Optimal control over inserted point source intensity for humidification of a two-dimensional porous medium

1Lyashko, SI
Klyushin, DA
Tymoshenko, AA
1Taras Shevchenko National University of Kyiv
Dopov. Nac. akad. nauk Ukr. 2019, 12:13-18
Section: Information Science and Cybernetics
Language: Ukrainian

  The humidity transfer process through an unsaturated porous medium with inserted point sources modeled by the Richards—Klute equation has calculation complexity and is unstable. The reason for that is a large number of diverse parameters for the equation used to describe the physical process. To reduce the difficulty, an approach is offered based on the Kirchhoff transformation, which allows one to bring down the quasilinear parabolic initialboundary problem to a linear dimensionless one. A two-dimensional quasilinear problem of optimal control using point sources for a rectangular unsaturated porous medium with zero initial conditions, zero humidity at the bounds, and the achievable given target humidity is considered, studied, and solved for the first time.

  The initial problem is transformed into the linear dimensionless optimal control problem of non-stationary moisture transport in an unsaturated porous medium using the Kirchhoff transformation. A variation algorithm identifying the optimal source power is used, which allows modeling the process with realistic assumptions. For this algorithm, the finite difference method is used for both direct and conjugate problems, followed by the numerical method application to solve the SLAE. The correctness of the linearized dimensionless problem of moisture transport is shown. In particular, the theorems of existence and uniqueness of the generalized solution are mentioned, as well as the existence and uniqueness of the optimal control over the source power.

  The current paper is devoted to the modeling of the moisture transport from an inserted source in a dry ground area. Results of numerical experiments demonstrating a high accuracy of the method are given. The proposed method allows one to solve actual problems of optimal parameter choice for a drop irrigation system, and to improve its effectiveness.

Keywords: control, finite difference method, optimization, porous medium, Richards—Klute equation

1. Vabishchevich, P. N (2003). Numerical solution of the problem of the identification of the right-hand side of a parabolic equation. Russian Math. (Iz. VUZ), 47, No. 1, pp. 29-37 (in Russian).
2. Lyashko, S. H., Klyushin, D. A., Semenov, V. V. & Shevchenko, K. V. (2007). Eulerian and Lagrangian ap proach to solving the inverse convection–diffusion problem. Dopov. Nac akad. nauk Ukr., No. 10, pp. 38-43 (in Ukrainian).
3. Tymoshenko, A., Klyushin, D. & Lyashko, S. (2019). Optimal Control of Point Sources in Richards-Klute equa tion. Advances in Intelligent Systems and Computing, 754, pp. 194-203. Doi: https://doi.org/10.1007/978-3-319-91008-6_20
4. Nikolaevskaya, E. A., Khimich, A. N. & Chistyakova, T. V. (2012). Solution of linear algebraic equations by gauss method. Studies in Computational Intelligence, 399, pp. 31-44. Doi: https://doi.org/10.1007/978-3-642-25673-8_3
5. Lyashko S.I., Klyushin D.A., Onotskyi V.V. & Lyashko N.I. (2018). Optimal Control of Drug Delivery from Microneedle Systems. Cybernetics and System Analysis, 54(3), P. 1–9. Doi: https://doi.org/10.1007/s10559-018-0037-9
6. Lyashko, S. I., Klyushin, D. A, Nomirovsky, D. A. & Semenov, V. V. (2013). Identification of age — structured contamination sources in ground water. Optimal control of age — structured populations in economy, demography, and the invironment. Eds. By R. Boucekkline et all.), London and New York: Routledge, pp. 277-292.
7. Lyashko, S. I. Klyushin, D. A. & Palienko, L. I. (2000). Simulation and generalized optimization in pseudohyperbolical systems. J. Automation and Inform. Sci., 32(5), pp. 108-117. Doi: https://doi.org/10.1615/JAutomatInfScien.v32.i5.80
8. Lyashko, S. I. (1995). Numerical solution of pseudoparabolic equations. Cybernetics and System Analysis, 31(5), pp. 718-722. Doi: https://doi.org/10.1007/BF02367730
9. Lyashko, S. I. (1991). Approximate solution of equations of pseudoparabolic type. Comput. Mathematics and Math. Physics, 31(12), pp. 107-111.
10. Shulgin, D. F. & Novoselskiy, S. N. (1986). Mathematical models and methods of calculation of humidity transfer during subsurface irrigation. Matematika I problemy vodnogo khozyajstva, Kyiv: Naukova Dumka, pp. 73-89 (in Russian).