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Linear Differential Games of Pursuit with

Integral Block of Control in its Dynamics

Kirill A. Chikrii, Anna V. Chikrii

January 3, 2013

Abstract

The game problem of bringing a trajectory of dynamic system to the terminalset, which has cylindrical form, is treated. Here the case is analyzed, whencontrols enter the system equation in integral form. Sufficient conditions forthe game termination in some guaranteed time are derived on the basis of theMethod of Resolving functions. The result is supported by a model example(see section ??) and is compared with the game simple motions.

1 Problem Statement

We consider the dynamical process of the form

dzdt =

A (t) z (t) +

tt0

B (t, s) (u (s) , (s)) ds, z (t0) = z0, (1)

evolving in condition of conflict. Here the phase vector z takes its values in thefinite-dimensional Euclidian space Rn; A (t) is n square matrix, continuouslydepending on t, t t0, and B (t, s) , t0 s t < is a matrix function, con-tinuous in all its variables. The block of control is defined by function (u, ),continuous on the direct product of compacts U and V. The pair (t, z) will bereferred to as a current state of the process, and (t0, z0) as its initial state.As admissible controls the players employ Lebesgue-measurable functions u (s)and (s) with values in the sets U and V respectively. By virtue of the aboveassumptions function (u, ) satisfies the condition on superpositional measur-ability and (s) = (u (s) , (s)) is a bounded measurable function.

In addition, the terminal set, having cylindrical form, is given:

M = M0 + M, (2)

Here M0 is a linear subspace in Rn and M is a convex compact from L,

which is the orthogonal complement to M0 in Rn.

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One can easy see that in the case B (t, s) = (t s) E, where (t s) is-function and E is a unit matrix, the conflict-controlled process (??), (??)

reduces to ordinary differentional game [?, ?, ?].We study the problem of bringing a trajectory of system (??) to the terminal

set (??) in a some guaranteed time. In so doing, the first player (u) employsquasi-strategies, that is, at each current instant of time he constructs his controlin the form

u (t) = u (t0, z0, t ()) ,

where t () = { (s) : s [t0, t]}, for any control of the second player [?, ?, ?].

2 Lemma

For any chosen admissible controls of the players solution of system (??

) maybe presented in the form

z (t) = (t, t0) z0 +

tt0

C(t, s) (u (s) , (s)) ds, (3)

where C(t, s) =ts

(t, ) B (, s) d, and (t, t0) is the fundamental matrix of

homogeneous system (??).Proof. From formula Cauchy [?] as applied to system (??) it follows

z (t) = (t, t0) z0 +

t

t0

(t, )

t0

B (, s) (u (s) , (s)) dsd.

Then, using Fubini theorem [?] we have

z (t) = (t, t0) z0 +

tt0

ts

(t, )B (, s) d

(u (s) , (s)) ds,

whence follows formula (??).Denote by the orthoprojector, acting from Rn onto L. Let us study the

set-valued mappings

W (t,s,) = C(t, s) (U, ) ,

W (t, s) =V

W(t,s,) ,

where (U, ) = { (u, v) : u U}, t s t0, V.

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In the sequel, Pontryagins condition is assumed to hold

W(t, s) = (t, s) = {(t, s) : t0 s t < }. (4)

By virtue of the assumptions on parameters of process (??), (??) and con-dition (??) the mapping W (t, s) has at least a single measurable selection(t, s) [?]. Fix it and set

(t , z, (t, )) = (t, t0) z +

tt0

(t, s) ds.

Let us introduce the resolving function by the formula

(t,s,z,,(t, )) = sup { 0 : [W(t,s,) (t, s)][M (t , z, (t, ))] = },(5)

Define the set-valued mapping

T (t0, z , (, )) =

t t0 :

tt0

infV

(t,s,z,,(t, )) ds 1

. (6)

The properties of similar functions are thoroughly studied in [?]. We onlynote that T (t0, z , (, )) = , if inequality in (??) fails for finite t t0.

3 Theorem

Let for the game problem (??), (??) Pontryagins condition hold.

Then, if a measurable selection (t, s) W(t, s) , (t, s) exists such thatT T (t0, z0, (, )) = then a trajectory of the process (??) may be broughtin a finite time from the initial state (t0, z0) to set (??). In so doing the firstplayer employs quasi-strategies.

The proof is conducted by the scheme, presented in [?].By way of illustration below is given a simple example.

4 Model Example

Let A (t) 0, B (t, s) E, (u, ) = u , M = {0}, U = aS, a > 1, V = S,where S is a square ball in Rn, centered at the origin. Set t0 = 0.

Thus, a trajectory of the process

dz

dt=

t0

(u (s) (s)) ds, u aS, S,

should be brought in a finite time into the origin.

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In our case M0 = {0} and M = {0} therefore L = Rn and the orthoprojector is an operator of identical transform and defined by the unit matrix. In the

turn, since A (t) 0 then (t, 0) = E.Then C(t, s) = (t s) E and the following presentation for the set-valued

mapping is true

W (t,s,) = (t s) (aS ) ,

W (t, s) = (t s) (a 1) S.

Therefore Pontryagins condition holds if a 1 and (t, s) . Since0 W (t, s), (t, s) then we can pick (t, s) 0. From formula (??) wededuce that function (t,s,z,, 0) appears as the greatest root of the quadratioequation

(t s) z = (t s) a

and has the form

(t,s,z,, 0) = (t s) (z, ) ,

where

(z, ) =

(z, ) +

(z, )

2+ z

a2 2

z2.

Minimum of function (z,) in is furnished by the element =

z

z .Then

min1

(t,s,z,, 0) = (t s)a 1

z

and therefore

t = min {t 0 : t T (0, z, 0)}

is a root of the equation

t

0

(t s)a 1

zds = 1.

Thus,

t =

2z

a 1

12

.

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Note [?] that in the case of simple motions

dzdt

= u , u aS, S,

the time of hitting the origin is given by the expression

t =z

a 1.

One can easy see that times t and t differ essentially.

5 References and Notes

References

[1] N. N. Krasovskii. Game Problems on Motions Encounter, Moscow, Nauka,1970.

[2] L. S. Pontryagin. Selected Scientific Works, Vol. 2, Moscow, Nauka, 1988.

[3] A. A. Chikrii. Conflict-Controlled Processes, Dordrecht/Boston/London,Kluwer Academic Publishers, 1997.

[4] I. P. Natanson. Theory of Functions of Real Variable, Moscow, Nauka, 1974.

[5] A. D. Ioffe, V. M. Tikhomirov. Theory of Extremal Problems, Moscow,Nauka, 1974.

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