Applied Mathematics Letters

Transcription

1 PERGAMON Applied Matheatics Letters 15 (2002) Applied Matheatics Letters Soe Peculiarities of the General Method of Lyapunov Functionals Construction V. KOLMANOVSKII Moscow Institute of Electronics and Matheatics Bolshoy Vusovsii, 3/12, Moscow , Russia volano~x, ii. rssi. ru L. SHAIKHET Departent of Matheatics, Inforatics and Coputing Donets State Acadey of Manageent Chelyusintsev str., 163-a, Donets 83015, Uraine donets, ua, leonid, shaihet usa, net (Received April 2000; accepted February ~001) Abstract--The general ethod of Lyapunov functionals construction for stability investigation of stochastic hereditary systes which was proposed and developed before is considered. Soe features of this ethod for difference systes which allow one to use the ethod ore effectively are discussed Elsevier Science Ltd. All rights reserved. Key~vords--Method of Lyapunov functionals construction, Stability of stochastic difference equations. Let {~, P, a} be a probability space, h be a given nonnegative nuber, i be a discrete tie, i E Z0 U Z, Z0 = {-h,...,0}, Z = {0,1,...}, fi E a, i Z, be a sequence of a-algebras, E be the atheatical expectation, ~0, ~1,... be a sequence of utually independent rando variables, ~ R, ~ be fi+l-adapted and independent on fi, E~i = 0, E~i~ = I, i Z, I be identity atrix, process xi R n be a solution of the equation xi+l = F(i,x-h,...,Xi) + ~ G(i,j,X_h,...,Xj)~j,,4=0 i i e z, (1) with initial function xi = ~i, i E Z0. Here F : Z * S ~ R n, G : Z Z * S :~ R n-, S is a space of sequences with eleents fro R n. It is assued that F(i,...) is independent on xj for j > i, G(i,j,...) is independent on x for > j, F(i,O,...,O) = O, G(i,j,O,...,O) = O /02/$ - see front atter (~) 2002 Elsevier Science Ltd. All rights reserved. Typeset by fl, h/~s-tex PII: S (01)

2 356 V. KOLMANOVSKII AND L. SHAIKHET DEFINITION. The zero solution of equation (1) is cmled ean square stable //for any e > 0 there exists a 6 > 0 such that E[xi[ 2 < e, i c Z, if [[~[[2 = supiez E[~[2 < 6. If, besides, lii-.oo E[xi[ 2 = 0 for 021 initial functions qo, then the zero solution of equation (1) is called asyptotically ean square stable. THEOREM. (See [1].) Let there exist a nonnegative functional Vi = V(i, Z-h,..., x~), i E Z, which satisfies the conditions EV(O,X-h,...,xo) <_ Clllqal[ 2, EAV~ _< -c2e[xd 2, i ~ Z, AV/= V~+I - V/, Cl > 0, c2 > 0. Then the zero solution of equation (1) is asyptoticmly ean square stable. Fro Theore 1 it follows that the stability investigation of stochastic equations can be reduced to construction of appropriate Lyapunov functionals. Following the general ethod of Lyapunov functionals construction, which was proposed and developed in [1-24], it is necessary to construct Lyapunov functional Vi in the for Vi -- Vii + V2i, the ain coponent Vii ust be chosen by a special way. This choice is not unique. Hence, for each choice of Vu we can construct other Lyapunov functionms and therefore to get other stability conditions. Besides choosing different ways of estiation of EAVli we can construct different Lyapunov functionals and as a result can again obtain different stability conditions. Let us deonstrate these peculiarities of the general ethod of Lyapunov functionais construction for the equation X,+l = ax, + b ~ ( j)x~_j + ~ ~ ( j)~i-j~,. (2) j=0 Here it is supposed that > 0, _> 0. Following [1], we will construct Lyapunov functional for equation (2) in the for V/= Vii + V2i, Vii = x 2. Calculating EAVIi by virtue of (2), we get rn EAVli = E 2 + b2e (~ ( j)xi-j + a2e ( \ j=~ j=0 < (.~ - 1)Ex~ + I.bl ~ (~ j) (~.~ + E~L~) + b 2 ~ ( + l - l) ~ ( + l - j)ex2_j I=1 +o 2 ~ ( + l - l) ~ ( + l - j)exlj /=0 j=0 = (a2+[abl (-bl)------~---ba2(+l)2(rn+2)-l)ex (l~bl + b~),l) ~ ( j)e~l~ + ~:~,~o ( j)exl~, 1 (+l-l)(+2-1). (a)

