Boundary flex control for the systems governed by Boussinesq equation with the nonstandard boundary conditions

TL;DR

This paper studies boundary flex control for a generalized Boussinesq system with nonstandard boundary conditions, proving existence of optimal control and deriving optimality conditions using Pontryagin's maximum principle.

Contribution

It introduces a novel boundary control problem for a Boussinesq system with temperature-dependent viscosity and heat conductivity, including nonstandard boundary conditions, and establishes optimal control conditions.

Findings

Existence of optimal boundary control proven.

Optimality conditions derived via Pontryagin's maximum principle.

Analysis applicable to systems with temperature-dependent properties.

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.. First, we have proved the existence of the optimal control. Then the optimal condition has been derived. Pontryagin's maximum principle in the special case has been derived.