I have started doing research in optimal control theory. My area of research is PDE constrained optimization.
I am facing significant difficulty in deriving the Gateaux Derivative of functionals required for optimization.
For example I have to find out $U(\sigma,u,w;C) = 1/2 \int (\sigma - C:\epsilon[u]):{C}^{-1}:(\sigma - C:\epsilon[u]))$, where $C$ is a 4th order constitutive tensor and $\sigma$ is 2nd order stress tensor. The operator $:$ is the double dot product operator. All quantities are function of space ($R^3$).
How can I calculate derivative of $U$ with respect to $\sigma$ ?? I have tried to expand this using index notation. But it made the task very difficult.
Could you provide a source where I can have exposure to such kind of mathematics?
Thank you.