### Journal Menu >> American Journal of Computational Mathematics, 2013, 3, 56-61 doi:10.4236/ajcm.2013.33B010 Published Online September 2013 (http://www.scirp.org/journal/ajcm) Differential Games of Pursung in the Systems with Distributed Parameters and Geometrical Restrictions M. Sh. Mamatov, E. B. Tashmanov, H. N. Alimov Department “Geometry”, National University of Uzbekistan Named After M. Ulugbek, Tashkent, Uzbekistan Email: mamatovmsh @ mail.ru Received 2013 ABSTRACT A problem of pursuit in the controlled systems of elliptic type without mixed derivativ es with variable coefficients was considered. The model of the considered system is described by partial differential equations. The players (opponents) control parameters occur on the right-hand side of the equation and are subjected to various constraints. The first player’s goal is to bring the system from one state into another desired state; the second player’s goal is to prevent this from happening. We represent new sufficient conditions for bringing the system from one state into another. The fi-nite-difference method is used to solve this problem. Keywords: Pursuit; Pursuer; Evader; Terminal Set; Pursuit Con trol; Evasion control 1. Introduction Some problem formulations in the theory of differential games may be illustrated by motion of two controlled objects, pursuer and evader. Let in the course of motion the objects continuously observe each other and at each time instant correct their motions depending on the in-formation about the adversary. Depending on the pur-suer’s aim, the problem of pursuit is then formulated as follows: using the information about the evader, at each time instant t select a cont rol such that coincid enc e of the objects’ spatial coordinates is reached as soon as possi-ble. The majority of studies consider the case where be-havior of the lumped-parameter model described by a system of ordinary differential equations. This scheme encompasses many problems of differential games aris-ing in diverse filed of the natural sciences. The mathe-matical issues of the differential games describing the lumped-parameter systems were developed in detail. In many applications, however, the lumped-parameter models describe phenomena inadequately. It often turns out that a system which is optimal in the sense of a sim-plified model does not use the additional designed-in potentialities of control. The distributed-parameter mod-els obeying the differential equ ations with partial deriva-tives offer a better, more adequate description. Use of these equations also gives rise to various game problems of which one is the subject matter of the present paper. It focuses only on the problem of pursuit. Therefore, we make an assumption about the nature of information for this problem. 2. Formulation of the Problem The operated distributed system described by the elliptic equations (see, for example, [1,2]) is considered 22 22(,) /(,) /((,),(,))axy zxbxy zy fuxyxy , (1) /(,)(,zxyzx)y , (,)xywhere (,)zzxy – unknown function, , – continuous functions in with border (,)axy):0 1,x(,)bxy0 1}y {( ,xy, (,)xy – smooth function on ,  – external normal. It is supposed that there is a positive constant  such that for any the inequal-ity, (, )xy(, )ybx,(, )uuxy, (,)xy2()L – operating func-tions is executed from a class . The first (pursuing) player, (pursued or escaping) the player, uP, Q, P and Q – nonempty compacts in disposes of function 1R(, )xy second of function . The ter-minal set (, )uxy11MR s allocated. Definition 1. In a task (1) it is possible  – comple-tion of (0) prosecutions from “boundary” situation (,), if exist function (,,)uxyP, Q, (, )xy, such that for any function 0(,)xyQ, (, )xy0((,),xy the solution of a task (1) where 0zx(,)t,)xyu u, 0(,)xy, gets on a set 1IM, at some (, )xy , 01(, ):(, )xyzxy  IM where (1I,1). Decompose the Euclidean space of variables 2R(, )xy by the planes ixih, , and 1/hr0,1, ,ijyjl, 1/l, 0,1, 2,,j into parallelepipeds Copyright © 2013 SciRes. AJCM M. Sh. MAMATOV ET AL. 57(, ){(,) :(1),(1) }ij ixyih xihjlyjl , and r being some natural numbers. The points (, )ijxyhl belonging to a set are the nodes of the grid . Each node has its neighbors. If all these neighbor nodes also belong to the grid, then the node hl(, )ijxy is referred to as “internal”, otherwise, (, )ihjl(, )ijxy21(),h22(),/,,,, ,, ,ij is called the “boundary” node. The set of all boundary nodes is called as border of net area and is designated through . h21121122 1122 1/(,)((,)(,))/2 (),/(,)((, )(, ))/2(),/ (,)((,)2(,)(,))// (,)((,)2(,)(,))/iji ji jij ijijiji jijijijijij ijzxxyzx yzx yhOhz yxyzxyzxylOlz xxyzxyzxyzxyzyx yzx yzx yzx yl    ,ijz,1,,1,,,1 ,,1(2)/ (20,1,,1; 0,1,,1.ijijij ijijijijijazz zhbzz zirj (, ):i jijReplace the internal nodes of the derivatives (1) dif-ferential second-order accuracy of approximation ratios with formulas 2OOlijf,ij ijh22)l, Substituting these ratios in (1), having rejected an error of approximation of derivatives, we will receive the dif-ferential equations for unknown (2) where the following designations of values of coeffi-cients and the right part in a hub xya,,,ij ijb dc gf,((,),(,)), (,)ijijijijhffuxy xyxy, for example are entered l,ijz,ijz1,0,0, 0,1,0,,1,,1,, ,,1,0,0 ,1,0,0,,1 ,,,1,()/()/2, (0,1,()/()/2,()/()/2, (0,1,2,()/()/2jj jjjjrjr jrjrjrjrjii iiiiii iiiizz hzzjzz hz zzzl zzizz lzz     2(r21)1)r Ratios (2) contains except unknown in internal nodes also unknown on border of net area. For boundary nodes we will write down a ratio ,,,)(3) Thus, we will receive system of r,1iz,ijz,iz equa-tions with the same number of unknown . Using boundary conditions (3), we will express ,  through , ,0iz,1iz,1,0,0 ,0,0,0,,,,1,(2)/(2)2 (2(2)/(2)2 (2ii iiiiiiiizl lzllzl lzl. Let's have ,).ii)l z z (4) Using these ratios, we will exclude in system (3) un-known ,1i, ,i. If to enter designation 2/hl21,00 ,0,11,0,01, ,1, ,11,,1, 1,2,, 11, 1,(2 2),2(1)(1,2,, 2),(2 2)(0,1,2,,1),iiiiiij ijij ijijijii iiizkzzzzz zzzFjzz kzzir     , we will receive system 1iiFF  (5) where 22,0 0,0,0,2,1 ,1,,2/(2); (1,2,, 2);2(2).ii iiijiii ii,jFhfllF hfjFhf l l   (6) This system can shortly be written down in a look 11, (0,1,2,,1).iiii izAzzFi r  (7) where ,0,1, 1,0,1, 1(,,,); (,,,)iiiii iiizzzzFFF F， ,0,2(1 )0002(1 )0002(1)00000 2(1)iiikAk (8) Boundary conditions (3) and (5) can be copied in a look 1,0,0,0,0,0,0, 0,0,1, , ,,, ,,, ,(2)/2(2 )/(2) (0,1,2,,1)(2 )/(2 )(2)/(2 ) (0,1,2,,1)jjjjjjjj jr jrjrjrjrjrjrjrjrjzh hzhhkz yjzh hzhhkz yj    ，， (9) where 0, 0, 0,0,0, 0,,,,,,(2)/(2) (2)/(2)(2)/(2) (2)/(2).jjjjjjrjrjrj rjrjrjkhhyh hkhhyh h,   ；；； (10) Having put 00,00,1 0,1,0,1,10,00,100, 1,0,11,1(,,,); (,,,).000 00000;000 0000 00000,000 0rrr rrrryyyyyyy ykkXkkkXk  (11) it is possible to write down systems (9) in such look: 100011,.rrzXzyzXzyr (12) Finally we have the following system of the equations: 10001111 (1,2,,1)iiiiirrrzXzyzAzzF irzXzy ，。 (13) Instead of game (13) we will consider more the gen-eral game described by system of the equations 0001 0111(,) 11nnnnnnn nnNNNN NCz BzfAzCzBzfunNAzCz f ，，，， (14) where mnzR, 0,nN, ,,nnnACB – constant square matrixes, mm,nnu – operating parameters, – nuCopyright © 2013 SciRes. AJCM M. Sh. MAMATOV ET AL. 58 prosecution parameter, n – beanie parameter, nnuP pR, qnnQR , n and – nonempty sets; nPnQf – the set function displaying pqRR in . Besides, in the terminal set is mRmRM allocated. Definition 2. We shall say that from “boundary” situa-tion 0(, )Nff it is possible to complete pursuit for steps if from any sequence N12 1N,,, of the values of evasion controls it is possible to construct a sequence 121 of values of the pursuit control values such that the solution ,, ,Nuuu012,z1{,,,, }NNzzz z of the equation 00111nnNNBz010 (,) 11nnnnn nnNN NCzzfAzCzBf uAzCz f ，，，.n N (15) Gets on :iMzM for some . Thus for finding of value inu it is allowed to use values n and nz. Note that the type of systems (14) is difference schemes for elliptic equations of second order with variable coef-ficients in any field of any number of dim ensions [3-14]. Solution of problem (14) will be sought in the form 11 , 1, 0,nnnzz nN1n2, ,N (16) where 1nmm – uncertain while a square matrix of the sizes , and 1n – a vector of dimension m. From a formula (16) and the equations of system (14) for there are recurrent ratios for calculation of matrixes 11nN n and vectors n. Really from a formula nnz(16) 1nzn substituting it in (14) we will receive ].nn1111() ,),11;()(, ;()(),)nnn nn nnnnnnnn nnnnnnnnn nnnnAzzBz fnNCA zzfuzCABz CAfA 1nn nnCB()[(nnnnuAu  1()()1, 2,nnn2,,)nnNA Equating now the right parts of the last and (16) equali-ties we will receive 111, 1,[(,, .nn nnnnnnnCAB nCA funN1;],   Further from (16) and the equations (14) for 0,nN, there are the initial values 1, 1 and , allowing beginning the account on recurrent ratios. From (14) and (16) for we will have Nz11,0n11100zC00 00, zCBfzz110CB0 And, therefore 11100, .Cf In the same way for we have nNN()NNNNN NAzCzfNAN or 1()(NN NNNzC fA). Uniting, we will write out final formulas 11110(), 1,2,,, nnnnnCAB nNCB0  (17) 111100()((,)1,2,,1. ,nnnnnnnnnCA fuAnNCf),  (18) 11 11(, ),1,2,,0, nnn nnnNNzzunN Nz  (19) It is clear, that if in game (17), (18), (19) nzM that in game (14) too game comes to the end. Therefore fur-ther instead of game (14) we will consider discrete game described by system of th e equations (17), (18), (19). Before giving determination of stability of algorithm (17), (18), (19), we will provide some data from linear algebra. Let A – any square matrix and ||mm||mx be norm of a vector in , then the norm A is defined by equal-ity mR0||||sup||||/|||| .mmxAAx x For a case of Euclidean norms in we have mR|| ||A, where  – maximum on the module own value of a matrix AA. Without the proof we will give the following known lemma (see ). Lemma 1. Let for some matrix norm the square matrix meet a condition || ||1Aq. Then there is a matrix 1()EA and || 1)|| 1/(1EA( )q. Let's say that the algorithm is steady if the assessment |||| 1j for 1jN is carried out. Lemma 2. If jC for 0jN – no degenerate ma-trixes and jA and jB – nonzero matrixes for 1j 1N also are satisfied conditions 110011||||1, ||||1,||||||||1, 11.NNjj jjCB CACA CBjN And at least in one of inequalities the strict inequality takes place, there are return to the .jjjCA matrix and |||| 1j, here 1100СB, 11(), 1jjjjjCA BjN1. Proof. 