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Towards a relaxation of the pursuit-evasion differential game
9th 9th IFAC IFAC Conference Conference on on Manufacturing Manufacturing Modelling, Modelling, Management Management and and Control 9th IFAC Conference on Manufacturing Modelling, Management and Control Available online at www.sciencedirect.com 9th IFAC Conference on Manufacturing Modelling, Management and Berlin, Germany, August 28-30, 2019 Control Berlin, Germany, August 28-30, 2019 Control Berlin, Germany, August 28-30, 2019 Berlin, Germany, August 28-30, 2019
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IFAC PapersOnLine 52-13 (2019) 2303–2307
Towards Towards Towards Towards
a a a a
relaxation relaxation of of the the pursuit-evasion pursuit-evasion relaxation of the pursuit-evasion differential game relaxation of the pursuit-evasion game differential differential game differential game ∗,∗∗ ∗,∗∗ Alexander Chentsov ∗,∗∗ ∗,∗∗ Daniel Khachay ∗,∗∗ ∗,∗∗
Alexander Chentsov ∗,∗∗ Daniel Khachay ∗,∗∗ Alexander Chentsov ∗,∗∗ Daniel Khachay ∗,∗∗ ∗,∗∗ Alexander Chentsov Daniel Khachay ∗,∗∗ ∗ ∗ Krasovsky Institute of Mathematics and Mechanics, 16 S.Kovalevskoy ∗ Krasovsky Institute of Mathematics and Mechanics, 16 S.Kovalevskoy ∗ ∗ Krasovsky Institute ofEkaterinburg Mathematics620990, and Mechanics, S.Kovalevskoy str. Russia str. Ekaterinburg 620990, Russia 16 ∗ Krasovsky Institute ofEkaterinburg Mathematics620990, and Mechanics, 16 S.Kovalevskoy (e-mail: [email protected], [email protected]). str. Russia (e-mail: [email protected], [email protected]). ∗∗ str. Ekaterinburg 620990, Russia ∗∗ Federal University, 51 Lenin ave. Ekaterinburg (e-mail: [email protected], [email protected]). ∗∗ Ural Ural Federal University, 51 Lenin ave. Ekaterinburg 620002, 620002, Russia Russia ∗∗ ∗∗ (e-mail: [email protected]). Ural [email protected], University, 51 Lenin ave. Ekaterinburg 620002, Russia ∗∗ Ural Federal University, 51 Lenin ave. Ekaterinburg 620002, Russia Abstract: We We consider consider aa special special case case of the the non-linear non-linear zero-sum zero-sum pursuit-evasion pursuit-evasion differential differential Abstract: Abstract: We consider agame special case of ofby thetwo non-linear zero-sum pursuit-evasion differential game. The instance of this is defined closed sets -- target set and one specifying state game. The instance of this game is defined by two closed sets target set and one specifying state Abstract: We consider agame special case ofby thetwo non-linear zero-sum pursuit-evasion differential constraints. We find an optimal non-anticipating strategy for player I (the pursuer). Namely, game. The instance of this is defined closed sets target set and one specifying state constraints. We find an optimal non-anticipating strategy for player I (the pursuer). Namely, game. The instance ofan this game isnon-anticipating defined two closed setsfor -function target set specifying state we construct construct his find successful solvability set by specified by limit limit ofI and the one iterative procedure constraints. We optimal strategy player (the pursuer). Namely, we his successful solvability set specified by function of the iterative procedure constraints. We find an optimal non-anticipating strategy for player I (the pursuer). Namely, we construct his successful solvability set specified by limit function of the iterative procedure in space space of of positions. positions. For For positions positions located located outside outside the the successful successful solvability solvability set, set, we we provide provide a in we construct his successful solvability set specified limit of the iterative procedure in space ofof positions. For positions located outside by the successful solvability set,two wementioned provide aa relaxation our game by determining the smallest size of aafunction neighborhoods of relaxation of our game by determining the smallest size of neighborhoods of two mentioned in space ofofpositions. For outside the solvability set,two wementioned provide a sets, for which which thegame pursuer can solve solvelocated his the problem successfully. Then, we construct construct his successful relaxation our by positions determining smallest sizesuccessful of a Then, neighborhoods of sets, for the pursuer can his problem successfully. we his successful relaxation of our game by determining the smallest size of a neighborhoods of two mentioned solvability set in terms of those neighborhoods. sets, for which theterms pursuer can solve his problem successfully. Then, we construct his successful solvability set of neighborhoods. sets, for which theterms pursuer can solve his problem successfully. Then, we construct his successful solvability set in in of those those neighborhoods. © 2019, IFAC Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. solvability set(International in terms of those neighborhoods. Keywords: Pursuit-Evasion Pursuit-Evasion differential differential game, game, non-anticipating non-anticipating strategies, strategies, program program iterations iterations Keywords: method. Keywords: Pursuit-Evasion differential game, non-anticipating strategies, program iterations method. Keywords: method. Pursuit-Evasion differential game, non-anticipating strategies, program iterations method. 1. be 1. INTRODUCTION INTRODUCTION AND AND RELATED RELATED WORK WORK be reduced reduced to to problem problem of of finding finding the the set set of of successful successful be reducedof toaa pursuer problem (player of finding the is, setthe of maximum successful solvability I) (that 1. INTRODUCTION AND RELATED WORK solvability of pursuer (player I) (that is, the maximum 1. INTRODUCTION AND RELATED WORK be reduced to problem of finding the set of successful stable bridge by N.N. Krasovskii), who is interested in solvability of a pursuer (player I) (that is, the maximum During bridge by N.N. Krasovskii), who isis, interested in the the During the the last last decades, decades, differential differential game game theory theory has has been been stable solvability of a pursuer (player I) (that the maximum guaranteed feasibility of approach. Various methods stable bridge by N.N. Krasovskii), who is interested inwere the an actively developing field of operations research and During the last decades, differential game theory has been guaranteed feasibility of approach. Various methods were an actively developing field of operations research and stable bridge by N.N. Krasovskii), who isal., interested inwere the guaranteed feasibility of approach. Various methods During the last decades, differential game theory has been used to build this set. In (Ushakov et 2015), special control theory. The first mention of such a differential an actively developing field of operations research and used to build this set.ofInapproach. (UshakovVarious et al., methods 2015), special control theory. The firstfield mention of such aresearch differential guaranteed feasibility were an actively developing of operations and kind of procedures constructed by dynamic programming used to build this set. In (Ushakov et al., 2015), special game back to In well-known study control theory. firstIsaacs. mention of such a differential of constructed by dynamic programming game goes goes backThe to R. R. Isaacs. In the the well-known study kind to procedures build this set. Inthat, (Ushakov et al., 2015), special control theory. The first mention of such differential was considered. Besides the constructions kind of procedures constructed by program dynamic programming game goes back to R. Isaacs. Ina the well-known study used (Isaacs, 1969), Isaacs overviewed a number number ofaapplications applications was considered. Besides that, the program constructions (Isaacs, 1969), Isaacs overviewed of kind of procedures constructed by dynamic programming game goes back to to R.a Isaacs. Indifferential the well-known study was can be used to find aa solutions for differential game (see considered. Besides that, the program constructions that be These (Isaacs, Isaacs number of game. applications be used to for differential game that can can1969), be reduced reduced tooverviewed a model model of ofa differential game. These can was considered. Besides that, the program constructions can be usedand to find find a solutions solutions for differential game (see (see (Isaacs, 1969), Isaacs overviewed adifferential number of game. applications (Krasovskii and Subbotin, 1987; Krasovskii, 1988a,b) and applications are of great practical importance. Later, this that can be reduced to a model of These (Krasovskii Subbotin, 1987; Krasovskii, 1988a,b) and applications are of great practical importance.game. Later, this can be used to find a solutions for differential game (see that can be reduced to a model of differential These others). (Krasovskii and Subbotin, 1987; Krasovskii, 1988a,b) and theory was significantly developed by Soviet mathematiapplications are of great practical importance. Later, this others). theory was significantly developed by Soviet mathemati(Krasovskii and Subbotin, 1987; Krasovskii, 1988a,b) and applications are of great practical importance. Later, this others). theory was significantly developed by Soviet mathematicians L.S. L.S. Pontryagin Pontryagin (Pontrjagin, (Pontrjagin, 1981), 1981), N.N. N.N. Krasovskii Krasovskii For the general case of the differential game in quescians others). For the general case of the differential game in questheory wasPontryagin significantly developed by Soviet mathemati(Krasovskii, 1970) improved by cians L.S. (Pontrjagin, N.N.Kurzhanski, Krasovskii For (Krasovskii, 1970) and and improved 1981), by A.B. A.B. Kurzhanski, the program iterations was introduced in the of the method differential in question, the general programcase iterations method wasgame introduced in cians L.S. Pontryagin (Pontrjagin, 1981), N.N.Kurzhanski, Krasovskii tion, Yu.S. Osipov and A.N. Subbotin. (Krasovskii, 1970) and improved by A.B. For the general case ofalso the(Chentsov, differential game inInquesYu.S. Osipov and A.N. Subbotin. (Chentsov, 1976) (see 2017a)). our tion, the program iterations method was introduced in 1976) (see also (Chentsov, 2017a)). In our (Krasovskii, andSubbotin. improved by A.B. Kurzhanski, (Chentsov, Yu.S. Osipov 1970) and A.N. tion, theuse program iterations method was introduced in case, we adaptation of the program iterations method, (Chentsov, 1976) (see also (Chentsov, 2017a)). In our In this paper, we consider the non-linear zero-sum pursuitwe use 1976) adaptation of the(Chentsov, program iterations method, Yu.S. A.N. Subbotin. In thisOsipov paper, and we consider the non-linear zero-sum pursuit- case, (Chentsov, (see also 2017a)). In our case, we use adaptation of the program iterations method, also known known as as stability stability iterations iterations technique technique (Chentsov, (Chentsov, evasion differential game, defined by two sets In this paper, we consider non-linear pursuitevasion differential game,the by zero-sum two closed closed sets also case, we This use adaptation the program iterations method, In this paper, we consider thedefined non-linear zero-sum pursuit2017b). provides aa way to solve pursuitalso known asapproach stabilityof iterations technique (Chentsov, evasion differential game, defined by two closed sets in space of positions. This setting was introduced in 2017b). This approach provides way to solve pursuitin space differential of positions. Thisdefined setting bywastwo introduced in also known as stability iterations technique (Chentsov, evasion game, closed sets evasion games with additional constraint on the number 2017b). This approach provides a way to solve pursuitin space of positions. This setting was introduced in (Krasovskii and Subbotin, 1970, 1987). For differential evasion games with additional constraint on the number (Krasovskii and Subbotin, 1970, 1987). For differential This approach providesconstraint a2017b). way toMore solve pursuitevasion games with additional on the number in space of and positions. This1970, setting was For introduced in 2017b). of control switchings (Chentsov, precisely, game considered in (Krasovskii and Subbotin, 1970), the (Krasovskii Subbotin, 1987). differential of control switchings (Chentsov, 2017b). More precisely, game considered in (Krasovskii and Subbotin, 1970), the evasion games with additional constraint on the number of control switchings (Chentsov, 2017b). More precisely, (Krasovskii and Subbotin, 1970, 1987). For differential at each stage of the iterative procedure, we construct fundamental Theorem on was game considered in (Krasovskii and Subbotin, 1970), The the at each stage of the iterative procedure, we construct the the fundamental Theorem on alternative alternative was established. established. The of control switchings (Chentsov, 2017b). More precisely, game considered in (Krasovskii and Subbotin, 1970), the successful solvability set. at each stage of the iterative procedure, we construct the fundamental Theorem on alternative was established. The important generalization of this result was obtained by successful solvability set. important generalization of this result obtainedThe by at each stage of the iterative procedure, we construct the fundamental Theorem on alternative waswas established. successful solvability set. important generalization of this result was obtained by (Kryazhimskii, 1978). According to theorem of alternative, (Kryazhimskii, 1978). According toresult theorem of alternative, we the successful solvability In this this study, study, we relax relaxset. the initial initial setting setting of of the the considered considered important generalization of thisto wassplit by In the set defining state constraints, can be into two (Kryazhimskii, 1978). According theorem ofobtained alternative, the set defining state constraints, can be split into two pursuit-evasion differential game by analyzing In this study, we relax the initial setting of thepossibilities considered (Kryazhimskii, 1978). According to theorem of alternative, pursuit-evasion differential game by analyzing possibilities disjoint subsets, specifying successful sets for the set defining state constraints, cansolvability be split into two In this study, wedifferential relax the initial setting of neighborhoods thepossibilities considered disjoint subsets, specifying successful solvability sets for of player I in terms of reachability of closed pursuit-evasion game by analyzing the set defining state constraints, cansolvability be splitcondition, into two of player II in terms of reachability of closed neighborhoods disjoint subsets, specifying successful sets for each player. If differential game satisfies Isaacs pursuit-evasion differential game by analyzing possibilities of player in terms of reachability of closed neighborhoods each player. If differential game satisfies Isaacs condition, target within corresponding neighborhoods of target Iset set within neighborhoods of state state disjoint subsets, specifying successful solvability sets for of then the can be in terms each If differential satisfies Isaacs player inset. terms ofcorresponding reachability offixed closed neighborhoods target set within corresponding neighborhoods of state then player. the alternative alternative can game be implemented implemented in condition, terms of of of constraints Moreover, for each position, our goal of constraints set. Moreover, for each fixed position, our goal each player. If differential game satisfies Isaacs condition, pure positional strategies and, of course, in terms of nonthen the alternative can be implemented in terms of of target set within corresponding neighborhoods of state pure positional strategies and, of course, in terms of nonis to find the minimal neighborhood, guaranteeing the constraints set. Moreover, for each fixed position, our goal is to find the minimal neighborhood, guaranteeing the then the alternative can be implemented in terms of anticipating strategies (Roxin, 1969; Elliott and Kalton, Kalton, pure positional strategies and, of course, in terms of non- constraints set.the Moreover, for each fixed position, our goal anticipating strategies (Roxin, 1969; Elliott solvability of pursuit problem. Also, such a neighis to find the minimal neighborhood, guaranteeing the and solvability of the pursuit problem. Also, such a neighpure positional strategies and, of course, in terms of nonanticipating strategies (Roxin, 1969; Elliott and Kalton, 1972). In In literature literature those those strategies strategies are are also also known known as as solvability is to findestimates the minimal neighborhood, guaranteeing the of the pursuit problem. Also, such a neigh1972). borhood possibilities of player II (evader) in estimates possibilities of player II (evader) in anticipating strategies (Roxin, 1969; Elliott and Kalton, Elliott-Kalton strategies (Elliott and Kalton, 1972) and 1972). In literature those strategies are also known as borhood solvability of the pursuit problem. Also,one such aperform neighElliott-Kalton strategies (Elliott and Kalton, 1972) and following way: for smaller neighborhood, can borhood estimates possibilities of player II (evader) in following way: for smaller neighborhood, one can perform 1972). In literature those strategies are also known as quasi-strategies (Subbotin and Chentsov, 1981). MultivalElliott-Kalton strategies (Elliott and Kalton, 1972) and borhood estimates possibilities of player II (evader) in quasi-strategies (Subbotin and Chentsov, evasion procedure with finite number of switchings. Based following way: for smaller neighborhood, one can perform 1981). MultivalElliott-Kalton strategies (Elliott and Kalton, 1972) and evasion procedure with finite number of switchings. Based ued cases of non-anticipating strategies were considered quasi-strategies (Subbotin and Chentsov, 1981). Multivalfollowing way: for smaller neighborhood, one can perform ued cases of non-anticipating strategies were considered on this define minimax function, which evasion procedure we withcan finite number of switchings. this approach, approach, we define minimax function, Based which quasi-strategies (Subbotin and and Chentsov, 1981). Multivalued(Chentsov, cases of non-anticipating strategies were considered in (Chentsov, 1976; Subbotin Chentsov, 1981). The on evasion procedure withcan finite number of switchings. Based on thiscan approach, we can define minimax function, which in 1976; Subbotin and Chentsov, 1981). The values can be represented as guaranteed result for special values be represented as guaranteed result for special ued casesofofconstruction non-anticipating strategies were considered problem of an alternative partition can in (Chentsov, 1976; Subbotin and Chentsov, 1981). The on thiscan approach, we can define minimax function, which problem of construction of an alternative partition can payoff. To define this function, we construct special iteravalues be represented as guaranteed result for special To define this function, we construct special iterain (Chentsov, 1976; Subbotin Chentsov, 1981). The problem of construction of anand alternative partition can payoff. values can be represented as guaranteed result forfunction special tive procedure in space of position functions; the payoff. To define this function, we construct special iteraproblem of construction of an alternative partition can tive procedure in space of position functions; the function payoff. To define this function, we construct special itera This itself is a fixed point of special conversion operator. This research research was was supported supported by by the the Russian Russian Foundation Foundation for for Basic Basic tive procedure in space of position functions; the function itself is aa fixed point of special conversion operator. Research, grant No. 19-01-00371. This research was supported by the Russian Foundation for Basic tive procedure in space of position functions; the function itself is fixed point of special conversion operator. Research, grant No. 19-01-00371. This research was supported by the Russian Foundation for Basic Research, grant No. 19-01-00371. itself is a fixed point of special conversion operator.
Research, No.IFAC 19-01-00371. 2405-8963 grant © 2019 2019, (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright © 2353 Copyright © under 2019 IFAC IFAC 2353Control. Peer review responsibility of International Federation of Automatic Copyright © 2019 IFAC 2353 10.1016/j.ifacol.2019.11.549 Copyright © 2019 IFAC 2353
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Alexander Chentsov et al. / IFAC PapersOnLine 52-13 (2019) 2303–2307
2. PROBLEM STATEMENT Consider following control system on finite time interval T [t0 , ϑ0 ]: x˙ = f (t, x, u, v), u ∈ P, v ∈ Q, (1) p where P and Q are non-empty compact sets in R and Rq , respectively, p, q ∈ N. As u(· ) we define control of player I and as v(· ) - control of player II. Also we assume that system 1 satisfies the conditions of generalized uniqueness and uniform boundedness of solutions, similar to one in (Chentsov, 2017b). We are given by two following sets: (M ∈ F) & (N ∈ F), where F is the family of all subsets of T × Rn , closed in topological space, M ⊂ N, M = ∅. First one is a target set for first player and second one defines state constraints in terms of set cross-sections: Nt {x ∈ Rn | (t, x) ∈ N}, ∀t ∈ T. We need to construct an approximation to M under state constraints Nt, t ∈ T . Most of the time (M, N)-approximation cannot be contructed, thus problem is intractable. However, we can relax this problem by defining some quality guarantee. Namely, let us consider (S0 (M, ε), S0 (N, ε))-approximation with guarantee ε ∈ [0, ∞[ (see (Chentsov and Khachay, 2018)). For fixated position (t∗ , x∗ ), we find the smallest number ε∗ ∈ [0, ∞[, for which (S0 (M, ε∗ ), S0 (N, ε∗ ))-approximation can be constructed in terms of admissible control procedures. Later, we show that this number exists. Thus, according to such admissible procedure, for every trajectory x(· ) = x(t), t∗ ≤ t ≤ ϑ0 , generating by given procedure, the following property ((θ, x(θ)) ∈ S0 (M, ε∗ )) & ((t, x(t)) ∈ S0 (N, ε∗ ) ∀t ∈ [t∗ , θ]) (2) where θ ∈ [t∗ , ϑ0 ], holds. Moreover, for trajectory x(· ) in (2), the equality x(t∗ ) = x∗ is fulfilled. Therefore, we can estimate ε∗ by two following values ρ((t∗ , x∗ ), M) and ρ((t∗ , x∗ ), N). In addition, by choice of M and N ρ((t∗ , x∗ ), N) ≤ ρ((t∗ , x∗ ), M), where ρ(· , H) was designated in (Chentsov and Khachay, (1) (2) 2018). We suppose that ε∗ ρ((t∗ , x∗ ), N) and ε∗ (1) (2) ρ((t∗ , x∗ ), M). Then, by (2), ε∗ ≤ ε∗ . Therefore, it is (2) (2) obvious that (S0 (M, ε∗ ), S0 (N, ε∗ ))-approximation can (2) be constructed. By the choice of ε∗ , we obtain ε∗ ≤ ε∗ .
On the other hand, from (2), for admissible procedure, which constructs (S0 (M, ε∗ ), S0 (N, ε∗ ))-approximation, we (1) obtain ε∗ ≤ ε∗ . This follows from (2) and definition (1) of ε∗ . Therefore, we obtain the chain of inequalities (1) (2) (1) (2) ε∗ ≤ ε∗ ≤ ε∗ . Using definition of ε∗ and ε∗ , we have ρ((t∗ , x∗ ), N) ≤ ε∗ ≤ ρ((t∗ , x∗ ), M).
