See this: icts.res.in/sites/default/files/…

Can you point out where in the Kushner/Yin book is this definition of stability given ?

I don't understand it. The term g(x)(x-y) cannot be positive for all y close to x, since suppose $y_1>x$, then it requires g(x)<0, but then for $y_2<x$ close to x, it gives $g(x)(x-y_2)<0$.

I still don't think it is correct.

There seems to be an error in your definition of stable points.

Well link #2 is presentation by a industrial firm on their use of TDA. So atleast they claim it is useful

And also contributed to the theory of turbulence in fluid mechanics.

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

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, $f=f(x,a(x,v,\nabla v,\Delta v,\sigma))$

You can read more here: anilvrao.com/Publications/ConferencePublications/…

Optimal control equations result in BVP to be solved on 1d (time) domain. So essentially, you are solving for intersection of solution of two Cauchy (IVP) problems with half of boundary values known on one end, and half on other end. By guesssing the rest, one integration is started from t=0, and one is started back from t=T (final time). That makes shooting a natural candidate,

It is still used in optimal control.scholar.google.com/…

note this thread:linkedin.com/pulse/… where the author gives reference as back as 1930!