Problems of group pursuit with integral constraints on controls of the players. I
- PDF / 150,192 Bytes
- 12 Pages / 594 x 792 pts Page_size
- 10 Downloads / 187 Views
PROBLEMS OF GROUP PURSUIT WITH INTEGRAL CONSTRAINTS ON CONTROLS OF THE PLAYERS. I UDC 518.9
B. T. Samatov
Abstract. The paper studies problems of group pursuit for linear differential games with integral constraints. The problems are analyzed on the basis of Chikrii’s method of resolving functions. The proposed method substantiates the parallel approach strategy, i.e., the Ï-strategy. The new sufficient solvability conditions are obtained for problems of group pursuit. As an example, two classes of problems are considered, namely, the Pontryagin control example and a group pursuit with a simple motion for the case of “l-catch.” Keywords: problem of group pursuit, integral constraint, resolving function, strategy, guaranteed time of pursuit. 1. PROBLEM STATEMENT Consider a linear differential game in a finite-dimensional Euclidean space, described by the system of equations z& i = A i z i + B i u i - C i u, z i ( 0) = z i0 ,
(1)
where z i Î R ni , u i Î R pi , u Î R q ; n i ³ 1, p i ³ 1, q ³ 1, i = 1, m is the set of integer numbers from 1 to m; A i , B i , and C i are constant rectangular n i ´ n i , n i ´ p i , and n i ´ q matrices, respectively; z i0 is the initial state of the ith object; u i is the control parameter of the ith pursuer; u is the control parameter of the evader. The realizations of the parameters u i , i = 1, m, and u at the end of the game should be measurable functions from the class L p [ 0, T ], p > 1, and satisfy the constraints
T
ò | ui (t ) |
p
dt £ r i , r i > 0, i = 1, m,
(2)
0
T
ò | u( t ) |
p
dt £ s , s ³ 0 ,
(3)
0
respectively, where T > 0 (the case T = +¥ is not excluded). In what follows, we will call such controls admissible and will denote their sets by UTi , i = 1, m , and VT , respectively. The terminal set consists of the union of sets M 1 , M 2 ,KK, M m , each having the form M i = M i0 + M i0 , where M i0 is a linear subspace from R ni and M 1i is a convex compact subset of the orthogonal complement Li to the subspace M i0 in R ni . Definition 1. In game (1)–(3) the set of mappings u i : [ 0, T ] ´ VT Þ UTi , i = 1, m , is called a strategy of the group of pursuers if the following conditions are satisfied: (i) admissibility: for each u( × ) ÎVT the inclusion u i (× ) = u i ( × , u(× )) ÎUTi , i = 1, m , holds; (ii) the property of being Volterrian: if for any t Î [ 0, T ], u1 (× ), u 2 (× ) ÎVT the equality u1 ( t ) = u 2 ( t ) holds for almost all t Î[ 0, t ] , then u1i ( t ) = u i2 ( t ) almost everywhere on [ 0, t ] , where u1i (× ) = u i ( × , u1 (× )) , u i2 (× ) = u i ( × , u 2 (× )) . Namangan State University, Namangan, Uzbekistan, [email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 5, September–October, 2013, pp. 132–145. Original article submitted November 26, 2012. 756
1060-0396/13/4905-0756
©
2013 Springer Science+Business Media New York
0 Definition 2. In game (1)–(3) from the initial position z 0 = { z10 , z 20 ,... , z m } it is possible to complete the pursuit in
time T = T ( z 0 ) if at least for one value of i, i =
Data Loading...