(3)
Thus, by (3), we defined the range for optimal value of ε∗ for every position from T × Rn . Therefore, ρ((· ), N) and ρ((· ), M) - are two boundary functions which define corresponding range in functional space. We should pay special attention to construction of function ε0 (see (Chentsov and Khachay, 2018)). In particular, we (s) use sequence of functions (ε0 )s∈N0 constructed on each iteration with number s. Later, we explain how those
functions relate in terms of iterations. Let us consider special kind of number sets. Each set is generated on respective stage of our iteration procedure. We denote them as follows (see (Chentsov and Khachay, 2018) and (Chentsov, 2017b)) (s)
Σ0 (t, x) {ε ∈ [0, ∞[|(t, x) ∈ Ws (S0 (M, ε), S0 (N, ε))} ∀s ∈ N0 ∀(t, x) ∈ T × Rn , (4) Σ0 (t, x) {ε ∈ [0, ∞[ | (t, x) ∈ W(S0 (M, ε), S0 (N, ε))} where according to (Chentsov, 2017b), to each sets M ∈ F and N ∈ F we assign the sequence of sets (Wk (M, N ))k∈N0 : N0 → P(T × Rn ) and limit set W(M, N ) = Wk (M, N ) ∈ P(T × Rn ). k∈N0
Following (Chentsov and Khachay, 2018), we obtain
ε0 (t, x) inf(Σ0 (t, x)) ∈ [0, ∞[. Proposition 1. If (t∗ , x∗ ) ∈ T × Rn , then ε0 (t∗ , x∗ ) ∈ Σ0 (t∗ , x∗ ). As a corollary, we note that (t, x) ∈ W(S0 (M, ε0 (t, x)), S0 (N, ε0 (t, x))) ∀(t, x) ∈ T × Rn . (5) Let us introduce another option for functional on trajectories of the process. If t∗ ∈ T, x(· ) ∈ Cn ([t∗ , ϑ0 ]) and θ ∈ [t∗ , ϑ0 ], then we assume that ω(t∗ , x(· ), θ) = sup ({ρ((θ, x(θ)), M); max ρ((t, x(t)), N)}), t∗ ≤t≤θ
(6)
ω(t∗ , x(· ), θ) ∈ [0, ∞[. As a corollary, we define for t∗ ∈ T special payoff function γt∗ : Cn ([t∗ , ϑ0 ]) → [0, ∞[, (7) by following conditions: ∀ x(· ) ∈ Cn ([t∗ , ϑ0 ]). γt∗ (x(· ))
min ω(t∗ , x(· ), θ).
θ∈[t∗ ,ϑ0 ]
(8)
Minimum in (8) is attainable, since ρ((· , x(· )), M) and ρ((· , x(· )), N) - are uniformly continuous functions (see also (6)). Proposition 2. Let t∗ ∈ T, x(· ) ∈ Cn ([t∗ , ϑ0 ]) and ε∗ ∈ [0, ∞[ be given. Then, following two conditions are equivalent: ((i)) ∃ϑ ∈ [t∗ , ϑ0 ] : ((ϑ, x(ϑ)) ∈ S0 (M, ε∗ )) & ((t, x(t)) ∈ S0 (N, ε∗ ) ∀t ∈ [t∗ , ϑ]); ((ii)) γt∗ (x(· )) ≤ ε∗ We recall that M ⊂ N. If ε ∈ [0, ∞[, then S0 (M, ε) ⊂ S0 (N, ε). Hereinafter, we will consider special multivalued nonanticipating strategies as admissible control procedures. We define considered strategies in terms of non-anticipating operators on control-measure spaces. For this, it is helpful to recall some of the properties of measurable spaces, used in paragraph 2. We should improve some constructions given there. For arbitrary t ∈ T , let’s fix compactum Zt [t, ϑ0 ] × Q and σ-algebra Dt of Borel subsets of Zt .
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Wherein, for moments of time t1 ∈ T and t2 ∈ [t1 , ϑ0 ] Dt2 = Dt1 |Zt2 = {D ∈ Dt1 |D ⊂ Zt2 }. Further, if t ∈ T , then following holds: Γ × Q ∈ Dt ∀Γ ∈ Tt . Following those constructions, we consider the set of measures, which are similar to Borel mappings from [t, ϑ0 ] to Q, namely: Et {ν ∈ (σ − add)+ [Dt ] | ν(Γ × Q) = λt (Γ) ∀Γ ∈ Tt }; we use Et as set of generalized controls of player II. Moreover, we note that D×P {(t, u, v) ∈ Ωt | (t, v) ∈ D} ∈ Ct ∀D ∈ Dt . Therefore, we have the following (see (Chentsov, 2017a))
2305
From Proposition 4 we obtain (s)
ε0 (t, x) ∈ Σ0 (t, x) ∀(t, x) ∈ T × Rn ∀s ∈ N0 .
In particular, if (t, x) ∈ T we have non-empty subset [0, ∞[ is defined. Hereinafter, we assume (s)
(s)
ε0 (t, x) inf(Σ0 (t, x)) ∀(t, x) ∈ T ×Rn ∀s ∈ N0 . (13) Using (13) with each s ∈ N0 , we can define function (s)
Πt (ν) {η ∈ Ht | η(D×P ) = ν(D) ∀D ∈ Dt } ∀ν ∈ Et . (9) Thus, the family of programs for second player on a interval [t, ϑ0 ] was introduced. We also need additional operations for generalized controls, namely gluing and constriction. Let us recall another notation and define corresponding designation: if X and ˜ ∈ P (X), then Y - nonempty sets, where h ∈ Y X and X ˜ ˜ (h(x)) ˜ ∈ Y X . As X a family can be used and (h|X) x∈X as h — set function. If t1 ∈ T and t2 ∈ [t1 , ϑ0 ], then (see (Chentsov, 2017a)) Ct2 = Ct1 |Ωt2 = {H ∈ Ct1 | H ⊂ Ωt2 }, Ctt12 = Ct1 |[t1 ,t2 [×P ×Q = {H ∈ Ct1 | H ⊂ [t1 , t2 [×P × Q}, Dtt12
= Dt1 |[t1 ,t2 [×Q = {D ∈ Dt1 | D ⊂ [t1 , t2 [×Q}. In this case we have σ-algebras of sets. It is useful to note that (see (Chentsov, 2017a)) Ht2 = {(η | Ct2 ) : η ∈ Ht1 }, Et2 = {(ν | Dt2 ) : ν ∈ Et1 }. If t∗ ∈ T , then we assume that A˜t∗ is the set of all generalized multivalued non-anticipating strategies (see (Chentsov, 2017a)) for first player on a line [t∗ , ϑ0 ] : A˜t∗ {α ∈
ν∈Et∗
(s)
(s)
(16) ε0 ε0 ∀s ∈ N0 . Proposition 5. If s ∈ N0 and (t∗ , x∗ ) ∈ T × Rn , then (s) (s) ε0 (t∗ , x∗ ) ∈ Σ0 (t∗ , x∗ ). (s)
From (4) we have property ε0 ∈ Σ0 (t∗ , x∗ ).
Also, from (15) we show, in particular, that exact upper bound is defined properly. That is, (s)
sup({ε0 (t, x) : s ∈ N0 }) ∈ [0, ε0 (t, x)] ∀(t, x) ∈ T × Rn . Proposition 6. If (t∗ , x∗ ) ∈ T × Rn , then ε0 (t∗ , x∗ ) = (s) sup({ε0 (t∗ , x∗ ) : s ∈ N0 }) We consider another proposition which will establish rela(s) (s+1) (t∗ , x∗ ). tions between ε0 (t∗ , x∗ ) and ε0 n Proposition 7. Let (t∗ , x∗ ) ∈ T × R and s ∈ N0 . Then (s)
(s+1)
ε0 (t∗ , x∗ ) ε0 (s)
Dtθ∗ )
= {(η | Ctθ∗ ) : η ∈ α(ν2 )})}.
(10)
If α ∈ A˜t∗ and ν ∈ Et∗ , then as α(ν) we have nonempty subset of Πt∗ (ν); in particular, α(ν) ⊂ Ht∗ . Thus, we defined following union-set (see (Chentsov, 2017a)) ˜ t (α) α(ν) ∈ P (Ht ), (11) Π ν∈Et∗
(14)
(15) ε0 (t, x) ≤ ε0 (t, x) ∀(t, x) ∈ T × Rn ∀s ∈ N0 . Let us designate point-wise order in the set of all functions from T × Rn into [0, ∞[ by . Then from (15) we have
(s+1)
ε0 ε0
= (ν2 | Dtθ∗ )) ⇒ ({(η | Ctθ∗ ) : η ∈ α(ν1 )}
∗
ε0 : T × Rn → [0, ∞[. Also, from (13) and Proposition 4, we get
(t∗ , x∗ ).
(17)
Finally, we have following property:
P (Πt∗ (ν)) | ∀ν1 ∈ Et∗
∀ν2 ∈ Et∗ ∀θ ∈ [t∗ , ϑ0 ] : ((ν1 |
(12)
(s) × Rn and s ∈ N0 as Σ0 (t, x), (s) of [0, ∞[, thus, inf(Σ0 (t, x)) ∈
∗
which is the set of all generalized controls-measures, generated by non-anticipating strategy α. From Proposition 1 we have following: Proposition 3. If (t∗ , x∗ ) ∈ T × Rn , then sup γt∗ (ϕ(· , t∗ , x∗ , η)), ε0 (t∗ , x∗ ) = inf ˜t α∈A ˜ t (α) ∗ η∈Π ∗
˜ t (α wherein ∃˜ α∗ ∈ A˜t∗ : ε0 (t∗ , x∗ ) = Π ˜ ∗ ). ∗ Therefore it is established that function ε0 : T × R → [0, ∞[, is equal to minimax of payoff function γ in terms of non-anticipating strategies for any fixed position. Proposition 4. If s ∈ N0 and (t, x) ∈ T × Rn , then (s) Σ0 (t, x) ⊂ Σ0 (t, x).
∀s ∈ N0 .
