Group pursuit in Pontryagin’s nonstationary example |
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Authors: | A I Blagodatskikh |
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Institution: | 1. Udmurt State University, Izhevsk, Russia
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Abstract: | On the interval t 0, ∞), we consider the following group pursuit problem with one evader: 1 $$ z_i^{(l)} + a_1 (t)z_i^{(l - 1)} + a_2 (t)z_i^{(l - 2)} + \cdots + a_l (t)z_i = u_i - v, u_i ,v \in V, z_i^{(q)} (t_0 ) = z_i^q , $$ where z i , u i , v ∈ R v , (v ≥ 2), V is a strictly convex compact set in R v , the functions a 1(t), a 2(t), …, a l (t) are continuous, i = 1, 2, …, n and q = 0, 1, …, l ? 1. Let ? q (t, s) be the solution of the Cauchy problem $$ \begin{gathered} \omega ^{(l)} + a_1 (t)\omega ^{(l - 1)} + a_2 (t)\omega ^{(l - 2)} + \cdots + a_l (t)\omega = 0, \omega ^{(q)} (s) = 1, \hfill \\ \omega ^{(r)} (s) = 0, r = 0, \ldots q - 1,q + 1, \ldots ,l - 1, \hfill \\ \end{gathered} $$ and let $$ \xi _\iota (t) = \varphi _0 (t,t_0 )Z_i^0 + \varphi _1 (t,t_0 )Z_i^1 + \cdots + \varphi _{l - 1} (t,t_0 )Z_i^{l - 1} . $$ We prove that if there exist continuous functions α i (t) and ξ i 1 (t) such that the ξ i 1 (t) are Bohr almost periodic on t 0, ∞), α i (t) > 0 for all t ≥ t 0, lim t→∞(ξ i 1 (t) ? α i (t)ξ i (t)) = 0, lim t→∞(min i α i (t) ∝ t0 t |? l?1(t, s)| ds) = ∞, and there exist points h i 0 ∈ H i 1 = {ξ i 1 (t), t ∈ 0, ∞)} such that 0 ∈ Int co{h i 0 }, then the pursuit problem with evader discrimination is solvable. |
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