I need help to understand a proof about invariant submanifolds. I wouldn't ask but I have been trying for a week unsuccessfully and I am truly getting crazy over it.
The system is the classic state-costate dynamics from optimal control theory attached in the picture here and I am trying to figure out why $\lambda - \nabla V_x = 0$ (although it makes sense and I am aware it is true). I am aware the trick consists of analysing $\dot{\lambda}-\frac{\partial^2 V(x)}{\partial x^2}\dot{x}$ and I guess you should also use the fact that the first derivative of the HJB equations $$\frac{1}{2}\nabla_x q(x) +\frac{\partial^2 V(x)}{\partial x^2}f(x) + \frac{\partial f^\top}{\partial x}\nabla V-\frac{1}{2}\nabla_x(\nabla V^\top g(x)R^{-1}g(x)^\top \nabla V)=0$$ and cancels terms out but I can not get it and there are always some extra terms that do not cancel out. Thank you so much in advance, I really appreciate the help.
