Pure & Applied Analysis, 2009, 8 (1) : 457-471. doi: 10.3934/cpaa.2009.8.457 [7] Jussi Korpela, Matti Lassas, Lauri Oksanen. Discrete regularization and convergence of the inverse problem for 1+1 dimensional wave equation. Inverse Problems & Imaging, 2019, 13 (3) : 575-596. doi: 10.3934/ipi.2019027 [8] Xiaoshan Chen, Fahuai Yi. Free boundary problem of Barenblatt equation in stochastic control. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1421-1434. doi: 10.3934/dcdsb.2016003 [9] Alexander Arguchintsev, Vasilisa Poplevko. An optimal control problem by parabolic equation with boundary smooth control and an integral constraint. Numerical Algebra, Control & Optimization, 2018, 8 (2) : 193-202. doi: 10.3934/naco.2018011 [10] Zhi-Xue Zhao, Mapundi K. Banda, Bao-Zhu Guo. Boundary switch on/off control approach to simultaneous identification of diffusion coefficient and initial state for one-dimensional heat equation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2020021 [11] Alexander Gladkov. Blow-up problem for semilinear heat equation with nonlinear nonlocal Neumann boundary condition. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2053-2068. doi: 10.3934/cpaa.2017101 [12] Masaru Ikehata, Mishio Kawashita. An inverse problem for a three-dimensional heat equation in thermal imaging and the enclosure method. Inverse Problems & Imaging, 2014, 8 (4) : 1073-1116. doi: 10.3934/ipi.2014.8.1073 [13] Xiangtuan Xiong, Jinmei Li, Jin Wen. Some novel linear regularization methods for a deblurring problem. Inverse Problems & Imaging, 2017, 11 (2) : 403-426. doi: 10.3934/ipi.2017019 [14] Luca Rondi. On the regularization of the inverse conductivity problem with discontinuous conductivities. Inverse Problems & Imaging, 2008, 2 (3) : 397-409. doi: 10.3934/ipi.2008.2.397 [15] Sergey V. Bolotin, Piero Negrini. Global regularization for the $n$-center problem on a manifold. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 873-892. doi: 10.3934/dcds.2002.8.873 [16] Peter I. Kogut. On approximation of an optimal boundary control problem for linear elliptic equation with unbounded coefficients. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2105-2133. doi: 10.3934/dcds.2014.34.2105 [17] Max Gunzburger, Sung-Dae Yang, Wenxiang Zhu. Analysis and discretization of an optimal control problem for the forced Fisher equation. Discrete & Continuous Dynamical Systems - B, 2007, 8 (3) : 569-587. doi: 10.3934/dcdsb.2007.8.569 [18] Gökçe Dİlek Küçük, Gabil Yagub, Ercan Çelİk. On the existence and uniqueness of the solution of an optimal control problem for Schrödinger equation. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 503-512. doi: 10.3934/dcdss.2019033 [19] Qing Tang. On an optimal control problem of time-fractional advection-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 761-779. doi: 10.3934/dcdsb.2019266 [20] Nicolas Fournier. A new regularization possibility for the Boltzmann equation with soft potentials. Kinetic & Related Models, 2008, 1 (3) : 405-414. doi: 10.3934/krm.2008.1.405

2018 Impact Factor: 0.545

## Tools

Article outline

Figures and Tables