Optimal control over inserted point source intensity for humidification of a two-dimensional porous medium - Lyashko, S.I.; Klyushin, D.A.; Tymoshenko, A.A. (Reports of the National Academy of Sciences of Ukraine, 2019)
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.