STABILITY ESTIMATES FOR SOLUTIONS OF A LINEAR NEUTRAL STOCHASTIC EQUATION

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1 TWMS J. Pure Appl. Math., V.4, N.1, 2013, pp STABILITY ESTIMATES FOR SOLUTIONS OF A LINEAR NEUTRAL STOCHASTIC EQUATION IRADA A. DZHALLADOVA 1 Abstract. A linear stochastic functional ifferential equation of neutral type is consiere. Sufficient conitions for the exponential stability are erive by usin Lyapunov-Krasovskii functionals of quaratic form with exponential factors. Upper boun estimates for the exponential rate of ecay are erive in terms on the equation s coefficients. Keywors: scalar linear stochastic ifferential elay equations of neutral type, asymptotic stability of zero solution, exponential stability an rate of ecay, stochastic Lyapunov functionals. AMS Subject Classification: 34K20, 34K40, 34K50, 34K6, 34K Introuction Theory an applications of functional ifferential equations form an important part of moern nonlinear ynamics. Those equations are natural mathematical moels for various real life phenomena where the aftereffects are intrinsic features of their functionin. In recent years functional ifferential equations have been use to moel processes in iverse areas such as population ynamics an ecoloy, physioloy an meicine, economics an other natural sciences [6, 8, 12, 16. In many of the moels the initial ata an parameters are subject to ranom perturbations, or the ynamical systems themselves represent stochastic processes. This leas to consieration of stochastic functional ifferential equations [7, 10, 17. One of the principal problems of the corresponin mathematical analysis of equations is a comprehensive stuy of their lobal ynamics an relate preiction of lon term behaviors in applie moels. Of those the problem of stability of a particular solution plays a sinificant role. The latter is typically reuce to the stuy of stability of the zero solution of a transforme system. In cases of complex systems the stuy of local stability is usually approache by consierin the corresponin system linearize about the zero solution. Therefore, the stuy of stability of linear equations represents the first natural an important step in the analysis of more complex nonlinear systems. When applyin the mathematical theory to real life problems the mere statement about the stability in the system is harly sufficient. In aition to the fact of the stability itself it is of sinificant importance to obtain constructive an verifiable estimates on the rate of converence of solutions in time. One of the principal available tools in the relate stuies is the secon Lyapunov metho. For the functional ifferential equations the metho has been evelopin in recent years in two main irections. The first one is the metho of finite Lyapunov functions with the aitional assumption of Razhumikhin [18. The secon one is the metho of Lyapunov-Krasovskii functionals [14, 15. For the stochastic functional ifferential equations some aspects of these two irections of research have been evelope in papers [1-[4, [11, 13, 14 1 Kiev National University of Economics, Kiev, Ukraine, iraa-05@mail.ru Manuscript receive April

2 62 TWMS J. PURE APPL. MATH., V.4, N.1, 2013 an [5, 9, 13, 14, 19, 20, respectively. In the present paper, by usin the metho of Lyapunov- Krasovskii functionals, we erive sufficient conitions for stability toether with the rate of converence to zero of solutions for a class of linear the stochastic functional ifferential equation of neutral type. 2. Main results Consier the followin linear stochastic ifferential-ifference equation of neutral type [x(t) c x(t τ) = [a 0 x(t) + a 1 x(t τ) t + [b 0 x(t) + b 1 x(t τ) w(t), (1) where x R, a 0, a 1, b 0, b 1, c are all real constants, τ > 0 is a constant elay, an w(t) is a stanar scalar Wiener process with M{ w(t) = 0, M{ w(t) 2 = t, M{ w(t 1 ) w(t 2 ), t 1 t 2 = 0. An F t -measurable ranom process {x(t) x(t, ω) is calle a solution of equation (1) if it satisfies with the probability one the followin interal equation x(t) = c x(t τ) + [x(0) cx( τ) [b 0 x(s) + b 1 x(s τ) w(s), t 0 0 [a 0 x(s) + a 1 x(s τ) s + an the initial conitions x(t) = ϕ(t), x (t) = ψ(t), t [ τ, 0. Here an for the remainer of the paper we will be assumin that the initial functions ϕ an ψ are continuous ranom processes. Uner those assumptions the solution to the corresponin initial value problem for equation (1) exists an is unique for all t 0, up to its stochastic equivalent solution on the space (Ω, F, P ) [20 We shall make use of the followin norms for solutions of equation (1) x(t) τ := 0 max { x(t + s), τ s 0 x(t) 2 τ,β := τ e βs x 2 (t + s) s. (2) As it is well known, the zero solution of equation (1) is calle stable in the square mean if for every ε > 0 there exists δ(ε) > 0 such that every solution x of equation (1) satisfies M{ x(t) 2 < ε provie the initial conitions x(s) ϕ(s), x (s) ψ(s), s [ τ, 0 are such that max τ s 0 ϕ(s) < δ an max τ s 0 ψ(s) < δ. If the zero solution is stable in the square mean an lim t + M{ x(t) 2 = 0 then it is calle asymptotically stable in the square mean. If there exist positive constants N an γ such that the inequality hols M{ x(t) 2 τ,β < N x(0) 2 τ e γt, then the zero solution is calle exponentially (γ, β)-interally stable in the square mean. In this paper we erive constructive estimates of the exponential (γ, β)-interable stability in the square mean of the ifferential-ifference equation with constant elay (1). We employ the metho of stochastic Lyapunov-Krasovskii functionals. In the papers [13, 14, 19, 20 the functional is chosen to be of the form V [x(t), t = h[x(t) cx(t τ) τ x 2 (t + s) s

