SOLVABILITY FOR CONTINUOUS CLASSICAL BOUNDARY OPTIMAL CONTROL OF COUPLE FOURTH ORDER LINEAR ELLIPTIC EQUATIONS
Abstract
In this paper, we study continuous classical boundary optimal control problem for the couple fourth order of linear elliptic system with variable coefficients. The existence theorem of a unique couple vector state solution of the weak form obtaining from the couple fourth order of linear elliptic system with Neumann conditions (NCs) is demonstrated for fixed continuous classical couple boundary control vector (CCCPBCTV) utilizing Hermite finite element method. The existence theorem of a couple continuous classical boundary optimal control vector dominated with the considered problem is proved. The existence and uniqueness of the solution of the couple adjoint equations (CPAEs) is discussed, when the classical couple optimal boundary control is given. Finally, the Fréchet derivative (FrD) of the Hamiltonian is obtained to establish the theorem of the necessary condition for optimality.
References
[1]N. L. Grigorenko, E. V. Grigorieva, P. K. Roi and E. N. Khailov, Optimal control problems for a mathematical model of the treatment of psoriasis, Comput. Math. Model. 30(4) (2019), 352-363.
[2]M. Derakhshan, Control theory and economic policy optimization: the origin, achievement and the fading optimism from a historical standpoint, International Journal of Business and Development Studies 7(1) (2015), 5-29.
[3]A. V. Kryazhimskii and A. M. Taras’ev, Optimal control for proportional economic growth, Proceedings of the Steklov Institute of Mathematics 293(1) (2016), 101-119.
[4]M. Chalak, Optimal control for a dispersing biological agent, Journal of Agricultural and Resource Economics 39(2) (2014), 271-289.
[5]Y. O. Aderinto, A. O. Afolabi and I. T. Issa, On optimal planning of electric power generation systems, Journal of Mathematics 50(1) (2017), 89-95.
[6]L. Kahina, P. Spiteri, F. Demim, A. Mohamed, A. Nemra and F. Messine, Application optimal control for a problem aircraft flight, Journal of Engineering Science and Technology Review 11(1) (2018), 156-164.
[7]J. Warga, Optimal Control of Differential and Functional Equations, Academic Press, New York and London, 1972.
[8]J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, New York, 1972.
[9]J. A. Al-Hawasy and N. A. Al-Ajeeli, The continuous classical boundary optimal control of triple nonlinear elliptic partial differential equations with state constraints, Iraqi Journal of Science 62(9) (2021), 3020-3030.
[10]J. A. Al-Hawasy and E. H. Al-Rawdhanee, The continuous classical optimal control of a coupled of nonlinear elliptic equations, Mathematical Theory and Modeling 4(14) (2014), 211-219.
[11]B. Vexler, Finite element approximation of elliptic Dirichlet optimal control problems, Numer. Funct. Anal. Optim. 28 (2007), 957-973.
[12]J. A. Al-Hawasy, Solvability for classical continuous boundary optimal control problem dominating by triple hyperbolic equations, IOP Conf. Ser. Material Science and Engineering, 2020. https://doi.org/10.1088/1757-899X/871/1/012056.
[13]J. A. Al-Hawasy and M. Jaber, The continuous classical boundary optimal control vector governing by triple linear partial differential equations of parabolic type, Ibn Al-Haitham J. for Pure and Appl. Sci. 33(3) (2020), 113-126.
[14]C. He and J. Li, Optimal boundary control for cascaded parabolic PDEs with constraints, Advances in Computer Science Research, Proceedings of the 2nd International Conference on Electronics, Network and Computer Engineering, Atlantis Press, 2016. https://doi.org/10.2991/icence-16.2016.62.
[15]F. Tröltzsch, Optimal control of partial differential equations, Theory, Methods and Applications, American Mathematical Society, 2010.
