LINEAR DIFFERENTIAL PURSUIT PROBLEM WITH MANY PLAYERS UNDER INTEGRAL AND GEOMETRIC RESTRICTIONS

Аннотация

We have considered a simple motion differential game of 𝑚 pursuers with integral restriction and one evader with geometric restriction in ℝ𝑛. To solve a pursuit problem, the attainability domain of each pursuer is constructed. Sufficient condition of pursuit is obtained by intersection of the attainability domains. Lifeline game is solved for the advantage of the pursuers. Keywords: Differential game, pursuer, evader, geometric restriction, integral restriction, attainability domain, lifeline. Consider the differential game when the Pursuers 𝑋 , 𝑖 = 1,2,…,𝑚, and the Evader 𝑌 𝑖 having radius vectors 𝑥 and 𝑦 respectively move in ℝ𝑛. If their velocity vectors are 𝑢 and 𝑣 then 𝑖 𝑖 the game will be described by the equations 𝑋 : 𝑥̇ +𝑎𝑥 = 𝑢 , 𝑥 (0) = 𝑥 , (1) 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖0 𝑌: 𝑦̇ +𝑎𝑦 = 𝑣, 𝑦(0) = 𝑦 , (2) 0 where 𝑥 ,𝑦,𝑢 ,𝑣 ∈ ℝ𝑛, 𝑛 ≥ 2; 𝑥 ,𝑦 are the initial positions of the objects 𝑋 and 𝑌 and 𝑖 𝑖 𝑖0 0 𝑖 it is assumed 𝑥 ≠ 𝑦 . Here the temporal variation of 𝑢 must be a measurable function 𝑖0 0 𝑖 𝑢 (∙):[0,∞) → ℝ𝑛, on which is imposed the constraint 𝑖 ∫ 𝑡 |𝑢 (𝑠)|2𝑑𝑠 ≤ 𝜌 for almost every 𝑡 ≥ 0, (3) 0 𝑖 𝑖0 where 𝑖 = 1,2,…,𝑚, and 𝜌 are the positive numbers. From the physical point of view, 𝑖0 the right-hand of (3) corresponds to the given resource of the Pursuer 𝑋 . We call the inequality 𝑖 (3) as integral restriction (briefly, 𝐼 −constraint) and denote by 𝑈𝑖 the class of admissible controls 𝐼 satisfying (3). Similar, the temporal variation of 𝑣 should be a measurable function 𝑣(∙):[0,∞) → ℝ𝑛 and on this function, we impose geometrical restriction (briefly, 𝐺 −constraint) |𝑣(𝑡)| ≤ 𝛽 for almost every 𝑡 ≥ 0, (4) where 𝛽 is a positve number which means the maximal velocity of the Evader. We denote by 𝑉 the class of the Evader’s admissible controls satisfying (4). 𝐺 8 RAQAMLI DUNYO: MATEMATIK VA INFORMATIK 2025 -Yil YONDASHUVLAR 13-Fevral Respublika ilmiy-uslubiy konferensiyasi We are going to study mainly the game with phase restrictions for the Evader being given by a subset 𝐴 of which is called the “Life-line” [1]. Definition 1. For each triple (𝜌 ,𝑢 (∙)), 𝑢 (∙) ∈ 𝑈𝑖, 𝑖 = 1,2,…,𝑚, the scalar function 𝑖0 𝑖 𝑖 𝐼 𝜌 (𝑡) = 𝜌 −∫ 𝑡 |𝑢 (𝑠)|2𝑑𝑠, 𝜌 (0) = 𝜌 , 𝑡 ≥ 0, is called the residual resource of the Pursuer 𝑋 𝑖 𝑖0 0 𝑖 𝑖 𝑖0 𝑖 at the time 𝑡. Let 𝑧 (𝑡) = 𝑥 (𝑡)−𝑦(𝑡), 𝑧 = 𝑥 −𝑦 , 𝜇 = 𝜌 /|𝑧 |. 𝑖 𝑖 𝑖0 𝑖0 0 𝑖0 𝑖0 𝑖0 Suppose that the pair (𝜇 ,𝛽) is a parametric state of the game (1)–(4) and denote it by 𝑝 . 𝑖0 𝑖 We introduce the following nonempty connected set of such states𝑝 : 𝑃𝑖 = {𝑝 : 𝜇 ≥ 4𝛽, 𝛽 > 𝑖 𝐼𝐺 𝑖 𝑖0 0}. Definition 2. The function 𝒖 (𝑣) = 𝑣−𝜆 (𝑣)𝜉 (5) 𝑖 𝑖 𝑖0 is called the parallel pursuit strategy (П𝑖 -strategy) for the Pursuer 𝑋 , where 𝜆 (𝑣) = 𝐼𝐺 𝑖 𝑖 𝜇 𝜇 𝑧 𝑖0+〈𝑣,𝜉 〉+√( 𝑖0+〈𝑣,𝜉 〉)2−|𝑣|2, 𝜉 = 𝑖0 . 2 𝑖0 2 𝑖0 𝑖0 𝑖0 Consider the set 𝑊𝑖 (𝑡) = {𝑤: |𝑤 −𝑥 (𝑡)|2 = 𝑒−𝑎𝑡𝛬 (𝑡,𝑣(∙))(𝑝 (𝑡)/𝛽)|𝑤 −𝑦(𝑡)|, 𝐼𝐺 𝑖 𝑖 𝑖 0 ≤ 𝑡 ≤ 𝑡∗}, where 𝑡∗ = 𝑚𝑖𝑛{𝑡: |𝑧 (𝑡)| = 0} and 𝑤 is a point where the Pursuers 𝑋 should meet the 𝑖 𝑖 Evader 𝑌. Theorem 1. Let 𝑝 ∈ 𝑃𝑖 be valid in the game (1)–(4). Then the relation 𝑊𝑖 (𝑡 ) ⊂ 𝑖 𝐼𝐺 𝐼𝐺 2 𝑊𝑖 (𝑡 )is satisfied for any 𝑡 < 𝑡 , 𝑡 ,𝑡 ∈ [0,𝑡∗]. 𝐼𝐺 1 1 2 1 2 Proposition 1. If 𝑝 ∈ 𝑃𝑖 , then inclusion 𝑦(𝑡) ∈ 𝑊𝑖 (0) is valid on [0,𝑡∗]. 𝑖 𝐼𝐺 𝐼𝐺 Theorem 2. If in the game (1)–(4), 𝑝 ∈ 𝑃𝑖 holds for some 𝑖 = 1,2,…,𝑚, then 𝑦(𝑡) ∈ 𝑖 𝐼𝐺 𝑊𝑖 is met on the time interval [0,𝑇 ], where 𝑊 = ⋂𝑚 𝑊𝑖 (0), 𝑇 = 𝑑/𝛽, 𝑑 = 𝐼𝐺 𝐼𝐺 𝐼𝐺 𝑖=1 𝐼𝐺 𝐼𝐺 max {|𝑤 −𝑤 |: 𝑤 ,𝑤 ∈ 𝑊 }. 1 2 1 2 𝐼𝐺 Theorem 3. If 𝑊 ∩𝐴 = ∅, then the Pursuers 𝑋 win on the time interval [0,𝑇 ] in the 𝐼𝐺 𝑖 𝐼𝐺 “Life-line” game (1)-(4).