3 Lyapunov Functionals Construction 357 Choosing the functional V2, in the for i--1 i--1 we get i l=i- l=i- Al = (lab[ + b2al) "~l, B l ~ O'2~rnOAl, (4) i l=i+ l- l=i+ l- i-1 = (Ax + Ba)x~ + y~ (A,+l-t - Ai-t)x~ - AXL l=i+l- i-1 + Z (Bi+l-t- B,-I)x~ - Bx2i_ l=i+ l-- = (A1 + B1)x2i - ( labl + b2al)" Z ( + 1-3)xi-j 2 - '2A0 Z ( j)xlj. Fro (3),(4), it follows that A1 = labl -= ( 1) b2 2(4 + 1)5 ' B1 = o.2 ( + 1)2(4 + 2) Therefore, for ~ = Vu + ~i we obtain + p - 1 Ex~ 2, t7 2 p = ~- ( + 1)~( + 2) 2. Pro here and Theore 1, it follows that the inequality + p < 1 (5) is a sufficient condition of asyptotic ean square stability of the zero solution of equation (2). Let us show that choosing another way of estiation of EAVli and supposing soe additional conditions on a and b, we can get stability conditions which differ fro (5). So let us use the functional Vii in the for Vii = x~ 2 again. Calculating EAVI~ and using (3), we get as previously E~Vl, = (a 2-1) E~ + 2ab ~ ( + ~ - j)e~,~,_j + b2e ~ ( j)~_, + o~e ~ (~ j)~_,~, \ 3=0 < (a2+a2(+l)2(+2)-l) 2 + b2al Z ( j)exlj + a2ao Z ( j)ex~_j. rn

4 358 V. KOLMANOVSKII AND L. SIIAIKHET Suppose now that ab < 0 aad put l-.~i- l=t- Bl are defined by (4), (3), and Cl = b2)~l)~z. Calculating EAV2i as previously, we get EAV2~ = lable'h + (C1 + B1)Ex~ -b2al E ( j)ex~_j - a2a0 E ( j)ex2 j, j~-t Note that +l ~i= E ] Xi ~- l~l Xi--I -- Xi--j -~ ~ Xi--I " j--1 ( 2 x i 4-2Z i E Xi-l -- X 2 i-j + 2Xi-j E 1=1 )] :~i-i It is easy to see that +l( E p~ = x~_j + 2xi_j +l ~-I j-i Pi = E x~_j+ex~_,ex~_l+ex~_, 2 3--=1 +l j 4-I ) E :~i-i +l E +l x~_j 1=1 l=j+l 1=1 \ It eans that Note that 7i < ( + 1)z~ + 2x~ ~ ( l)xi_~- ~2 ( + I)2( + 2) +B1 =p. 2

