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doi: 10.3934/dcdss.2020124

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:
[1]

A. Atangana, On the new fractional derivative and application to nonlinear Fisher's reaction-diffusion equation, Applied Mathematics and Computation, 273 (2016), 948-956.  doi: 10.1016/j.amc.2015.10.021.  Google Scholar

[2]

A. Atangana and D. Baleanu, New fractional derivatives with non-local and non-singular kernel theory and application to heat transfer model, Thermal Science, 20 (2016), 763-769.   Google Scholar

[3]

S. EffatiA. Nazemi and H. Shabani, Time optimal control problem of the heat equation with thermal source, IMA Journal of Mathematical Control and Information, 31 (2014), 384-402.  doi: 10.1093/imamci/dnt016.  Google Scholar

[4]

H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Mathematics and its Applications, 375. Kluwer Academic Publishers Group, Dordrecht, 1996.  Google Scholar

[5]

M. Goebel, On existence of optimal control, Mathematische Nachrichten, 93 (1979), 67-73.  doi: 10.1002/mana.19790930106.  Google Scholar

[6]

O. A. Ladyzhenskaya, Boundary value problems in mathematical physics, Izdat. "Nauka", Moscow, (1973), 407 pp.  Google Scholar

[7]

M. Subașı, Optimal control of heat source in a heat conductivity problem, Optimization Methods and Software, 17 (2002), 239-250.  doi: 10.1080/1055678021000012444.  Google Scholar

[8]

R. K. Tagiyev, Optimal coefficient control in parabolic systems, Differential Equations, 45 (2009), 1526-1535.  doi: 10.1134/S0012266109100164.  Google Scholar

[9]

R. K. Tagiyev, Optimal control for the coefficients of a quasilinear parabolic equation, Automation and Remote Control, 70 (2009), 1814-1826.  doi: 10.1134/S0005117909110058.  Google Scholar

[10]

R. K. Tagiyev, Optimal control problem for a quasilinear parabolic equation with controls in the coefficients and with state constraints, Differential Equations, 49 (2013), 369-381.  doi: 10.1134/S0012266113030129.  Google Scholar

[11]

R. K. Tagiyev and S. A. Hashimov, On optimal control of the coefficients of a parabolic equation involving phase constraints, Proceedings of IMM of National Academy of Sciences of Azerbaijan, 38 (2013), 131-146.   Google Scholar

[12]

R. Teymurov, Optimal control of mobile sources for heat conductivity processes, International Journal of Control, 90 (2017), 923-931.  doi: 10.1080/00207179.2016.1187306.  Google Scholar

[13]

F. P. Vasil'ev, Methods for solving extremal problems: Minimization problems in function spaces, regularization, approximation, "Nauka", Moscow, (1981), 400 pp.  Google Scholar

[14]

H. Yetișkin and M. Subașı, On the optimal control problem for Schrödinger equation with complex potential, Applied Mathematics and Computation, 216 (2010), 1896-1902.  doi: 10.1016/j.amc.2010.03.039.  Google Scholar

[15]

G. YagubovF. Toyoǧlu and M. Subașı, An optimal control problem for two-dimensional Schrödinger equation, Applied Mathematics and Computation, 218 (2012), 6177-6187.  doi: 10.1016/j.amc.2011.12.028.  Google Scholar

show all references

References:
[1]

A. Atangana, On the new fractional derivative and application to nonlinear Fisher's reaction-diffusion equation, Applied Mathematics and Computation, 273 (2016), 948-956.  doi: 10.1016/j.amc.2015.10.021.  Google Scholar

[2]

A. Atangana and D. Baleanu, New fractional derivatives with non-local and non-singular kernel theory and application to heat transfer model, Thermal Science, 20 (2016), 763-769.   Google Scholar

[3]

S. EffatiA. Nazemi and H. Shabani, Time optimal control problem of the heat equation with thermal source, IMA Journal of Mathematical Control and Information, 31 (2014), 384-402.  doi: 10.1093/imamci/dnt016.  Google Scholar

[4]

H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Mathematics and its Applications, 375. Kluwer Academic Publishers Group, Dordrecht, 1996.  Google Scholar

[5]

M. Goebel, On existence of optimal control, Mathematische Nachrichten, 93 (1979), 67-73.  doi: 10.1002/mana.19790930106.  Google Scholar

[6]

O. A. Ladyzhenskaya, Boundary value problems in mathematical physics, Izdat. "Nauka", Moscow, (1973), 407 pp.  Google Scholar

[7]

M. Subașı, Optimal control of heat source in a heat conductivity problem, Optimization Methods and Software, 17 (2002), 239-250.  doi: 10.1080/1055678021000012444.  Google Scholar

[8]

R. K. Tagiyev, Optimal coefficient control in parabolic systems, Differential Equations, 45 (2009), 1526-1535.  doi: 10.1134/S0012266109100164.  Google Scholar

[9]

R. K. Tagiyev, Optimal control for the coefficients of a quasilinear parabolic equation, Automation and Remote Control, 70 (2009), 1814-1826.  doi: 10.1134/S0005117909110058.  Google Scholar

[10]

R. K. Tagiyev, Optimal control problem for a quasilinear parabolic equation with controls in the coefficients and with state constraints, Differential Equations, 49 (2013), 369-381.  doi: 10.1134/S0012266113030129.  Google Scholar

[11]

R. K. Tagiyev and S. A. Hashimov, On optimal control of the coefficients of a parabolic equation involving phase constraints, Proceedings of IMM of National Academy of Sciences of Azerbaijan, 38 (2013), 131-146.   Google Scholar

[12]

R. Teymurov, Optimal control of mobile sources for heat conductivity processes, International Journal of Control, 90 (2017), 923-931.  doi: 10.1080/00207179.2016.1187306.  Google Scholar

[13]

F. P. Vasil'ev, Methods for solving extremal problems: Minimization problems in function spaces, regularization, approximation, "Nauka", Moscow, (1981), 400 pp.  Google Scholar

[14]

H. Yetișkin and M. Subașı, On the optimal control problem for Schrödinger equation with complex potential, Applied Mathematics and Computation, 216 (2010), 1896-1902.  doi: 10.1016/j.amc.2010.03.039.  Google Scholar

[15]

G. YagubovF. Toyoǧlu and M. Subașı, An optimal control problem for two-dimensional Schrödinger equation, Applied Mathematics and Computation, 218 (2012), 6177-6187.  doi: 10.1016/j.amc.2011.12.028.  Google Scholar

Figure 2.  The effect of α param eter to the x-m inim um norm control with different starting elem ents
Figure 4.  The effect of α param eter to the 0.1-minimum norm control with different starting elements
Figure 1.  J(k) and ‖k-k+L2(0, 1)2 values for some α numbers
Figure 3.  J(k) and ‖k-k+L2(0, 1)2 valuesfor some α numbers
Table 1.  Some α values and corresponding k*(x) controls
Table 2.  Some α values and corresponding k*(x) controls
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