(18)
According to Proposition 3 in (Chentsov and Khachay, 2018), ε0 is the minimax of special payoff in terms of nonanticipating strategies. 3. MAIN RESULTS In this section we construct program operator, which (s) (s+1) . will define for s ∈ N0 conversion from ε0 to ε0 To achieve this, we will use special type of construction, which is similar to one introduced in (Chentsov, 1978). The modification of program iterations method described in (Chentsov, 1978) corresponds to differential game with non-fixed moment of termination. Also, let us introduce new notations according to those in (Chentsov, 1978). First of all, we introduce the function ψ : T × Rn → [0, ∞[ by condition: (19) ψ(t, x) ρ((t, x), M) ∀(t, x) ∈ T × Rn . According to the properties of the distance function from a point to a nonempty set M we have that ψ ∈ C(T × Rn ), where C(T × Rn ) - set of all continuous real-valued functions on T × Rn . We note that (20) ψ −1 ([0, c]) ∈ F ∀c ∈ [0, ∞[.
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Thus, according to (20) and due to non-negativity of ψ, we have lower semi-continuity property. Subsequently, for every non-empty set H as R+ (H) we denote the set of all non-negative real-valued functions on H. In terms of (20) we define, according to (Chentsov, 1978), the following set M {g ∈ R+ (T × Rn )| g −1 ([0, c]) ∈ F ∀c ∈ [0, ∞[}. (21) Besides, we require set with point-wise order (22) Mψ {g ∈ M| g ψ}; where ψ ∈ Mψ . Thus, (22) is non-empty subset of M. 3.1 Convergence operator In this subsection, we construct special convergence operator for ε0 . For the sake of brevity, we skip the rigorous mathematical proofs of the presented results. Interested reader can find them in (Chentsov and Khachay, 2018). If g ∈ Mψ and (t∗ , x∗ ) ∈ T × Rn , then (since g ψ) we obtain for some c ∈ [0, ∞[ {g(t, x(t)) : t ∈ [t∗ , ϑ]} ∈ P ([0, c]) ∀v ∈ Q ∀x(· ) ∈ Xπ (t∗ , x∗ , v) ∀ϑ ∈ [t∗ , ϑ0 ], (23) where Xπ (t∗ , x∗ , v) is used in the sense of (Chentsov and Khachay, 2018). Following property (23), we have useful corollary: sup g(t, x(t)) ≤ c ∀v ∈ Q
t∈[t∗ ,ϑ]
∀x(· ) ∈ Xπ (t∗ , x∗ , v) ∀ϑ ∈ [t∗ , ϑ0 ]. (24) By (24), with g ∈ Mψ and (t∗ , x∗ ) ∈ T × Rn , we obtain ∃˜ c ∈ [0, ∞[: ¯ x ¯(t)); ψ(ϑ, ¯(ϑ))}) ∈ [0, c˜] sup({ sup g(t, x ¯ t∈[t∗ ,ϑ]
∀v ∈ Q ∀¯ x(· ) ∈ Xπ (t∗ , x∗ , v) ∀ϑ¯ ∈ [t∗ , ϑ0 ].
Then for g ∈ Mψ and (t∗ , x∗ ) ∈ T ×Rn , we obtain following sup
inf
v∈Q (x(·),ϑ)∈Xπ (t∗ ,x∗ ,v)×[t∗ ,ϑ0 ]
sup({ sup g(t, x(t)); t∈[t∗ ,ϑ]
ψ(ϑ, x(ϑ))}) ∈ [0, ∞[.
(25)
Considering (25), we define the convergence operator Γ : Mψ → R+ [T × Rn ] as follows: Γ(g)(t∗ , x∗ ) sup
inf
v∈Q (x(·),ϑ)∈Xπ (t∗ ,x∗ ,v)×[t∗ ,ϑ0 ]
sup({ sup t∈[t∗ ,ϑ]
g(t, x(t)); ψ(ϑ, x(ϑ))}) ∀g ∈ Mψ ∀(t∗ , x∗ ) ∈ T × Rn .
If g ∈ Mψ , (t∗ , x∗ ) ∈ T × Rn and v ∈ Q, then by h[g; t∗ ; x∗ ; v] we denote the functional (x(· ), ϑ) −→ sup({ sup g(t, x(t)); ψ(ϑ, x(ϑ))}) :
(k+1)
Theorem 8. If k ∈ N0 , then ε0
3.2 Fixed point property of operator Γ In the sequel, we consider an important property of operator Γ, namely its fixed point. Theorem 9. Function ε0 is the fixed point of operator Γ: ε0 = Γ(ε0 ). Let us consider the set of all fixed points of operator Γ, (Γ) namely Mψ {g ∈ Mψ | g = Γ(g)}. In regards to (Γ) ˜ (Γ) {g ∈ previous theorem, we obtain ε0 ∈ M . Let M ψ
(Γ)
(0)
ψ
(Γ)
Mψ | ε0 g} = {g ∈ Mψ | ρ(· , N) g}.
˜ (Γ) , Theorem 10. Function ε0 is the smallest element of M ψ ˜ (Γ) and ε0 g ∀g ∈ M ˜ (Γ) . in other words, ε0 ∈ M ψ ψ
Let us consider one special case of our setting. Namely, let (M T × M) & (N T × N ), (27) where M and N are closed non-empty sets in Rn with usual topology generated by euclidean norm · . Moreover, let M ⊂ N . If H ∈ P (Rn ) and x ∈ Rn , then we introduce ( · − inf)[x; H] inf({ x − h : h ∈ H}) ∈ [0, ∞[. (28) We obtain the following property: for H ∈ P (Rn ) and x∗ ∈ R n , ρ((t, x∗ ), T × H) = ( · − inf)[x∗ ; H] ∀t ∈ T. (29) Indeed, we fixate t∗ ∈ T . Then, ρ((t∗ , x∗ ), T ×H) = inf({ρ((t∗ , x∗ ), (t, h)) : (t, h) ∈ T ×H}). In addition, for h ∈ H, we obtain that (t, h) ∈ T × H, where t ∈ T , and x∗ − h ≤ ρ((t∗ , x∗ ), (t, h)). (30) Also, (t∗ , h) ∈ T × H and ρ((t∗ , x∗ ), (t∗ , h)) = x∗ − h . Wherein, ρ((t∗ , x∗ ), T × H) ≤ ρ((t∗ , x∗ ), (t∗ , h)). As a corrollary, ρ((t∗ , x∗ ), T × H) ≤ x∗ − h . (31) Since the selection of h was arbitrary, by (28) ρ((t∗ , x∗ ), T × H) ≤ ( · − inf)[x∗ ; H]. (32) On the other hand, from (30), we have that ˜ ∀t ∈ T ∀h ˜ ∈ H. ( · − inf)[x∗ ; H] ≤ ρ((t∗ , x∗ ), (t, h)) Then ( · − inf)[x∗ ; H] ≤ ρ((t∗ , x∗ ), T × H), and by (32), we obtain ρ((t∗ , x∗ ), T × H) = ( · − inf)[x∗ ; H]. Since the selection of t∗ was arbitrary, the property (29) is established. So, by (27) and (29), ∀(t, x) ∈ T × Rn (ρ((t, x), M) = ( · − inf)[x; M]) & (ρ((t, x), N) = ( · − inf)[x; N ]). Therefore, we obtain that
ω(t, x(· ), θ) sup({( · −inf)[x(θ); M]; max ( · −inf) t≤ξ≤θ
[x(ξ); N ]}) ∀t ∈ T ∀x(· ) ∈ Cn ([t, ϑ0 ]) ∀θ ∈ [t, ϑ0 ]. As a corollary, in considered case (27), we obtain
t∈[t∗ ,ϑ]
γt (x(· )) min sup({( · − inf)[x(θ); M];
Xπ (t∗ , x∗ , v) × [t∗ , ϑ0 ] → [0, ∞[; (26) also, we consider following Lebesgue sets: for b ∈ [0, ∞[ Yb [g; t∗ ; x∗ ; v] {(x(· ), ϑ) ∈ Xπ (t∗ , x∗ , v) × [t∗ , ϑ0 ]| h[g; t∗ ; x∗ ; v](x(· ), ϑ) ≤ b}.
(k)
= Γ(ε0 ).
θ∈[t,ϑ0 ]
max ( · − inf)[x(ξ); N ]}) ∀t ∈ T ∀x(· ) ∈ Cn ([t, ϑ0 ]).
t≤ξ≤θ
Thus we have natural functional, which defines quality of pursuit. 2356
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CONCLUSION In this paper, we have considered a relaxation of the non-linear zero-sum pursuit-evasion differential game. For player I, we have constructed the optimal strategy, namely, we have defined minimax function, which values are represented as guaranteed result for special type of payoff. Then, we have constructed special iterative procedure in space of positions by finding convergence operator Γ and proving that minimax function is its fixed point. REFERENCES Chentsov, A. (1976). On a game problem of guidance with information memory. Soviet Math. Dokl., 17, 411–414. Chentsov, A. (1978). On the game problem of convergence at a given moment of time. Math. USSR-Izv., 12(2), 426–437. doi:10.1070/IM1978v012n02ABEH001985. Chentsov, A. (2017a). The program iteration method in a game problem of guidance. Proceedings of the Steklov Institute of Mathematics, 297(1), 43–61. doi: 10.1134/S0081543817050066. Chentsov, A. (2017b). Stability iterations and an evasion problem with a constraint on the number of switchings (in russian). Trudy Inst. Mat. i Mekh. UrO RAN, 23(2), 285–302. doi:10.21538/0134-4889-2017-23-2-285-302. Chentsov, A. and Khachay, D. (2018). Relaxation of the pursuit-evasion differential game and iterative methods (in russian). Trudy Inst. Mat. i Mekh. UrO RAN, 24(4), 246–269. doi:10.21538/0134-4889-2018-24-4-246-269. Elliott, R.J. and Kalton, N.J. (1972). Values in differential games. Bull. Amer. Math. Soc., 78(3), 427–431. Isaacs, R. (1969). Differential games: Their scope, nature, and future. J. Optim. Theory Appl.,
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3(5), 283–295. doi:10.1007/BF00931368. URL https://doi.org/10.1007/BF00931368. Krasovskii, N.N. and Subbotin, A. (1987). Gametheoretical Control Problems. Springer-Verlag, Berlin, Heidelberg. Krasovskii, N. (1970). Game problems on the encounter of motions (in Russian). Nauka. Krasovskii, N. (1988a). A differential game of approach and evasion. i. Eng Cybern, 11(2), 189–203. Krasovskii, N. (1988b). A differential game of approach and evasion. ii. Eng Cybern, 11(3), 376–394. doi: 10.1007/978-1-4612-3716-7 2. Krasovskii, N. and Subbotin, A.I. (1970). An alternative for the game problem of convergence. Journal of Applied Mathematics and Mechanics, 34(6), 948–965. doi: 10.1016/0021-8928(70)90158-9. Kryazhimskii, A. (1978). On the theory of positional differential games of approach-evasion. Soviet Math. Dokl., 19(2), 408–412. Pontrjagin, L. (1981). Linear differential games of pursuit. Mathematics of the USSR-Sbornik, 40, 285. doi: 10.1070/SM1981v040n03ABEH001815. Roxin, E. (1969). Axiomatic approach in differential games. Journal of Optimization Theory and Applications, 3(3), 153–163. doi:10.1007/BF00929440. Subbotin, A. and Chentsov, A. (1981). Optimization of guarantee in control problems (in Russian). Nauka. Ushakov, V.N., Ukhobotov, V.I., Ushakov, A.V., and Parshikov, G.V. (2015). On solving approach problems for control systems. Proceedings of the Steklov Institute of Mathematics, 291(1), 263–278. doi: 10.1134/S0081543815080210.
