Gradient of a (Lyapunov) function

Question

For $x\in\mathbb{R}^n$, define $\hat{r}(x) = \left\{ \begin{array}{ll} \vec{0} ,& x = 0\\ \frac{x}{||x||} ,& x \ne0\\ \end{array} \right. $

and for some $r>0$, $x_0\in\mathbb{R}^n$: $$B_r(x_0) = \{x \in \mathbb{R^{n}}: ||x-x_0|| < r \}$$

Consider the differential equation: $$\frac{dx(t)}{dt} = f(x(t)),\text{ with } x(0) = k$$ $k \in B_r(0)$ and f is Lipschitz continuous in $B_r(0)$

Does there exist a scalar function $V(x) \in C^{1}$ such that:

1. $\nabla{V}(x) \cdot \hat{r}(x) < 0$, for some $x\in\mathbb{R}^n$ such that $||x||\in(0,r)$
2. $\frac{dV(x(t))}{dt} = \nabla{V}(x(t)) \cdot \frac{dx(t)}{dt} = \nabla{V}(x(t)) \cdot f(x(t)) < 0$ for all $x \in B_r(0)-\{0\}$
3. $V(x) > 0$ for all $x \in B_r(0)-\{0\}$
4. $V(0) = 0$

Can you give me an example of a function that satisfies all of the above?
An example for 2-D vector x would be ok.

Answer 1

Suppose $x_0 \in \mathbb{R}^n$ such that:

I. $||x_0|| \in (0,r)$ and
II. $\nabla{V}(x_0) \cdot \hat{r}(x_0) < 0 \iff \nabla{V}(x_0) \cdot x_0 < 0$

Define $g(\lambda) = V(\lambda x_0)$, $\lambda \in \mathbb{R}$
$g(\lambda) \in C^1$, thus $$g^\prime(\lambda) = \nabla V(\lambda x_0) \cdot \frac{d(\lambda x_0)}{d\lambda} = \nabla V(\lambda x_0) \cdot x_0$$

$g^\prime(1) = \nabla V(x_0) \cdot x_0 < 0$, due to condition II.

$g(0) = 0$ from condition 4.

Let's apply the Mean Value Theorem for derivatives, for the function $g$ in the interval $[0,1]$

• g is continuous in [0,1]
• g is differentiable in (0,1)

Hence, $\exists \xi \in (0,1):$ $$g^\prime(\xi) = \frac{g(1)-g(0)}{1-0} = g(1) = V(x_0) > 0$$ by condition 3.

Now, let's apply Bolzano's Theorem for $g^\prime$ in $[\xi,1]$:

• $g^\prime$ is continuous in $[\xi,1]$ (since $g \in C^1$)
• $g^\prime(\xi) \cdot g^\prime(1) < 0$

Hence, $\exists \lambda_0 \in (\xi, 1): g^\prime(\lambda_0) = 0 \iff \nabla V(\lambda_0 x_0) \cdot x_0 = 0 \iff \nabla V(\lambda_0 x_0) \perp x_0$