I am attempting to find an optimal feedback control using the HJB equation. I am trying to stabilize an n-dimensional state vector $x$ using a scalar control $u$ as follows:
$$\dot{x} = Ax+uBx$$
where $A, B$ are $N$ by $N$ matrices.
My cost functional is $$\int_{0}^{\infty} x^TQx +ru^2 \ dt$$
with no terminal cost, and infinite horizon, so I believe the process goes as follows:
$$ 0 = min_u [x^TQx +ru^2 + V_x^T(Ax+uBx)]$$
where $V_x$ is the gradient of the value function (cost-to-go) $V$.
$$\frac{\partial}{\partial u } [x^TQx +ru^2 + V_x^T(Ax+uBx)] = 0$$ $$ 2ru + V_x^TBx = 0$$ $$u = -\frac{V_x^TBx}{2r} $$
plugging this into the HJB equation:
$$V_x^T(Bx)^2V_x\bigg(\frac{1}{4r}-\frac{1}{2r}\bigg) + V_x^TAx+x^TQx = 0 $$
What I see here is that in order to solve for the vector $V_x$ which I need for the feedback law, I have to find the intersection of a hyperellipse with a hyperplane (for any given x). My problem is conceptual, because I can't understand how this solution wouldn't be unique. I have tried to find results on the uniqueness of HJB solutions and, in my limited understanding, it seems as though for simple problems such as this there should exist a unique optimal control.
A second problem I have is how to effectively solve "vector quadratic equations" such as these. I hope that's the correct language to use. But if granted I could use a numerical method to do so, the greater problem of which solution $V_x$ to choose would still exist. I would appreciate any thoughts on the matter. Thank you.