I am studying optimal control and I was wondering about boundary conditions for a stationary Hamilton-Jacobi-Bellman equation arising from an infinite horizon regulation. Consider minimizing a quadratic objective $\ell(x,u) = x^\intercal Q x + \mu^\intercal(x) R \mu(x)$, subject to the
dynamics $\dot x(t) = f(x(t)) + g(x(t)) u(t)$ overIf we have a problem of an infinite time horizon.
The value function $V(x_0) = \int^\infty_0 \ell(x^\star(t),u^\star(t)) {\rm d}t$ must satisfy the HJB equation
$$x^\intercal Q x + f(x)^\intercal \nabla_xV(x) - \frac{1}{4} g(x)^\intercal \nabla_xV(x) R^{-1}\left(g(x)^\intercal\nabla_x V(x)\right)^\intercal = 0$$ with V(0)=0. Here, the associated optimal control is given by $\mu(x)= -\frac{1}{2}R^{-1}g(x)^\intercal \nabla_x V(x)$. I am interested in trying to get a FEM method or any off-the-shelf numerical method to get an accurate solution of this PDE. But, typically, FEM methods require boundary conditions. From the optimality conditions I only know generically that V is 0 at 0 and that V is a positive definite solution. As the latter is just an inequality I can not see how this can be used. If it helps, I was trying to do this for an inverted pendulum on a cart---neglecting the cart dynamics and taking the cart acceleration as the control. I also tried applying the method of characteristics, but with only the condition at zero I get a zero solution, so I feel there must be other conditions I am missing.
Any pointer would be appreciated.