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How can I prove this theorem on differential inclusions?

  1. Oct 20, 2013 #1
    Consider the following differential equations with initial conditions at time $t_0$ specified:

    [tex]\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 \\[/tex]
    [tex]\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 \\[/tex]

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

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

    Then every solution [itex]z(t)[/itex] to the differential inclusion:

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

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