# Optimal Control of Point Sources in Richards-Klute Equation

Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 754)

## Abstract

This article represents an approach to humidity control in porous media, which combines linearization with numerical methods. The main problems and several ideas to solve them are mentioned. The main interest for our research is humidity regulation, described by Richards-Klute equation. The optimization problem is to minimize the difference between a reached state and a desired state.

Modelling moisture transport in porous media requires taking into account the processes of heat transfer, chemical and physical processes, as they have a considerable influence on characteristics of the medium.

First of all, a mathematical model is developed with a number of simplifications: moisture incompressibility, constant external pressure, limitations on transfer or isothermal requirements. Then, it is often preferable to make a transition to linear problem, as this case is explored and it allows us to use more theoretical background for the research. After that, numerical models and time and space discretization are constructed according to the related problem. When the process is represented in a suitable way, control and optimization problems arise and should be solved.

## Keywords

Control Optimization Richards-Klute equation

## References

1. 1.
Pullan, A.J.: The quasilinear approximation for unsaturated porous media flow. Water Resour. Res. 26(6), 1219–1234 (1990)
2. 2.
Van Genuchten, M.T.: A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci. Soc. Am. J. 44, 892–898 (1980)
3. 3.
Kirk, D.E.: Optimal Control Theory. An Introduction Optimal Control Theory: An Introduction. Dover Books on Electrical Engineering (2004)Google Scholar
4. 4.
List, F., Radu, F.: A Study on Iterative Methods for Richards’ Equation. Math.NA. http://arxiv.org/abs/1507.07837
5. 5.
Shatkovskiy, A.P.: Parameters of the mods of drip irrigation and productivity of sugar beets in the area of Ukrainian Steppes (in Ukrainian). Tsukrovi Buryaky 3, 15–17 (2016)Google Scholar
6. 6.
Lyashko, S.: Generalized Control of Linear Systems. Naukova Dumka, Kyiv (1998). (in Russian)Google Scholar
7. 7.
Lyashko, S., Klyushin, D., Semenov, V., Shevchenko, K.: Identification of point contamination source in ground water. Int. J. Ecol. Develop. 5(F06), 36–43 (2006)Google Scholar
8. 8.
Vabishchevich, P.N.: Numerical solution of the problem of the identification of the right-hand side of a parabolic equation (in Russian). Russ. Math. (Iz. VUZ) 47(1) (2003)Google Scholar
9. 9.
Novoselskiy, S.N.: Solution of some boundary value problems of moisture transport with irrigation sources (in Russian). Dissertation for the academic degree of a Candidate in Mathematics and Physics, 01 February 2005, Kalinin Polytechnic Institute, Kalinin (1981)Google Scholar
10. 10.
Jaddu, H., Majdalawi, A.: An iterative technique for solving a class of nonlinear quadratic optimal control problems using Chebyshev polynomials. Int. J. Intell. Syst. Appl. (IJISA) 6(6), 53–57 (2014).
11. 11.
Noori Skandari, M.H., Erfanian, H.R., Kamyad, A.V., Farahi, M.H.: Solving a class of non-smooth optimal control problems. Int. J. Intell. Syst. Appl. (IJISA) 5(7), 16–22 (2013).
12. 12.
Karabutov, N.: Structural identification of nonlinear dynamic systems. Int. J. Intell. Syst. Appl. (IJISA) 7(9), 1–11 (2015).
13. 13.
Zhang, X.: Single component, multiphase fluids flow simulation in porous media with Lattice Boltzmann method. Int. J. Eng. Manuf. (IJEM) 2(2), 44–49 (2012).