Assume I have a system:
$\dot{\mathbf{x}}=-\frac{\partial}{\partial\mathbf{x}}L(\mathbf{x},\mathbf{y}) \\ \dot{\mathbf{y}}=\frac{\partial}{\partial\mathbf{y}}L(\mathbf{x},\mathbf{y})$
where $\mathbf{x}\in\mathbb{R}^m$, $\mathbf{y}\in\mathbb{R}^n$, $L$ is Lipchitz continuous. The equilibria are $(\mathbf{x}^*,\mathbf{y})$ for any $\mathbf{y}\in\mathbb{R}^n$ and isolated in terms of $\mathbf{x}^*$. That is, for each $\mathbf{x}^*$ and any $\mathbf{y}$, $(\mathbf{x}^*,\mathbf{y})$ is an equilibrium, and for a small neighbourhood $\mathbb{B}_{\mathbf{x}^*}-\{\mathbf{x}^*\}$ around $\mathbf{x}^*$, there does not exist another $\mathbf{x}'$ such that $(\mathbf{x}',\mathbf{y})$ is equilibrium. I can further show that at equilibrium, $\ddot{\mathbf{x}}$ is negative definite, $\ddot{\mathbf{y}}=\mathbf{0}$, $\frac{\partial^2 L(\mathbf{x},\mathbf{y})}{\partial\mathbf{x}\partial\mathbf{y}}=\mathbf{0}$, $\frac{\partial^2 L(\mathbf{x},\mathbf{y})}{\partial\mathbf{y}\partial\mathbf{x}}=\mathbf{0}$.
I am wondering if I can say this system is asymptotically stable. If not, what should I do to discribe its stability?
Please share your advice. Many thanks.