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Problem 1

# Let $$V: \mathcal{O} \rightarrow \mathbb{R}$$ be a continuous and positive definite function defined on some neighborhood $$\mathcal{O}$$ of $$x^{0}$$, where $$X=\mathbb{R}^{n}$$. Assume that $$V$$ satisfies property (3) in Definition 5.7.1. Show that, then, there exists a local controlLyapunov function $$W$$ for the same system, with $$W=V$$ on some neighborhood of $$x^{0}$$. (Hint: For $$\delta>0$$ small enough, let $$W=V$$ on the ball $$\mathcal{B}_{\delta}\left(x^{0}\right)$$ and $$W(x):=V\left(\delta\left(x-x^{0}\right) /\left\|x-x^{0}\right\|\right)$$ outside. Verify the definition of properness for $$\varepsilon$$ less than $$\left.\inf \left\\{V(x),\left\|x-x^{0}\right\|=\delta\right\\} .\right)$$

Expert verified
In summary, to prove the existence of a local control Lyapunov function $$W$$, we define $$W$$ using the given hint. We then verify the properness of $$W$$ by showing it admits a lower bound $$\varepsilon$$ less than $$\inf \{V(x) : \|x - x^{0}\| = \delta\}$$ when $$x$$ is outside the ball $$\mathcal{B}_{\delta}(x^{0})$$. Finally, since $$W$$ is positive-definite, proper, and satisfies the assumed properties of $$V$$, it can be concluded that $$W$$ is a local control Lyapunov function for the same system with $$W = V$$ on some neighborhood of $$x^{0}$$.
See the step by step solution

## Step 1: Define the Function W

Use the hint to define the new function $$W$$. For $$\delta > 0$$ small enough, let $$W$$ be equal to $$V$$ on the ball $$\mathcal{B}_{\delta}(x^{0})$$ and let $$W(x) := V\left(\delta\frac{x - x^{0}}{\left\|x - x^{0}\right\|}\right)$$ for $$x$$ outside of the ball $$\mathcal{B}_{\delta}(x^{0})$$.

## Step 2: Verify the Properness of W

Keeping in mind the goal is to verify the 'properness' of $$W$$, you need to show it admits a lower bound $$\varepsilon$$ less than $$\inf \{V(x) : \|x - x^{0}\| = \delta\}$$. Begin by noting that for $$x \in \mathcal{B}_{\delta}(x^{0})$$, $$W(x)$$ is equal to $$V(x)$$ and thus is certainly positive definite as $$V$$ was given to be so. Next, consider $$x$$ outside of $$\mathcal{B}_{\delta}(x^{0})$$. In this case $$W(x) = V\left(\delta\frac{x - x^{0}}{\left\|x - x^{0}\right\|}\right)$$. Since $$V$$ is positive definite and the norm of $$\delta\frac{x - x^{0}}{\left\|x - x^{0}\right\|}$$ is always less than or equal to $$\delta$$, $$W(x)$$ will have a lower bound $$\varepsilon$$ less than $$\inf \{V(x) : \|x - x^{0}\| = \delta\}$$. Therefore, $$W$$ is a proper function.

## Step 3: W is a Local Control Lyapunov Function

We've shown that $$W$$ is positive-definite and proper. If $$V$$ satisfies property (3) in the given definition, assuming this refers to the control Lyapunov function properties, then $$W$$ satisfies these properties and is a local control Lyapunov function for the same system with $$W = V$$ on some neighborhood of $$x^{0}$$. Therefore, the statement is proved.

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