Optimal Control for a Nonlinear Stochastic Parabolic Model of Language Competition

TL;DR

This paper develops an optimal control framework for a stochastic nonlinear parabolic model of language competition, incorporating environmental uncertainties to improve model reliability and deriving explicit optimal controls.

Contribution

It introduces a stochastic model for language competition with control variables and proves the existence and uniqueness of an optimal control using advanced mathematical techniques.

Findings

Existence and uniqueness of stochastic optimal control established.

Explicit form of optimal control derived in terms of adjoint states.

Model accounts for environmental uncertainties in language dynamics.

Abstract

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.