# Deriving adjoint equation of an Optimal Control Problem

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1. Sep 21, 2015

### Chung

Dear all,

I am investigating a Transient Optimal Heating Problem with distributed control and Dirichlet condition. The following are the mathematical expression of the problem:

Where Ω is the domain,
Γ is the boundary,
y is the temperature distribution,
u is the control,
yΩ is the optimal temperature distribution,
yD is some known temperature on Γ (i.e. Dirichlet condition),
λ and κ are some real constant.

I want to find the adjoint equation for the above problem, I found on some articles that I need to use Lagrangian Function and Divergence Theorem with Integration by Parts to derive the adjoint equation.

In other words, consider d/dε [L(y+εz,u,λ)] =0 and put ε=0, where L(y,u,λ) is the Lagrangian Function.

However, I could not keep going and derive the adjoint equation. I do not know how to apply Divergence Theorem with Integration by Parts to get the adjoint equation.

Can anyone help me to derive the adjoint equation?

2. Sep 26, 2015