Timeline for Confusion optimal control abuse notation

5 events

when toggle format	what		by	license	comment
Dec 13, 2019 at 21:14	history	edited	YCor	CC BY-SA 4.0	removed capitals from title
Dec 13, 2019 at 20:56	comment	added	Piyush Grover		Well, the optimality condition gives you HJB, where the Hamiltonian term (see notes I linked to) is maximized over all possible controls, i.e., to do the sup, you have to allow control to have arbitary dependence on $v,x,\sigma$.. So you cannot prescribe the optimal control to be constant: it comes out of the maximization (sup ) process. Now depending upon the system, it may depend just on $\nabla v$, .e.g., in linear quadratic case (example given in the paper you mention), and yes, then you can omit other terms
Dec 13, 2019 at 20:40	comment	added	ABIM		In other words if my control doesn't depend on the value function explicitly then those terms may be ommited? Ex if it is constant?
Dec 13, 2019 at 16:52	comment	added	Piyush Grover		Agree it is horrible notation. f is still the same f, but now implicitly depends on $\sigma$, $v$,and its gradient because the $a$ is taken to be $a(x,v,\nabla v,\Delta v,\sigma)$. See these notes e.g.: columbia.edu/~dl3133/MFGSpring2018.pdf pages 63-70. In other words, $f=f(x,a(x,v,\nabla v,\Delta v,\sigma))$
Dec 13, 2019 at 16:29	history	asked	ABIM	CC BY-SA 4.0