- #1
Chung
- 1
- 0
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?
I am investigating a Transient Optimal Heating Problem with distributed control and Dirichlet condition. The following are the mathematical expression of the problem:
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?