5 Therefore, for V~ = Vii + V2~, we have Lyapunov Functionals Construction 359 EAV~ < (a 2 + labl( + 1) + b 2 2( 1)2 - \ 4 + p- 1) Ez~. Thus, the inequalities a 2 + labi( + 1) + b 2 2( + 1)2 4 + p < 1, ab < O, (6) are a sufficient condition of asyptotic ean square stability of the zero solution of equation (2). Rewriting condition (5) in the for a s + ]ab]( + 1) + b 2 2( + 1)2 4 +p<l and collating conditions (5) and (6) by ab < 0, it is easy to see that by = 1 condition (6) coincides with (5), but by > 1 condition (6) is better than (5). REMARK. As it follows fro [1] constructing other Lyapunov functionals we can obtain sufficient conditions of asyptotic ean square stability of the zero solution of equation (2) in other fors, for exaple, p< (l_a_b(+l)) (l+aq_b(+l) ( l)(+2)) a (+l) 2 2 Ibl 3, + b 2 < 1, or Ib[ (( - 1)/2) [21a [ + (1 - b)lb I (( + 3)/2)] + (1 - b)p <1, Ibl < l, lal < l-b. (1 + b) [(1 - b) 2 - a s] REFERENCES 1. V.B. Kolanovsii and L.E. Shaihet, General ethod of Lyapunov functionals construction for stability investigations of stochastic difference equations, In Dynaical Systes and Applications, Volue 4, PP , World Scientific Series in Applicable Analysis, (1995). 2. V.B. Kolanovsii and L.E. Shaihet, Stability of stochastic hereditary systes, Avtoatia i Teleehania 7, 66-85, (1993). 3. V.B. Kolanovsii and L.E. Shaihet, On one ethod of Lyapunov functional construction for stochastic hereditary systes, Differentialniye Uravneniya 11, , (1993). 4. V.B. Kolanovsii and L.E. Shaihet, New results in stability theory for stochastic functional differential equations (SFDEs) and their applications, In Proceedings of Dynaic Systes and Applications, Volue 1, pp , Dynaic Publishers, (1994). 5. L.E. Shaihet, Stability in probability of nonlinear stochastic systes with delay, Mateatichesiye Zaeti 57 (1), , (1995). 6. L.E. Shaihet, Stability in probability of nonlinear stochastic hereditary systes, Dynaic Systes and Applications 4 (2), , (1995). 7. V.B. Kolanovsii and A.M. Rodionov, On the stability of soe discrete Volterra processes, Avtoatia i Teleehania 2, 3-13, (1995). 8. V.B. Kolanovsii and L.E. Shalhet, A ethod of Lyapunov functional construction for stochastic differential equations of neutral type, Differentialniye Uravneniya 11, , (1995). 9. L.E. Shaihet, Modern state and developent perspectives of Lyapunov functionals ethod in the stability theory of stochastic hereditary systes, Theory of Stochastic Processes 2 (18, No. 1/2), , (1996). 10. L.E. Shalhet, Stability of stochastic hereditary systes with Marov switching, Theory of Stochastic Processes 2 (18, No. 3/4), , (1996). 11. V.B. Kolanovsii and L.E. Shaihet, Asyptotic behaviour of soe systes with discrete tie, Avtoatia i Teleehania 12, 58-66, (1996). 12. L.E. Shaihet, Soe probles of stability for stochastic difference equations, In 15 th World Congress on Scientific Coputation, Modelling and Applied Matheatics, IMACS97, Berlin, August, 1997; Coputational Matheatics 1, , (1997). 13. L.E. Shaihet, Probles of the stability for Stochastic difference equations, Theory of Stochastic Processes (Proceedings of the 2 nd Scandinavian-Urainian Conference, June 8-13, 1997 Uea, Sweden) 3 (19, No. 3/4), , (1997).

6 360 V. KOLMANOVSKII AND L. SHAIKHET 14. L.E. Shalhet, Necessary and sufficient conditions of asyptotic ean square stability for stochastic linear difference equations, Appl. Math. Left. 10 (3), , (1997). 15. V.B. Kolanovsii and L.E. Shaihet, About stability of soe stochastic Volterra equations, Differentialniye Ura~eniya 11, , (1997). 16. N.J. Ford, J.T. Edwards, J.A. Roberts and L.E. Shaihet, Stability of a difference analogue for a nonlinear integro differential equation of convolution type, Nuerical Analysis Report, No. 312, University of Manchester, (1997). 17. V.B. Kolanovsii and L.E. Shalhet, Matrix Riccati equations and stability of stochastic linear systes with nonincreasing delays, b-~nctional Differential Equations 4 (3/4), , (1997). 18. E. Beretta, V. Kolanovsii and L. Shalhet, Stability of epideic odel with tie delays influenced by stochastic perturbations, Matheatics and Coputers in Siulation (Special Issue "Delay Systes") 45 (3/4), , (1998). 19. V.B. Kolanovsii and L.E. Shaihet, Riccati equations in stability of stochastic linear systes with delay, Avtoatia i Teleehania 10, 35-54, (1998). 20. V.B. Kolanovsii and L.E. Shaihet, Riccati equations and stability of stochastic linear systes with distributed delay, In Advances in Systes, Signals, Control and Coputers, Vol. L (Edited by V. Bajic), pp , IAAMSAD and SA Branch of the Acadey of Nonlinear Sciences, Durban, South Africa, (1998). 21. G. Shalhet and L. Shalhet, Stability of stochastic linear difference equations with varying delay, In Advances in Systes, Signals, Control and Coputers, Vol. L (Edited by V. Bajic), pp , IAAMSAD and SA Branch of the Acadey of Nonlinear Sciences, Durban, South Africa, (1998). 22. L.E. Shaihet, Stability of predator-prey odel with aftereffect by stochastic perturbations, Stability and Control: Theory and Application 1 (1), 3-13, (1998). 23. B. Paternoster and L. Shaihet, Stability in probability of nonlinear stochastic difference equations, Stability and Control: Theory and Application 2 (1-2), 25-39, (1999). 24. B. Paternoster and L. Shalhet, About stability of nonlinear stochastic difference equations, Appl. Math. Left. 13 (5), 27-32, (2000).