1100|||| |||| 1CB, suppose, that |||| 1j also we will show 1|||| 1j. After a course the proof of this fact we will receive existence of a matrix 1()jjjCA. Really from conditions of a lemma we will have 11 11||||||||||||||||1 ||||1.jjj jjj jjjjCACACA CB   As 1jjjCA square matrix that owing to a lemma 1 there are return to 1jjjECA and jjjCA matrixes and 1||)|| 11/|| ||(jjjj jCBECA. From here and from (17) we will receive 1111111|||| ||()||||()|| ||||1.jjjjjjjjjjECA CBECA CBj The proof of the lemma is complete. Copyright © 2013 SciRes. AJCM M. Sh. MAMATOV ET AL. 593. Main Results Everywhere further it is supposed that 01MMM , where 0M – linear subspace , 1mRM – a subset a subspace, – orthogonal complement of 0LM in . Denote we will designate a matrix of orthogonal design from on . nRmRLLet , (0){0}W11011()( ,)()(), 1.NkiNkikNkNki NkiNkiNkiiQWk PWkWkkN     (20) Theorem 1. Let N be the smallest of the numbers k, such that 1().NkNkN NzWk1 (21) Then from “boundar y” situation 0(, )Nff it is possible to complete pursuit for N steps. Let now , 21(0)WM111122 1122 111(1)[(0)(,)],()[ (1)(,)]Nk NkNNNk NkNkQNk NkNNNNQWWPWk WkP   1 (22) Theorem 2. If N be smallest of those numbers, for each of which takes place inclusion k()1zWkNkNkNN 2  (23) that of “boundary” situation 0(, )Nff it is possible to complete pursuit for N steps. Let 101 10(),,,:0,1kkkii i], and 1110313() (())(,) 0,(0), ()(()), 0.Nki MkikkkiNkNki NkiNkiNkiiQkWMPkNWMWkW kN  (24) Theorem 3. If 1M – a convex set and N be small-est of those numbers . For each of which inclusion takes place k1().NkNkN NzWk3 (25) That of “boundary” situation 0(, )Nff it is possible to complete pursuit for N steps. It is easy to be convinced  that the solution of differential task (2) meets to the solu tion ,ijzz of an initial task (1), the following assessment of speed of conver-gence takes place 2,12||( )||,hlhli jzz KhKl2 (26) where – values of the exact decision a task (1) in grid functions, ()hlzhl – spaces of net functions, || ||hl – is its norm and, 1K and 2K constants. Theorem 4. Let in an inequality (26) 2212Kh Kl, and in game (13) from a “boundary” situation 0(, )Nff 0(, )Nyy/(zxcompletion of prosecution that is definitions 2 be possible. Then in game (1) fro m “boundary” situation ,)(,yzx)y , it is possible to complete pursuit that are definitions 1. (, )xy4. Proof of Theorem Proof of Theorem 1. Let 12 1,,,N, iQi, 1iN1 – any sequence. Instead of inclusion (21) we will consider other inclusion equivalent to it 1111111(1)(,NNNkNkN NNk NkNNNNQzWkP1)  Means, exists 1Na 11111(, )NNNNkNkNNNQaP11.N Such that 11(1) .NkNkN NNzWk a1  (27) Now control of the pursuing player 1Nu, the relevant control of the escaping player 1N, we will construct as the solution of the following control 1111(, )Nk NkNNNNNua1.  It is clear, that the equation has the decision. From here owing to (27) we have 1111(1)(,)NkNkN NNk NkNNNNzWk u  11  We write down this inclusion in other look. 11 111[(,)](1)NkNkNNNNNNzu Wk.  (28) As a result from equalities (18) and (28) we will re-ceive 1111(1)Nk NkNNzWk (29) Done above a reasoning allow us to construct on the set control 1N providing inclusion (29). If now the control 2N becomes known that, we above can receive in the stated way control 1Nu providing inclusion 1221(2).Nk NkNNzWk Repeating this process, further we can construct step by step control iu, proceeding from becoming known controls i, therefore, that in any step inclusion takes place 111(0).NkzW It means that Copyright © 2013 SciRes. AJCM M. Sh. MAMATOV ET AL. 60 1Nkz As we set out to prove. Proof of Theorem 2. Let 121,,,N, iiQ, – any sequence. For concrete 1iN 11N owing to (22) and (23) we will receive inclusion 12111(1)(, )NkNkN NNk NkNNNNzWkP 1  (30) Now as 1Nu we take that element from 1NP for which inclusion (30) remained. Then we will