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Towards Towards Towards Towards
a a a a
relaxation relaxation of of the the pursuit-evasion pursuit-evasion relaxation of the pursuit-evasion differential game relaxation of the pursuit-evasion game differential differential game differential game ∗,∗∗ ∗,∗∗ Alexander Chentsov ∗,∗∗ ∗,∗∗ Daniel Khachay ∗,∗∗ ∗,∗∗
Alexander Chentsov ∗,∗∗ Daniel Khachay ∗,∗∗ Alexander Chentsov ∗,∗∗ Daniel Khachay ∗,∗∗ ∗,∗∗ Alexander Chentsov Daniel Khachay ∗,∗∗ ∗ ∗ Krasovsky Institute of Mathematics and Mechanics, 16 S.Kovalevskoy ∗ Krasovsky Institute of Mathematics and Mechanics, 16 S.Kovalevskoy ∗ ∗ Krasovsky Institute ofEkaterinburg Mathematics620990, and Mechanics, S.Kovalevskoy str. Russia str. Ekaterinburg 620990, Russia 16 ∗ Krasovsky Institute ofEkaterinburg Mathematics620990, and Mechanics, 16 S.Kovalevskoy (e-mail: [email protected], [email protected]). str. Russia (e-mail: [email protected], [email protected]). ∗∗ str. Ekaterinburg 620990, Russia ∗∗ Federal University, 51 Lenin ave. Ekaterinburg (e-mail: [email protected], [email protected]). ∗∗ Ural Ural Federal University, 51 Lenin ave. Ekaterinburg 620002, 620002, Russia Russia ∗∗ ∗∗ (e-mail: [email protected]). Ural [email protected], University, 51 Lenin ave. Ekaterinburg 620002, Russia ∗∗ Ural Federal University, 51 Lenin ave. Ekaterinburg 620002, Russia Abstract: We We consider consider aa special special case case of the the non-linear non-linear zero-sum zero-sum pursuit-evasion pursuit-evasion differential differential Abstract: Abstract: We consider agame special case of ofby thetwo non-linear zero-sum pursuit-evasion differential game. The instance of this is defined closed sets -- target set and one specifying state game. The instance of this game is defined by two closed sets target set and one specifying state Abstract: We consider agame special case ofby thetwo non-linear zero-sum pursuit-evasion differential constraints. We find an optimal non-anticipating strategy for player I (the pursuer). Namely, game. The instance of this is defined closed sets target set and one specifying state constraints. We find an optimal non-anticipating strategy for player I (the pursuer). Namely, game. The instance ofan this game isnon-anticipating defined two closed setsfor -function target set specifying state we construct construct his find successful solvability set by specified by limit limit ofI and the one iterative procedure constraints. We optimal strategy player (the pursuer). Namely, we his successful solvability set specified by function of the iterative procedure constraints. We find an optimal non-anticipating strategy for player I (the pursuer). Namely, we construct his successful solvability set specified by limit function of the iterative procedure in space space of of positions. positions. For For positions positions located located outside outside the the successful successful solvability solvability set, set, we we provide provide a in we construct his successful solvability set specified limit of the iterative procedure in space ofof positions. For positions located outside by the successful solvability set,two wementioned provide aa relaxation our game by determining the smallest size of aafunction neighborhoods of relaxation of our game by determining the smallest size of neighborhoods of two mentioned in space ofofpositions. For outside the solvability set,two wementioned provide a sets, for which which thegame pursuer can solve solvelocated his the problem successfully. Then, we construct construct his successful relaxation our by positions determining smallest sizesuccessful of a Then, neighborhoods of sets, for the pursuer can his problem successfully. we his successful relaxation of our game by determining the smallest size of a neighborhoods of two mentioned solvability set in terms of those neighborhoods. sets, for which theterms pursuer can solve his problem successfully. Then, we construct his successful solvability set of neighborhoods. sets, for which theterms pursuer can solve his problem successfully. Then, we construct his successful solvability set in in of those those neighborhoods. © 2019, IFAC Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. solvability set(International in terms of those neighborhoods. Keywords: Pursuit-Evasion Pursuit-Evasion differential differential game, game, non-anticipating non-anticipating strategies, strategies, program program iterations iterations Keywords: method. Keywords: Pursuit-Evasion differential game, non-anticipating strategies, program iterations method. Keywords: method. Pursuit-Evasion differential game, non-anticipating strategies, program iterations method. 1. be 1. INTRODUCTION INTRODUCTION AND AND RELATED RELATED WORK WORK be reduced reduced to to problem problem of of finding finding the the set set of of successful successful be reducedof toaa pursuer problem (player of finding the is, setthe of maximum successful solvability I) (that 1. INTRODUCTION AND RELATED WORK solvability of pursuer (player I) (that is, the maximum 1. INTRODUCTION AND RELATED WORK be reduced to problem of finding the set of successful stable bridge by N.N. Krasovskii), who is interested in solvability of a pursuer (player I) (that is, the maximum During bridge by N.N. Krasovskii), who isis, interested in the the During the the last last decades, decades, differential differential game game theory theory has has been been stable solvability of a pursuer (player I) (that the maximum guaranteed feasibility of approach. Various methods stable bridge by N.N. Krasovskii), who is interested inwere the an actively developing field of operations research and During the last decades, differential game theory has been guaranteed feasibility of approach. Various methods were an actively developing field of operations research and stable bridge by N.N. Krasovskii), who isal., interested inwere the guaranteed feasibility of approach. Various methods During the last decades, differential game theory has been used to build this set. In (Ushakov et 2015), special control theory. The first mention of such a differential an actively developing field of operations research and used to build this set.ofInapproach. (UshakovVarious et al., methods 2015), special control theory. The firstfield mention of such aresearch differential guaranteed feasibility were an actively developing of operations and kind of procedures constructed by dynamic programming used to build this set. In (Ushakov et al., 2015), special game back to In well-known study control theory. firstIsaacs. mention of such a differential of constructed by dynamic programming game goes goes backThe to R. R. Isaacs. In the the well-known study kind to procedures build this set. Inthat, (Ushakov et al., 2015), special control theory. The first mention of such differential was considered. Besides the constructions kind of procedures constructed by program dynamic programming game goes back to R. Isaacs. Ina the well-known study used (Isaacs, 1969), Isaacs overviewed a number number ofaapplications applications was considered. Besides that, the program constructions (Isaacs, 1969), Isaacs overviewed of kind of procedures constructed by dynamic programming game goes back to to R.a Isaacs. Indifferential the well-known study was can be used to find aa solutions for differential game (see considered. Besides that, the program constructions that be These (Isaacs, Isaacs number of game. applications be used to for differential game that can can1969), be reduced reduced tooverviewed a model model of ofa differential game. These can was considered. Besides that, the program constructions can be usedand to find find a solutions solutions for differential game (see (see (Isaacs, 1969), Isaacs overviewed adifferential number of game. applications (Krasovskii and Subbotin, 1987; Krasovskii, 1988a,b) and applications are of great practical importance. Later, this that can be reduced to a model of These (Krasovskii Subbotin, 1987; Krasovskii, 1988a,b) and applications are of great practical importance.game. Later, this can be used to find a solutions for differential game (see that can be reduced to a model of differential These others). (Krasovskii and Subbotin, 1987; Krasovskii, 1988a,b) and theory was significantly developed by Soviet mathematiapplications are of great practical importance. Later, this others). theory was significantly developed by Soviet mathemati(Krasovskii and Subbotin, 1987; Krasovskii, 1988a,b) and applications are of great practical importance. Later, this others). theory was significantly developed by Soviet mathematicians L.S. L.S. Pontryagin Pontryagin (Pontrjagin, (Pontrjagin, 1981), 1981), N.N. N.N. Krasovskii Krasovskii For the general case of the differential game in quescians others). For the general case of the differential game in questheory wasPontryagin significantly developed by Soviet mathemati(Krasovskii, 1970) improved by cians L.S. (Pontrjagin, N.N.Kurzhanski, Krasovskii For (Krasovskii, 1970) and and improved 1981), by A.B. A.B. Kurzhanski, the program iterations was introduced in the of the method differential in question, the general programcase iterations method wasgame introduced in cians L.S. Pontryagin (Pontrjagin, 1981), N.N.Kurzhanski, Krasovskii tion, Yu.S. Osipov and A.N. Subbotin. (Krasovskii, 1970) and improved by A.B. For the general case ofalso the(Chentsov, differential game inInquesYu.S. Osipov and A.N. Subbotin. (Chentsov, 1976) (see 2017a)). our tion, the program iterations method was introduced in 1976) (see also (Chentsov, 2017a)). In our (Krasovskii, andSubbotin. improved by A.B. Kurzhanski, (Chentsov, Yu.S. Osipov 1970) and A.N. tion, theuse program iterations method was introduced in case, we adaptation of the program iterations method, (Chentsov, 1976) (see also (Chentsov, 2017a)). In our In this paper, we consider the non-linear zero-sum pursuitwe use 1976) adaptation of the(Chentsov, program iterations method, Yu.S. A.N. Subbotin. In thisOsipov paper, and we consider the non-linear zero-sum pursuit- case, (Chentsov, (see also 2017a)). In our case, we use adaptation of the program iterations method, also known known as as stability stability iterations iterations technique technique (Chentsov, (Chentsov, evasion differential game, defined by two sets In this paper, we consider non-linear pursuitevasion differential game,the by zero-sum two closed closed sets also case, we This use adaptation the program iterations method, In this paper, we consider thedefined non-linear zero-sum pursuit2017b). provides aa way to solve pursuitalso known asapproach stabilityof iterations technique (Chentsov, evasion differential game, defined by two closed sets in space of positions. This setting was introduced in 2017b). This approach provides way to solve pursuitin space differential of positions. Thisdefined setting bywastwo introduced in also known as stability iterations technique (Chentsov, evasion game, closed sets evasion games with additional constraint on the number 2017b). This