  In this investigation, an optimal control problem for a stochastic
mathematical model of language competition is studied. We have considered the
stochastic model of language competition by adding the stochastic terms to the
deterministic model to take into account the random perturbations and
uncertainties caused by the environment to have more reliable model. The model
has formulated the population densities of the speakers of two languages, which
are competing against each other to be saved from destruction, attract more
speakers and so on, using two nonlinear stochastic parabolic equations. Four
factors including the status of the languages and the growth rates of the
populations are considered as the control variables (which can be controlled by
the speakers of the populations or policy makers who make decisions for the
populations for particular purposes) to control the evolution of the population
densities. Then, the optimal control problem for the stochastic model of
language competition is studied. Employing the tangent-normal cone techniques,
the Ekeland variational principle and other theorems proved throughout the
paper, we have shown there exists unique stochastic optimal control. We have
also presented the exact form of the optimal control in terms of stochastic
adjoint states.
