How can I prove this theorem on differential inclusions?

1. Oct 20, 2013

Only a Mirage

Consider the following differential equations with initial conditions at time $t_0$ specified:

$$\dot{x}_1 = f_1(x_1,t) ; \,\,\,x_1: [t_0,T]\to\mathbb{R}^n, f_1:\mathbb{R}^n\times[t_0,T]\to\mathbb{R}^n \\ \vdots \\$$
$$\dot{x}_k = f_k(x_k,t); \,\,\,x_1: [t_0,T]\to\mathbb{R}^n, f_1:\mathbb{R}^n\times[t_0,T]\to\mathbb{R}^n \\$$

Let $C(p,t)$ denote the convex hull of $\{f_1(p,t), f_2(p,t),...,f_k(p,t) \}$, where $p\in\mathbb{R}^n,t\in[t_0,T]$

Let $C_t(t)$ denote the convex hull of trajectories, that is, the convex hull of $\{x_1(t),...x_k(t)\}$

Then every solution $z(t)$ to the differential inclusion:

$$\dot{y}(t) \in C(y,t); \,\,\, y: [t_0,T]\to\mathbb{R}^n$$

satisfies the following property: If $z(t_0)\in C_t(t_0), then z(t)\in C_t(t)\forall t \in[t_0,T]$