approach provides a way to solve pursuitin space of positions. This setting was introduced in (Krasovskii and Subbotin, 1970, 1987). For differential evasion games with additional constraint on the number (Krasovskii and Subbotin, 1970, 1987). For differential This approach providesconstraint a2017b). way toMore solve pursuitevasion games with additional on the number in space of and positions. This1970, setting was For introduced in 2017b). of control switchings (Chentsov, precisely, game considered in (Krasovskii and Subbotin, 1970), the (Krasovskii Subbotin, 1987). differential of control switchings (Chentsov, 2017b). More precisely, game considered in (Krasovskii and Subbotin, 1970), the evasion games with additional constraint on the number of control switchings (Chentsov, 2017b). More precisely, (Krasovskii and Subbotin, 1970, 1987). For differential at each stage of the iterative procedure, we construct fundamental Theorem on was game considered in (Krasovskii and Subbotin, 1970), The the at each stage of the iterative procedure, we construct the the fundamental Theorem on alternative alternative was established. established. The of control switchings (Chentsov, 2017b). More precisely, game considered in (Krasovskii and Subbotin, 1970), the successful solvability set. at each stage of the iterative procedure, we construct the fundamental Theorem on alternative was established. The important generalization of this result was obtained by successful solvability set. important generalization of this result obtainedThe by at each stage of the iterative procedure, we construct the fundamental Theorem on alternative waswas established. successful solvability set. important generalization of this result was obtained by (Kryazhimskii, 1978). According to theorem of alternative, (Kryazhimskii, 1978). According toresult theorem of alternative, we the successful solvability In this this study, study, we relax relaxset. the initial initial setting setting of of the the considered considered important generalization of thisto wassplit by In the set defining state constraints, can be into two (Kryazhimskii, 1978). According theorem ofobtained alternative, the set defining state constraints, can be split into two pursuit-evasion differential game by analyzing In this study, we relax the initial setting of thepossibilities considered (Kryazhimskii, 1978). According to theorem of alternative, pursuit-evasion differential game by analyzing possibilities disjoint subsets, specifying successful sets for the set defining state constraints, cansolvability be split into two In this study, wedifferential relax the initial setting of neighborhoods thepossibilities considered disjoint subsets, specifying successful solvability sets for of player I in terms of reachability of closed pursuit-evasion game by analyzing the set defining state constraints, cansolvability be splitcondition, into two of player II in terms of reachability of closed neighborhoods disjoint subsets, specifying successful sets for each player. If differential game satisfies Isaacs pursuit-evasion differential game by analyzing possibilities of player in terms of reachability of closed neighborhoods each player. If differential game satisfies Isaacs condition, target within corresponding neighborhoods of target Iset set within neighborhoods of state state disjoint subsets, specifying successful solvability sets for of then the can be in terms each If differential satisfies Isaacs player inset. terms ofcorresponding reachability offixed closed neighborhoods target set within corresponding neighborhoods of state then player. the alternative alternative can game be implemented implemented in condition, terms of of of constraints Moreover, for each position, our goal of constraints set. Moreover, for each fixed position, our goal each player. If differential game satisfies Isaacs condition, pure positional strategies and, of course, in terms of nonthen the alternative can be implemented in terms of of target set within corresponding neighborhoods of state pure positional strategies and, of course, in terms of nonis to find the minimal neighborhood, guaranteeing the constraints set. Moreover, for each fixed position, our goal is to find the minimal neighborhood, guaranteeing the then the alternative can be implemented in terms of anticipating strategies (Roxin, 1969; Elliott and Kalton, Kalton, pure positional strategies and, of course, in terms of non- constraints set.the Moreover, for each fixed position, our goal anticipating strategies (Roxin, 1969; Elliott solvability of pursuit problem. Also, such a neighis to find the minimal neighborhood, guaranteeing the and solvability of the pursuit problem. Also, such a neighpure positional strategies and, of course, in terms of nonanticipating strategies (Roxin, 1969; Elliott and Kalton, 1972). In In literature literature those those strategies strategies are are also also known known as as solvability is to findestimates the minimal neighborhood, guaranteeing the of the pursuit problem. Also, such a neigh1972). borhood possibilities of player II (evader) in estimates possibilities of player II (evader) in anticipating strategies (Roxin, 1969; Elliott and Kalton, Elliott-Kalton strategies (Elliott and Kalton, 1972) and 1972). In literature those strategies are also known as borhood solvability of the pursuit problem. Also,one such aperform neighElliott-Kalton strategies (Elliott and Kalton, 1972) and following way: for smaller neighborhood, can borhood estimates possibilities of player II (evader) in following way: for smaller neighborhood, one can perform 1972). In literature those strategies are also known as quasi-strategies (Subbotin and Chentsov, 1981). MultivalElliott-Kalton strategies (Elliott and Kalton, 1972) and borhood estimates possibilities of player II (evader) in quasi-strategies (Subbotin and Chentsov, evasion procedure with finite number of switchings. Based following way: for smaller neighborhood, one can perform 1981). MultivalElliott-Kalton strategies (Elliott and Kalton, 1972) and evasion procedure with finite number of switchings. Based ued cases of non-anticipating strategies were considered quasi-strategies (Subbotin and Chentsov, 1981). Multivalfollowing way: for smaller neighborhood, one can perform ued cases of non-anticipating strategies were considered on this define minimax function, which evasion procedure we withcan finite number of switchings. this approach, approach, we define minimax function, Based which quasi-strategies (Subbotin and and Chentsov, 1981). Multivalued(Chentsov, cases of non-anticipating strategies were considered in (Chentsov, 1976; Subbotin Chentsov, 1981). The on evasion procedure withcan finite number of switchings. Based on thiscan approach, we can define minimax function, which in 1976; Subbotin and Chentsov, 1981). The values can be represented as guaranteed result for special values be represented as guaranteed result for special ued casesofofconstruction non-anticipating strategies were considered problem of an alternative partition can in (Chentsov, 1976; Subbotin and Chentsov, 1981). The on thiscan approach, we can define minimax function, which problem of construction of an alternative partition can payoff. To define this function, we construct special iteravalues be represented as guaranteed result for special To define this function, we construct special iterain (Chentsov, 1976; Subbotin Chentsov, 1981). The problem of construction of anand alternative partition can payoff. values can be represented as guaranteed result forfunction special tive procedure in space of position functions; the payoff. To define this function, we construct special iteraproblem of construction of an alternative partition can tive procedure in space of position functions; the function payoff. To define this function, we construct special itera This itself is a fixed point of special conversion operator. This research research was was supported supported by by the the Russian Russian Foundation Foundation for for Basic Basic tive procedure in space of position functions; the function itself is aa fixed point of special conversion operator. Research, grant No. 19-01-00371. This research was supported by the Russian Foundation for Basic tive procedure in space of position functions; the function itself is fixed point of special conversion operator. Research, grant No. 19-01-00371. This research was supported by the Russian Foundation for Basic Research, grant No. 19-01-00371. itself is a fixed point of special conversion operator.
Research, No.IFAC 19-01-00371. 2405-8963 grant © 2019 2019, (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright © 2353 Copyright © under 2019 IFAC IFAC 2353Control. Peer review responsibility of International Federation of Automatic Copyright © 2019 IFAC 2353 10.1016/j.ifacol.2019.11.549 Copyright © 2019 IFAC 2353
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2. PROBLEM STATEMENT Consider following control system on finite time interval T [t0 , ϑ0 ]: x˙ = f (t, x, u, v), u ∈ P, v ∈ Q, (1) p where P and Q are non-empty compact sets in R and Rq , respectively, p, q ∈ N. As u(· ) we define control of player I and as v(· ) - control of player II. Also we assume that system 1 satisfies the conditions of generalized uniqueness and uniform boundedness of solutions, similar to one in (Chentsov, 2017b). We are given by two following sets: (M ∈ F) & (N ∈ F), where F is the family of all subsets of T × Rn , closed in topological space, M ⊂ N, M = ∅. First one is a target set for first player and second one defines state constraints in terms of set cross-sections: Nt {x ∈ Rn | (t, x) ∈ N}, ∀t ∈ T. We need to construct an approximation to M under state constraints Nt, t ∈ T . Most of the time (M, N)-approximation cannot be contructed, thus problem is intractable. However, we can relax this problem by defining some quality guarantee. Namely, let us consider (S0 (M, ε), S0 (N, ε))-approximation with guarantee ε ∈ [0, ∞[ (see (Chentsov and Khachay, 2018)). For fixated position (t∗ , x∗ ), we find the smallest number ε∗ ∈ [0, ∞[, for which (S0 (M, ε∗ ), S0 (N, ε∗ ))-approximation can be constructed in terms of admissible control procedures. Later, we show that this number exists. Thus, according to such admissible procedure, for every trajectory x(· ) = x(t), t∗ ≤ t ≤ ϑ0 , generating by given procedure, the following property ((θ, x(θ)) ∈ S0 (M, ε∗ )) & ((t, x(t)) ∈ S0 (N, ε∗ ) ∀t ∈ [t∗ , θ]) (2) where θ ∈ [t∗ , ϑ0 ], holds. Moreover, for trajectory x(· ) in (2), the equality x(t∗ ) = x∗ is fulfilled. Therefore, we can estimate ε∗ by two following values ρ((t∗ , x∗ ), M) and ρ((t∗ , x∗ ), N). In addition, by choice of M and N ρ((t∗ , x∗ ), N) ≤ ρ((t∗ , x∗ ), M), where ρ(· , H) was designated in (Chentsov and Khachay, (1) (2) 2018). We suppose that ε∗ ρ((t∗ , x∗ ), N) and ε∗ (1) (2) ρ((t∗ , x∗ ), M). Then, by (2), ε∗ ≤ ε∗ . Therefore, it is (2) (2) obvious that (S0 (M, ε∗ ), S0 (N, ε∗ ))-approximation can (2) be constructed. By the choice of ε∗ , we obtain ε∗ ≤ ε∗ .
