We consider the differential inclusion problem given by $d = 2$ and $\mathcal{F} : \mathbb{R}^2 \rightarrow \mathcal{P}_c(\mathbb{R}^2)$ defined for all $x = (x_1, x_2) \in \mathbb{R}^2$ by: $$\mathcal{F}(x) = \begin{cases} \{v^-\} & \text{if } x_1 < 0 \\ \{v^+\} & \text{if } x_1 > 0 \\ [v_1^+, v_1^-] \times [v_2^+, v_2^-] & \text{if } x_1 = 0 \end{cases}$$ where $v^- = (v_1^-, v_2^-) \in \mathbb{R}^2$ and $v^+ = (v_1^+, v_2^+) \in \mathbb{R}^2$ with $v_1^- \geqslant v_1^+$ and $v_2^- \geqslant v_2^+$. We set $v^- = (1, 2)$ and $v^+ = (-1, 2)$. (a) Show that $\mathcal{F}$ satisfies condition (3). (b) We choose $y_{\text{init}} = (0, 0)$. Find all maximal solutions of problem (2). (c) We choose $y_{\text{init}} = (1, 0)$. Find all maximal solutions of problem (2).
We consider the differential inclusion problem given by $d = 2$ and $\mathcal{F} : \mathbb{R}^2 \rightarrow \mathcal{P}_c(\mathbb{R}^2)$ defined for all $x = (x_1, x_2) \in \mathbb{R}^2$ by:
$$\mathcal{F}(x) = \begin{cases} \{v^-\} & \text{if } x_1 < 0 \\ \{v^+\} & \text{if } x_1 > 0 \\ [v_1^+, v_1^-] \times [v_2^+, v_2^-] & \text{if } x_1 = 0 \end{cases}$$
where $v^- = (v_1^-, v_2^-) \in \mathbb{R}^2$ and $v^+ = (v_1^+, v_2^+) \in \mathbb{R}^2$ with $v_1^- \geqslant v_1^+$ and $v_2^- \geqslant v_2^+$.
We set $v^- = (1, 2)$ and $v^+ = (-1, 2)$.
(a) Show that $\mathcal{F}$ satisfies condition (3).
(b) We choose $y_{\text{init}} = (0, 0)$. Find all maximal solutions of problem (2).
(c) We choose $y_{\text{init}} = (1, 0)$. Find all maximal solutions of problem (2).