Stabilization, Regulation and Robust Control of Uncertain processes and parameters in Porous Medium systems with Applications
Résumé
This chapter investigates a robust control approach of uncertain complex systems in porous medium described by a class of degenerate nonlinear partial differential equations. The concept consists in setting the problem in the worst-case disturbances which leads to the game theory in which the controls and the disturbances (which destabilize the dynamical behavior of the system) play antagonistic roles. The aim of the approach is then to control these instabilities, either by acting on certain parameters to maintain the system in a desired state, or by calculating the limit of these parameters before the system becomes unstable (details of our approach are reported in Belmiloudi's book 2008).
A review of some practical applications and examples (which are mainly focused on the optimization of oil production and groundwater extraction) are presented. The dynamical models occurring in porous media are predicted by using some generalized uncertain evolution nonlinear porous medium equations with nonlinear Robin boundary conditions (with and without coupling with generalized Darcy's law).
The control and robust stabilization problems are formulated, in different situations, in order to reconstitute
simultaneously, different parameters, data functions and sources/sinks appearing in the state system and in the boundary conditions,
and to control and stabilize desired states provided by e.g., measurements (as pressure, production rates, time average of the moisture, etc). This work includes results concerning the existence of optimal solutions, sensitivity problems, adjoint problems, necessary optimality conditions (necessary to develop numerical optimization methods) and optimization problems. Applications to some problematics as optimal management of underground resources, control of oil reservoir production, etc, are presented; the case when data are measured in only some points in space-time domain, and the case when the objective is to find the best locations of a finite number of wells, which are drilled, are also considered.