Optimal condition of boundary flex control for the systems governed by Boussinesq equation with the press boundary condition and mixed boundary condition

TL;DR

This paper investigates optimal boundary control for a non-stationary Boussinesq system with temperature-dependent viscosity and heat conductivity, deriving optimality conditions and applying Pontryagin's maximum principle.

Contribution

It introduces a novel boundary control problem for a generalized Boussinesq system with mixed boundary conditions and derives the optimality conditions including Pontryagin's maximum principle.

Findings

Derived the optimal boundary control conditions.

Established Pontryagin's maximum principle for the system.

Provided insights into controlling coupled mass and heat flow.

Abstract

In this paper, the boundary flex control problem of non stationary equation governing the coupled mass and heat flow of a viscous incompressible fluid in a generalized Boussinesq approximation by assuming that viscosity and heat conductivity are dependent on temperature has been studied. The boundary condition for velocity of fluid is non -standard boundary condition: specifically the case where dynamical pressure is given on some part of the boundary and the boundary condition for temperature of fluid is mixed boundary condition has been considered. The optimal condition has been derived. Then, Pontryagin's maximum principle in the special case has been derived.