Neumann boundary optimal control problems governed by parabolic variational equalities

Abstract

We consider a heat conduction problem S with mixed boundary conditions in an n-dimensional domain Ω with regular boundary and a family of problems Sα with also mixed boundary conditions in Ω, where α > 0 is the heat transfer coefficient on the portion of the boundary Γ1. In relation to these state systems, we formulate Neumann boundary optimal control problems on the heat flux q which is definite on the complementary portion Γ2 of the boundary of Ω. We obtain existence and uniqueness of the optimal controls, the first order optimality conditions in terms of the adjoint state and the convergence of the optimal controls, the system state and the adjoint state when the heat transfer coefficient α goes to infinity. Furthermore, we formulate particular boundary optimal control problems on a real parameter λ, in relation to the parabolic problems S and Sα and to mixed elliptic problems P and Pα . We find an explicit form for the optimal controls, we prove monotony properties and we obtain convergence results when the parameter time goes to infinity.

Parameterization optimal control of an unsteady partial differential equations with convection term by an improved three-term spectrum conjugate gradient algorithm

“…We suppose that equation ( 56) is not true, so there exist constants g 1 , g 2 , which satisfy g 1 \ rJ k 2 \ g 2 . According to equation (36), we have…”

“…To solve this problem, scholars have carried out many research works and proposed various optimal control algorithms. 29–32 Gavrikov et al 33 study the optimal boundary control problem with linear quadratic cost function in the heat transfer process of a cylinder; Mechelli and Volkwein 34 study an optimal boundary and bilateral control problem of heat transfer equation with convection term; Dekhkonov 35 recently studies the boundary control problem, based on the heat transfer model, for the heat exchange process; Bollo et al 36 solve OCPs of Neumann boundary for a n -dimensional heat equation; Abbasi and Malek 37 solve pointwise optimal control problem on and inside a tissue subject to thermal wave model with Dirichlet and Rubin boundary conditions.…”

“…Theorem 4.1. If sequence v k and d k are generated by equations (35) and(36), we have the following equation…”

On Optimal Control of the Heat Flux at the Left-Hand Side in a Heat Conductivity System

“…The boundary control problem for heat transfer systems is one of the most addressed control problems for PDEs. Some detailed works of problems in these areas can be found in [2,3,5,9,10,14,15,17,19,20].…”