Time Optimal Control Problem for Differential Inclusions

Abstract

Let E ⁿ be Euclidean space of state-vectors x = (x ₁ ,...,x _n ) with the norm $\left\| x \right\| = \sqrt {\sum\nolimits_{i = 1}^n {x_i^2} }$ and Ω(E ⁿ ) be metric space of all nonempty compact subsets of E ⁿ with Hausdorff metric

$$h\left( {F,G} \right) = \min \left\{ {d:F \subset {S_d}\left( G \right),\;G \subset {S_d}\left( F \right)} \right\} $$

(1)

where S _d ^′ (M) denotes a d — neighborhood of a set M in the space E ⁿ .