On the other hand, from (2), for admissible procedure, which constructs (S0 (M, ε∗ ), S0 (N, ε∗ ))-approximation, we (1) obtain ε∗ ≤ ε∗ . This follows from (2) and definition (1) of ε∗ . Therefore, we obtain the chain of inequalities (1) (2) (1) (2) ε∗ ≤ ε∗ ≤ ε∗ . Using definition of ε∗ and ε∗ , we have ρ((t∗ , x∗ ), N) ≤ ε∗ ≤ ρ((t∗ , x∗ ), M).
(3)
Thus, by (3), we defined the range for optimal value of ε∗ for every position from T × Rn . Therefore, ρ((· ), N) and ρ((· ), M) - are two boundary functions which define corresponding range in functional space. We should pay special attention to construction of function ε0 (see (Chentsov and Khachay, 2018)). In particular, we (s) use sequence of functions (ε0 )s∈N0 constructed on each iteration with number s. Later, we explain how those
functions relate in terms of iterations. Let us consider special kind of number sets. Each set is generated on respective stage of our iteration procedure. We denote them as follows (see (Chentsov and Khachay, 2018) and (Chentsov, 2017b)) (s)
Σ0 (t, x) {ε ∈ [0, ∞[|(t, x) ∈ Ws (S0 (M, ε), S0 (N, ε))} ∀s ∈ N0 ∀(t, x) ∈ T × Rn , (4) Σ0 (t, x) {ε ∈ [0, ∞[ | (t, x) ∈ W(S0 (M, ε), S0 (N, ε))} where according to (Chentsov, 2017b), to each sets M ∈ F and N ∈ F we assign the sequence of sets (Wk (M, N ))k∈N0 : N0 → P(T × Rn ) and limit set W(M, N ) = Wk (M, N ) ∈ P(T × Rn ). k∈N0
Following (Chentsov and Khachay, 2018), we obtain
ε0 (t, x) inf(Σ0 (t, x)) ∈ [0, ∞[. Proposition 1. If (t∗ , x∗ ) ∈ T × Rn , then ε0 (t∗ , x∗ ) ∈ Σ0 (t∗ , x∗ ). As a corollary, we note that (t, x) ∈ W(S0 (M, ε0 (t, x)), S0 (N, ε0 (t, x))) ∀(t, x) ∈ T × Rn . (5) Let us introduce another option for functional on trajectories of the process. If t∗ ∈ T, x(· ) ∈ Cn ([t∗ , ϑ0 ]) and θ ∈ [t∗ , ϑ0 ], then we assume that ω(t∗ , x(· ), θ) = sup ({ρ((θ, x(θ)), M); max ρ((t, x(t)), N)}), t∗ ≤t≤θ
(6)
ω(t∗ , x(· ), θ) ∈ [0, ∞[. As a corollary, we define for t∗ ∈ T special payoff function γt∗ : Cn ([t∗ , ϑ0 ]) → [0, ∞[, (7) by following conditions: ∀ x(· ) ∈ Cn ([t∗ , ϑ0 ]). γt∗ (x(· ))
min ω(t∗ , x(· ), θ).
θ∈[t∗ ,ϑ0 ]
(8)
Minimum in (8) is attainable, since ρ((· , x(· )), M) and ρ((· , x(· )), N) - are uniformly continuous functions (see also (6)). Proposition 2. Let t∗ ∈ T, x(· ) ∈ Cn ([t∗ , ϑ0 ]) and ε∗ ∈ [0, ∞[ be given. Then, following two conditions are equivalent: ((i)) ∃ϑ ∈ [t∗ , ϑ0 ] : ((ϑ, x(ϑ)) ∈ S0 (M, ε∗ )) & ((t, x(t)) ∈ S0 (N, ε∗ ) ∀t ∈ [t∗ , ϑ]); ((ii)) γt∗ (x(· )) ≤ ε∗ We recall that M ⊂ N. If ε ∈ [0, ∞[, then S0 (M, ε) ⊂ S0 (N, ε). Hereinafter, we will consider special multivalued nonanticipating strategies as admissible control procedures. We define considered strategies in terms of non-anticipating operators on control-measure spaces. For this, it is helpful to recall some of the properties of measurable spaces, used in paragraph 2. We should improve some constructions given there. For arbitrary t ∈ T , let’s fix compactum Zt [t, ϑ0 ] × Q and σ-algebra Dt of Borel subsets of Zt .
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Wherein, for moments of time t1 ∈ T and t2 ∈ [t1 , ϑ0 ] Dt2 = Dt1 |Zt2 = {D ∈ Dt1 |D ⊂ Zt2 }. Further, if t ∈ T , then following holds: Γ × Q ∈ Dt ∀Γ ∈ Tt . Following those constructions, we consider the set of measures, which are similar to Borel mappings from [t, ϑ0 ] to Q, namely: Et {ν ∈ (σ − add)+ [Dt ] | ν(Γ × Q) = λt (Γ) ∀Γ ∈ Tt }; we use Et as set of generalized controls of player II. Moreover, we note that D×P {(t, u, v) ∈ Ωt | (t, v) ∈ D} ∈ Ct ∀D ∈ Dt . Therefore, we have the following (see (Chentsov, 2017a))
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From Proposition 4 we obtain (s)
ε0 (t, x) ∈ Σ0 (t, x) ∀(t, x) ∈ T × Rn ∀s ∈ N0 .
In particular, if (t, x) ∈ T we have non-empty subset [0, ∞[ is defined. Hereinafter, we assume (s)
(s)
ε0 (t, x) inf(Σ0 (t, x)) ∀(t, x) ∈ T ×Rn ∀s ∈ N0 . (13) Using (13) with each s ∈ N0 , we can define function (s)
Πt (ν) {η ∈ Ht | η(D×P ) = ν(D) ∀D ∈ Dt } ∀ν ∈ Et . (9) Thus, the family of programs for second player on a interval [t, ϑ0 ] was introduced. We also need additional operations for generalized controls, namely gluing and constriction. Let us recall another notation and define corresponding designation: if X and ˜ ∈ P (X), then Y - nonempty sets, where h ∈ Y X and X ˜ ˜ (h(x)) ˜ ∈ Y X . As X a family can be used and (h|X) x∈X as h — set function. If t1 ∈ T and t2 ∈ [t1 , ϑ0 ], then (see (Chentsov, 2017a)) Ct2 = Ct1 |Ωt2 = {H ∈ Ct1 | H ⊂ Ωt2 }, Ctt12 = Ct1 |[t1 ,t2 [×P ×Q = {H ∈ Ct1 | H ⊂ [t1 , t2 [×P × Q}, Dtt12
= Dt1 |[t1 ,t2 [×Q = {D ∈ Dt1 | D ⊂ [t1 , t2 [×Q}. In this case we have σ-algebras of sets. It is useful to note that (see (Chentsov, 2017a)) Ht2 = {(η | Ct2 ) : η ∈ Ht1 }, Et2 = {(ν | Dt2 ) : ν ∈ Et1 }. If t∗ ∈ T , then we assume that A˜t∗ is the set of all generalized multivalued non-anticipating strategies (see (Chentsov, 2017a)) for first player on a line [t∗ , ϑ0 ] : A˜t∗ {α ∈
ν∈Et∗
(s)
(s)
(16) ε0 ε0 ∀s ∈ N0 . Proposition 5. If s ∈ N0 and (t∗ , x∗ ) ∈ T × Rn , then (s) (s) ε0 (t∗ , x∗ ) ∈ Σ0 (t∗ , x∗ ). (s)
From (4) we have property ε0 ∈ Σ0 (t∗ , x∗ ).
Also, from (15) we show, in particular, that exact upper bound is defined properly. That is, (s)
sup({ε0 (t, x) : s ∈ N0 }) ∈ [0, ε0 (t, x)] ∀(t, x) ∈ T × Rn . Proposition 6. If (t∗ , x∗ ) ∈ T × Rn , then ε0 (t∗ , x∗ ) = (s) sup({ε0 (t∗ , x∗ ) : s ∈ N0 }) We consider another proposition which will establish rela(s) (s+1) (t∗ , x∗ ). tions between ε0 (t∗ , x∗ ) and ε0 n Proposition 7. Let (t∗ , x∗ ) ∈ T × R and s ∈ N0 . Then (s)
(s+1)
ε0 (t∗ , x∗ ) ε0 (s)
Dtθ∗ )
= {(η | Ctθ∗ ) : η ∈ α(ν2 )})}.
(10)
If α ∈ A˜t∗ and ν ∈ Et∗ , then as α(ν) we have nonempty subset of Πt∗ (ν); in particular, α(ν) ⊂ Ht∗ . Thus, we defined following union-set (see (Chentsov, 2017a)) ˜ t (α) α(ν) ∈ P (Ht ), (11) Π ν∈Et∗
(14)
(15) ε0 (t, x) ≤ ε0 (t, x) ∀(t, x) ∈ T × Rn ∀s ∈ N0 . Let us designate point-wise order in the set of all functions from T × Rn into [0, ∞[ by . Then from (15) we have
(s+1)
ε0 ε0
= (ν2 | Dtθ∗ )) ⇒ ({(η | Ctθ∗ ) : η ∈ α(ν1 )}
∗
ε0 : T × Rn → [0, ∞[. Also, from (13) and Proposition 4, we get
(t∗ , x∗ ).
(17)
Finally, we have following property:
P (Πt∗ (ν)) | ∀ν1 ∈ Et∗
∀ν2 ∈ Et∗ ∀θ ∈ [t∗ , ϑ0 ] : ((ν1 |
(12)
(s) × Rn and s ∈ N0 as Σ0 (t, x), (s) of [0, ∞[, thus, inf(Σ0 (t, x)) ∈
∗
which is the set of all generalized controls-measures, generated by non-anticipating strategy α. From Proposition 1 we have following: Proposition 3. If (t∗ , x∗ ) ∈ T × Rn , then sup γt∗ (ϕ(· , t∗ , x∗ , η)), ε0 (t∗ , x∗ ) = inf ˜t α∈A ˜ t (α) ∗ η∈Π ∗
˜ t (α wherein ∃˜ α∗ ∈ A˜t∗ : ε0 (t∗ , x∗ ) = Π ˜ ∗ ). ∗ Therefore it is established that function ε0 : T × R → [0, ∞[, is equal to minimax of payoff function γ in terms of non-anticipating strategies for any fixed position. Proposition 4. If s ∈ N0 and (t, x) ∈ T × Rn , then (s) Σ0 (t, x) ⊂ Σ0 (t, x).