receive 12111(1)(, )NkNkN NNk NkNNNNzWku 1  From this it follows that 11 112[(,)](NkNkNN NNNNzu Wk1). And therefore, owing to (19) we have 1112(1).Nk NkNNzWk If now the control 1N becomes the stated way known that we above us can constru ct control 1Nu pro-viding inclusion 1222(2).Nk NkNNzWk Further arguing similarly in any step we will receive 12 1(0) ,NkzW that is 1.Nkz The theorem is proved completely. Proof of Theorem 3. Instead of inclusion (25) mean-ing (24) we will consider inclusion equiv a lent to it 1(()).NkNkN NzW  Existence101 10(),,,,0,1kkiii   follows from (24). From here follows 11210111111 1[(,[(,)Nki NkiNNkNkNkN NiQiNkNki NkiNkiNkikNkNNNNQzPP )]]. (31) Let now 12 1,,,N, iiQ, – any se-quence. Owing to (31) exists such that 1iN 1Na111111 1121011[[(NNNki NkiNkNkNNNQkNkNkN NiQiNkNki NkiNkiNkiNPzPa  11(,)],,)]N (32) Therefore, controls 1Nu we will construct as the so-lution of the following equation 111111 1,11(,), .kNkNNNNNkmuam  Further owi n g t o (32) we have 2101111111,1[((, ).Nki NkikNkNkN NiQiNkNki NkiNkiNkiNkNN NNkkzPum ,)] It is equivalent to the fo llowing 11 1111,12110[(,)][(Nki NkiNkNkNN NNNNkkkiNkNki NkiNkiNkiQizu mP  ,)  Therefore owing to (32) we have 11111,12110[(Nki NkiNk NkNNkkkiNkNki NkiNkiNkiQizmP  ,)]. In the same way, if the control 2N becomes the stated way known that we above us can construct con-trols 2Nu providing inclusion 31 221122011[(,NkiNkikNk NkNNkkkkQiiNkNki NkiNkiNkizmmP )] etc. Thus, we will receive 31 221122011[(,NkiNkikNk NkNNkkkkQiiNkNki NkiNkiNkizmmP )] from here we receive 1.Nkz The theorem is proved completely. Proof of Theorem 4. Let in game (13) one be able to complete the pursuit from “boundary” situation 0(, )Nff0(, )Nyy in steps. Then, it follows from Defini-tion 2 that from any sequence N01, ,...,1, ,NkQ 0kN1, of the evasion control it is possible to con-struct a sequence 01 1, ,...,,,Nkuuuu P of pursuit control such that the solution 01kN,01 1( ,,...,,)NNzzz z of the equation 100zXzy0, , 11nnnzzF1nnnzA 1N, 11zNzXNNy, for some hits dN:dMz M. Let now in game (2) (,),(, )xy Q xy 2(), be an arbitrary control of an evader from the class L. With the knowledge of the evader control (, )xy, it is possible to determine ,ik as the values of this func-tion at the node points of the grid , that is, hl1, 2,1,( ,,...,).kkk krk  Whence it follows that in virtue of Theorem 4 we can construct the pursuer control in game (13) providing completion of pursuit 1, 2,1,( ,,...,).kkkkrkuu uuu Now in game (2) we construct the pursuer control (, )uuxy as follows: ,,(, ){:(1),iki kiuxyuuih xih 0,1,...,1,(1) ,0,1,...,1}irklyklk uP. Obviously,  and 2(, )()uxy L. By substituting (,)xy and (, )yuux in (2), we obtain a differen-Copyright © 2013 SciRes. AJCM M. Sh. MAMATOV ET AL. Copyright © 2013 SciRes. AJCM 61 O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural’tseva, “Lineinye I Kvazilineinye Uravneniya Parabolicheskogo Tipa,” (Linear and Quasi linear Func-tions of Parabolic Type), Moscow, Nauka, 1967. tial equation. Similarly, by substitu ting ,ik and ,iku in (3), we obtain a grid equation approximating equation (2). Let ()hlz be the value of the exact solution corre-sponding to the controls (, )xy and (, )uuxy of problem (2) at the nodes of the grid,,hli kz be the solu-tion corresponding to the controls ,ik and ,iku of the difference problem (3). Then, we obtain from (13) and the condition of Theorem 4 that  V. A. Il’in, “Boundary Control of String Oscillations at One End with Other End Fixed, Provided that Finite En-ergy Exists,” Dokl. Ross. Akad. Nauk, Vol. 378, No. 6, 2001, pp. 743-747.  