An existence theorem for a control problem in oil recovery

This paper studies a Sturm-Liouville system, arising from the stability of interfaces in secondary recovery process : the oil is obtained from a porous medium by displacing it with a second fluid (water). A polymer solute contained in an intermediate region (between water and oil) is used to minimize the ''fingering'' phenomenon (the instability of interfaces). The eigenvalues of the above system are the growth constants (in time) of the perturbations. The growth constants may be controlled by the viscosity in the intermediate region. An existence theorem for an optimal viscosity in the intermediate region is given, which allows us to minimize the maximum growth constant. The Rayleigh's quotient and the properties of the weakly continuous functionals on a bounded domain of a Hilbert space are used. A characterization is given for the domain of the wavenumbers for which we have only positive growth constants.