Deriving adjoint equation of an Optimal Control Problem

[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,
y_Ω is the optimal temperature distribution,
y_D 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?

 

[jvicens]

Dear fellow researcher,

Thank you for sharing your problem with us. The Transient Optimal Heating Problem with distributed control and Dirichlet condition is a complex and interesting topic. As you have mentioned, the adjoint equation is an important part of solving this problem.

To derive the adjoint equation, we need to follow the steps you have mentioned. First, we need to consider the Lagrangian Function L(y,u,λ) and take its derivative with respect to ε, as shown in your equation d/dε [L(y+εz,u,λ)] =0. Then, we need to put ε=0 to obtain the adjoint equation.

To continue, we need to use the Divergence Theorem and Integration by Parts. The Divergence Theorem states that the surface integral of a vector field over a closed surface is equal to the volume integral of the divergence of that vector field over the enclosed volume. This theorem can be applied to our problem by considering the temperature distribution y and its gradient ∇y as a vector field.

Next, we need to use Integration by Parts to transform the volume integral into a surface integral. This step involves applying the product rule to the function u and its derivative ∇u. This will result in a surface integral involving the control u and its adjoint λ.

By combining the results of the Divergence Theorem and Integration by Parts, we can obtain the adjoint equation in terms of the temperature distribution y, the control u, and the adjoint λ. This equation will have a similar form to the original problem, but with the roles of y and λ interchanged.

I hope this explanation helps you in deriving the adjoint equation for your problem. If you need further assistance, please do not hesitate to ask for help from your colleagues or consult with an expert in this field. I wish you all the best in your research.