Pursuit problem in distributed control systems
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PURSUIT PROBLEM IN DISTRIBUTED CONTROL SYSTEMS M. Sh. Mamatova† and M. Tukhtasinova‡
UDC 517.95
A pursuit problem in distributed parabolic control systems without mixed derivatives with variable coefficients is considered. The finite-difference method is used to solve this problem. Necessary conditions for the termination of a pursuit are obtained. Keywords: pursuit, pursuer, evader, terminal set, pursuit control, evasion control.
We consider the distributed control system described by the parabolic equations z t - Az = -u + u,
z t = 0 = f ( x ),
z S = 0,
(1)
T
where z = z ( x, t ) is an unknown function, x = ( x1 , x 2 , K , x n ) Î W Ì R n , n ³ 1,
W = {x Î R n : 0 £ x1 £ 1, 0 £ x 2 £ 1, K ,
0 £ x n £ 1} is an n-dimensional unit cube with faces parallel to the coordinate planes, t Î[ 0, T ] , T is an arbitrary positive constant, n ¶ æ ¶z ö çç a a ( x ) ÷, Az = å ¶x a ÷ø a = 1 ¶x a è and a a ( x ) is a continuous bounded function in W. It is assumed that there is a positive constant n such that, for some arbitrary x ÎW, the following inequality is fulfilled: a a ( x ) ³ n , a = 1,2, K , n, u = u( x, t ), and u = u( x, t ) are control functions from the class L2 (QT ) , QT = {( x, t )| x Î W , t Î ( 0, T )} is an open cylinder in R n+ 1 , f ( x ) Î L2 (W ), S T = {( x, t )| x Î ¶W, t Î ( 0, T )} is the lateral surface of the cylinder QT , and ¶W — is the boundary of the domain W that is considered to be piecewise-smooth. The function u( x, t ) is used by the first player (pursuer), and the function u( x, t ) is used by the second player (evader), u Î P , u ÎQ, and P and Q are nonempty compacts in R 1 . A terminal set M 1 Ì R 1 is selected. Next, we recall that W21 (W ) is a Hilbert space consisting of elements L2 (W ) that have generalized first-order 0
derivatives quadrically summable in W, W 21 (W ) is a subspace of W21 (W ) in which the set of all smooth compactly supported functions is dense, W21,0 (QT ) is a Hilbert space consisting of elements of the space L2 (QT ) that have generalized derivatives 0
z xa , a = 1,2, K , n, quadratically summable in QT , and W 12,0 (QT ) is a subspace of W21,0 (QT ) in which smooth functions equal to zero in the vicinity of S T form a dense set. It is well known [1, 2] that, under the above conditions, problem (1) has a unique generalized solution z = z ( x, t ) in 0
the class W 21,0 (QT ) for any u( x, t ), u( x, t ) Î L2 (QT ) and f ( x ) Î L2 (W ) . In most cases, the obtaining of sufficient conditions of pursuit termination for problem (1) is impossible. In this connection, approximate methods of its solution assume a great significance. The mostly used method of numerical solution of problem (1) is the finite-difference method. In this work, the finite-difference method is applied to the solution of the pursuit problem described by equations of the form (1) with distributed parameters. a
National University of Uzbekistan, Tashkent, Uzbekistan, †[email protected]; ‡[email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 153–158, March–Apr
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