V. A. Il’in and V. V. Tikhomirov, “Wave Equation with Boundary Control at Two Ends and Problem of Complate Oscillation Damping ,” Diff. Uravn., Vol. 35, No. 5, 1999, pp. 692-704. 2,12() .hlhli kzz KlKh From this fact and ,ikzM1, we obtain ,() ,hli kzz S ,() ,()hli khlzSzzSM  Yu. S. Osipov and S. P. Okhezin, “On the Theory of Dif-ferential Games in Parabolic Systems,” Dokl. Akad. Nauk SSSR, Vol. 226, No. 6, 1976, pp. 1267-1270. 1n, which proves the theorem. 5. Conclusions  F. L. Chernous’ko, “Bounded Controls in Distrib-uted-parameter Systems,” Prikl. Mat. Mekh., Vol. 56, No. 5, 1992, pp. 810-826. Thus, to solve the game problem of pursu it in th e fo r m (1) we pass to the discrete game (13) or (14), and Theorems 1-3 establish the sufficient condition for such problems. Theorem 4 establishes the sufficient conditions for solv-ing the problem of pursuit (1). Here, the difference (see Section 3) plays the main part in the solu-tion of problem and implies that the solutions of the grid equation (2) are stable. ,()hli jzz N. Satimov and M. Sh. Mamatov, “On a Class of Linear Differential and Discrete Games between Groups of Pur-suers and Evaders,” Diff. Uravn., Vol. 26, No. 9, 1990, pp. 1541-1551.  N. Satimov and M. Tukhtasinov, “On some Game Prob-lems in the Distributed Controlled Systems,” Prikl. Mat. Mekh., Vol. 69, No. 6, 2005, pp. 997-1003. The problem of stability of the grid equation (2) lies in determining the conditions under which the numerical error tends to zero with growing j uni-formly in all , or at least remains bounded. ,()ijhl ijpz z,0i,i N. Satimov and M. Tukhtasinov, “On some Game Prob-lems in Controlled First-order Evolutionary Equations,” Diff. Uravn., Vol. 41, No. 8, 2005, pp. 1114 -1121.  M. Sh. Mamatov, “On the Theory of Differential Pursuit Games in Distributed Parameter Systems,” Automatic Control and Computer Sciences, Vol. 43, No. 1, 2009, pp. 1-8. doi:10.3103/S0146411609010015 Equation (2) is called stable if the round off errors generated in the course of calculations have tendency to decrease or at least not to increase. Otherwise, the accu-mulated errors may reach a value such that the numerical solution has nothing in common with the exact solution of the grid problem (2). It goes without saying that such unstable grid equations cannot be used for nu-merical solution of the differential games. ()hlz M. Sh. Mamatov, “About Application of a Method of Final Differences to the Decision a Prosecution Problem in Systems with the Distributed Parameters,” Automation and Remote Control, Vol. 70, No. 8, 2009, pp. 1376-1384. doi:10.1134/S0005117909080104  M. Tukhtasinov and M. Sh. Mamatov, “On Pursuit Prob-lems in Controlled Distributed Systems,” Mathematical notes, Vol. 84, No. 2, 2008, pp. 273-280. Theorems 1-4 are easily generalized to a wider class of differential games, for example, when 21212 1221(,,...,)(( ,,...,),(,,...,))nnnzaxxxfuxxx xxxxn  M. Tukhtasinov and M. Sh. Mamatov, “About Transition Problems in Operated Systems,” Diff. Uravn., Vol. 45, No.3, 2009, pp. 1-6. with discontinuous coefficients.  M. Sh. Mamatov and M. Tukhtasinov, “Pursuit Problem in Distributed Control Systems,” Cybernetics and Systems Analysis, Vol. 45, No. 2, 2009, pp. 297-302. doi:10.1007/s10559-009-9100-x REFERENCES  O. A. Ladyzhenskaya, “Kraevye Zadachi Mate-maticheskoi Fiziki,” (Boundary Problems of Mathemati-cal Physics), Moscow, Nauka, 1973.  G. I. Marchuk, “Metody Vychislitel’noi MateMatiki,” (Methods of computational Mathematics), Moscow, Nauka, 1989.