∀s ∈ N0 .
(18)
According to Proposition 3 in (Chentsov and Khachay, 2018), ε0 is the minimax of special payoff in terms of nonanticipating strategies. 3. MAIN RESULTS In this section we construct program operator, which (s) (s+1) . will define for s ∈ N0 conversion from ε0 to ε0 To achieve this, we will use special type of construction, which is similar to one introduced in (Chentsov, 1978). The modification of program iterations method described in (Chentsov, 1978) corresponds to differential game with non-fixed moment of termination. Also, let us introduce new notations according to those in (Chentsov, 1978). First of all, we introduce the function ψ : T × Rn → [0, ∞[ by condition: (19) ψ(t, x) ρ((t, x), M) ∀(t, x) ∈ T × Rn . According to the properties of the distance function from a point to a nonempty set M we have that ψ ∈ C(T × Rn ), where C(T × Rn ) - set of all continuous real-valued functions on T × Rn . We note that (20) ψ −1 ([0, c]) ∈ F ∀c ∈ [0, ∞[.
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Thus, according to (20) and due to non-negativity of ψ, we have lower semi-continuity property. Subsequently, for every non-empty set H as R+ (H) we denote the set of all non-negative real-valued functions on H. In terms of (20) we define, according to (Chentsov, 1978), the following set M {g ∈ R+ (T × Rn )| g −1 ([0, c]) ∈ F ∀c ∈ [0, ∞[}. (21) Besides, we require set with point-wise order (22) Mψ {g ∈ M| g ψ}; where ψ ∈ Mψ . Thus, (22) is non-empty subset of M. 3.1 Convergence operator In this subsection, we construct special convergence operator for ε0 . For the sake of brevity, we skip the rigorous mathematical proofs of the presented results. Interested reader can find them in (Chentsov and Khachay, 2018). If g ∈ Mψ and (t∗ , x∗ ) ∈ T × Rn , then (since g ψ) we obtain for some c ∈ [0, ∞[ {g(t, x(t)) : t ∈ [t∗ , ϑ]} ∈ P ([0, c]) ∀v ∈ Q ∀x(· ) ∈ Xπ (t∗ , x∗ , v) ∀ϑ ∈ [t∗ , ϑ0 ], (23) where Xπ (t∗ , x∗ , v) is used in the sense of (Chentsov and Khachay, 2018). Following property (23), we have useful corollary: sup g(t, x(t)) ≤ c ∀v ∈ Q
t∈[t∗ ,ϑ]
∀x(· ) ∈ Xπ (t∗ , x∗ , v) ∀ϑ ∈ [t∗ , ϑ0 ]. (24) By (24), with g ∈ Mψ and (t∗ , x∗ ) ∈ T × Rn , we obtain ∃˜ c ∈ [0, ∞[: ¯ x ¯(t)); ψ(ϑ, ¯(ϑ))}) ∈ [0, c˜] sup({ sup g(t, x ¯ t∈[t∗ ,ϑ]
∀v ∈ Q ∀¯ x(· ) ∈ Xπ (t∗ , x∗ , v) ∀ϑ¯ ∈ [t∗ , ϑ0 ].
Then for g ∈ Mψ and (t∗ , x∗ ) ∈ T ×Rn , we obtain following sup
inf
v∈Q (x(·),ϑ)∈Xπ (t∗ ,x∗ ,v)×[t∗ ,ϑ0 ]
sup({ sup g(t, x(t)); t∈[t∗ ,ϑ]
ψ(ϑ, x(ϑ))}) ∈ [0, ∞[.
(25)
Considering (25), we define the convergence operator Γ : Mψ → R+ [T × Rn ] as follows: Γ(g)(t∗ , x∗ ) sup
inf
v∈Q (x(·),ϑ)∈Xπ (t∗ ,x∗ ,v)×[t∗ ,ϑ0 ]
sup({ sup t∈[t∗ ,ϑ]
g(t, x(t)); ψ(ϑ, x(ϑ))}) ∀g ∈ Mψ ∀(t∗ , x∗ ) ∈ T × Rn .
If g ∈ Mψ , (t∗ , x∗ ) ∈ T × Rn and v ∈ Q, then by h[g; t∗ ; x∗ ; v] we denote the functional (x(· ), ϑ) −→ sup({ sup g(t, x(t)); ψ(ϑ, x(ϑ))}) :
(k+1)
Theorem 8. If k ∈ N0 , then ε0
3.2 Fixed point property of operator Γ In the sequel, we consider an important property of operator Γ, namely its fixed point. Theorem 9. Function ε0 is the fixed point of operator Γ: ε0 = Γ(ε0 ). Let us consider the set of all fixed points of operator Γ, (Γ) namely Mψ {g ∈ Mψ | g = Γ(g)}. In regards to (Γ) ˜ (Γ) {g ∈ previous theorem, we obtain ε0 ∈ M . Let M ψ
(Γ)
(0)
ψ
(Γ)
Mψ | ε0 g} = {g ∈ Mψ | ρ(· , N) g}.
˜ (Γ) , Theorem 10. Function ε0 is the smallest element of M ψ ˜ (Γ) and ε0 g ∀g ∈ M ˜ (Γ) . in other words, ε0 ∈ M ψ ψ
Let us consider one special case of our setting. Namely, let (M T × M) & (N T × N ), (27) where M and N are closed non-empty sets in Rn with usual topology generated by euclidean norm · . Moreover, let M ⊂ N . If H ∈ P (Rn ) and x ∈ Rn , then we introduce ( · − inf)[x; H] inf({ x − h : h ∈ H}) ∈ [0, ∞[. (28) We obtain the following property: for H ∈ P (Rn ) and x∗ ∈ R n , ρ((t, x∗ ), T × H) = ( · − inf)[x∗ ; H] ∀t ∈ T. (29) Indeed, we fixate t∗ ∈ T . Then, ρ((t∗ , x∗ ), T ×H) = inf({ρ((t∗ , x∗ ), (t, h)) : (t, h) ∈ T ×H}). In addition, for h ∈ H, we obtain that (t, h) ∈ T × H, where t ∈ T , and x∗ − h ≤ ρ((t∗ , x∗ ), (t, h)). (30) Also, (t∗ , h) ∈ T × H and ρ((t∗ , x∗ ), (t∗ , h)) = x∗ − h . Wherein, ρ((t∗ , x∗ ), T × H) ≤ ρ((t∗ , x∗ ), (t∗ , h)). As a corrollary, ρ((t∗ , x∗ ), T × H) ≤ x∗ − h . (31) Since the selection of h was arbitrary, by (28) ρ((t∗ , x∗ ), T × H) ≤ ( · − inf)[x∗ ; H]. (32) On the other hand, from (30), we have that ˜ ∀t ∈ T ∀h ˜ ∈ H. ( · − inf)[x∗ ; H] ≤ ρ((t∗ , x∗ ), (t, h)) Then ( · − inf)[x∗ ; H] ≤ ρ((t∗ , x∗ ), T × H), and by (32), we obtain ρ((t∗ , x∗ ), T × H) = ( · − inf)[x∗ ; H]. Since the selection of t∗ was arbitrary, the property (29) is established. So, by (27) and (29), ∀(t, x) ∈ T × Rn (ρ((t, x), M) = ( · − inf)[x; M]) & (ρ((t, x), N) = ( · − inf)[x; N ]). Therefore, we obtain that
ω(t, x(· ), θ) sup({( · −inf)[x(θ); M]; max ( · −inf) t≤ξ≤θ
[x(ξ); N ]}) ∀t ∈ T ∀x(· ) ∈ Cn ([t, ϑ0 ]) ∀θ ∈ [t, ϑ0 ]. As a corollary, in considered case (27), we obtain
t∈[t∗ ,ϑ]
γt (x(· )) min sup({( · − inf)[x(θ); M];
Xπ (t∗ , x∗ , v) × [t∗ , ϑ0 ] → [0, ∞[; (26) also, we consider following Lebesgue sets: for b ∈ [0, ∞[ Yb [g; t∗ ; x∗ ; v] {(x(· ), ϑ) ∈ Xπ (t∗ , x∗ , v) × [t∗ , ϑ0 ]| h[g; t∗ ; x∗ ; v](x(· ), ϑ) ≤ b}.
(k)
= Γ(ε0 ).
θ∈[t,ϑ0 ]
max ( · − inf)[x(ξ); N ]}) ∀t ∈ T ∀x(· ) ∈ Cn ([t, ϑ0 ]).
t≤ξ≤θ
Thus we have natural functional, which defines quality of pursuit. 2356
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CONCLUSION In this paper, we have considered a relaxation of the non-linear zero-sum pursuit-evasion differential game. For player I, we have constructed the optimal strategy, namely, we have defined minimax function, which values are represented as guaranteed result for special type of payoff. Then, we have constructed special iterative procedure in space of positions by finding convergence operator Γ and proving that minimax function is its fixed point. REFERENCES Chentsov, A. (1976). On a game problem of guidance with information memory. Soviet Math. Dokl., 17, 411–414. Chentsov, A. (1978). On the game problem of convergence at a given moment of time. Math. USSR-Izv., 12(2), 426–437. doi:10.1070/IM1978v012n02ABEH001985. Chentsov, A. (2017a). The program iteration method in a game problem of guidance. Proceedings of the Steklov Institute of Mathematics, 297(1), 43–61. doi: 10.1134/S0081543817050066. Chentsov, A. (2017b). Stability iterations and an evasion problem with a constraint on the number of switchings (in russian). Trudy Inst. Mat. i Mekh. UrO RAN, 23(2), 285–302. doi:10.21538/0134-4889-2017-23-2-285-302. Chentsov, A. and Khachay, D. (2018). Relaxation of the pursuit-evasion differential game and iterative methods (in russian). Trudy Inst. Mat. i Mekh. UrO RAN, 24(4), 246–269. doi:10.21538/0134-4889-2018-24-4-246-269. Elliott, R.J. and Kalton, N.J. (1972). Values in differential games. Bull. Amer. Math. Soc., 78(3), 427–431. Isaacs, R. (1969). Differential games: Their scope, nature, and future. J. Optim. Theory Appl.,
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