3 I.A. DZHALLADOVA: STABILITY ESTIMATES FOR SOLUTIONS where constants h > 0 an > 0 are such that the total stochastic ifferential of the functional alon solutions is neative efinite. In the present paper we a exponential factors to the functional so that it has the followin form 0 V [x(t), t = e γt h[x(t) cx(t τ)2 + e βs x 2 (t + s) s, where constants h > 0, > 0, γ > 0, β > 0 are to be etermine later. This allows us not only to erive sufficient conitions for the stability of the zero solution but also obtain coefficient estimates on the rate of the exponential ecay of solutions. Since the functional is homoeneous in both h an we can set h = 1. By chanin variables in the interal by t + s = ξ we obtain V [x(t), t = e γt h[x(t) cx(t τ)2 + e β(t ξ) x 2 (ξ) ξ. (3) By usin the earlier introuce norms (2) the functional (3) allows for the followin two-sie estimates Introuce the followin notations τ e γt M{ x(t) 2 τ,β M{V [x(t), t eγt (1 + c 2 ) (4) M{[x 2 (t) + x 2 (t τ) + e γt M{ x(t) 2 τ,β. A := 2a 0 b 2 0, B := a 1 + a 0 c b 0 b 1, C := 2a 1 c b 2 1, (5) S = S(, β, γ) := [ A γ B + γc B + γc e βτ γc 2 + C, (6) an let λ min = λ min (S(, β, γ)) be the smallest eienvalue of the matrix S = S(, β, γ). The followin result establishes the stability an the rate of rowth of the solutions of equation (1) as t +. Theorem 2.1. Suppose there exist positive constants β, γ, with β > γ such that the followin inequalities are satisfie 1 = A γ > 0, (7) 2 = (A γ )( e βτ γc 2 + C) (B + γc) 2 > 0, where constants A, B, an C are efine by (5). Then the zero solution of equation (1) is exponentially (γ, β)-interal stable in the square mean. Moreover, every solution x(t) satisfies the followin converence estimate for all t 0 M { x(t) 2 [ 1 + c 2 τ,β ( x(0) + x( τ) + x(0) τ,β ) e θ(β,γ)t, (8) where θ is efine by { θ = θ(, β, γ) = min β, λ min(s) 1 + c 2 + γ. (9)

4 64 TWMS J. PURE APPL. MATH., V.4, N.1, 2013 Proof. In orer to arrive at conitions (7) an the rate of solutions ecay (8) we shall apply the metho of Lyapunov-Krasovskii functionals to the one iven by formula (3). By usin the Ito formula we evaluate the stochastic ifferential of functional (3) as follows V [x(t), t = γe γt [x(t) cx(t τ)2 + e β(t s) x 2 (s) s t + +e γt {2[x(t) cx(t τ) [x(t) cx(t τ)+ [b 0 x(t) + b 1 x(t τ) 2 (w) x 2 (t) e βτ x 2 (t τ) β e β(t s) x 2 (s) s t. By takin the mathematical expectation we obtain M t {V [x(t), t = eγt M{γ[x(t) cx(t τ) 2 + 2[x(t) cx(t τ) [a 0 x(t) + a 1 x(t τ) + [b 0 x(t) + b 1 x(t τ) 2 + +x 2 (t) e βγ x 2 (t τ) (β γ) e β(t s) x 2 (s) s. By usin notations (5) an (6) the last expression can be rewritten in the followin vector-matrix form t M {V [x(t), t = e γt M {(x(t), x(t τ)) S(, β, γ) ( ) x(t) (β γ)e γt M e β(t s) x 2 (s) s x(t τ). With the notations (5) an (6) an accorin to the Silvester criterion [20 assumptions (7) imply that matrix S is positive efinite. Assume now (7) to hol. Then the smaller eienvalue of matrix S is positive, that is λ min (S) = 1 [σ σ > 0, σ = A + C γ(1 + c 2 ) + (e βτ 1), an the full erivative of the mathematical expectation of the Lyapunov functional is neative efinite. The zero solution of equation (1) is then asymptotically stable. We shall show next that solutions of equation (1) ecay exponentially by calculatin the corresponin exponential rate. Choose the constants β an γ to satisfy β γ > 0. Then the full erivative of the mathematical expectation for the Lyapuvov functional satisfies the inequality t M{V [x(t), t eγt λ min M{[x 2 (t) + x 2 (t τ) (10) (β γ)e γt M{ x(t) 2 τ,β. Let us erive conitions that are necessary to impose on the coefficients of equation (1) an parameters of the Lyapunov-Krasovskii functional (3) in orer for the followin inequality to hol M{V [x(t), t ζm{v [x(t), t, for some ζ > 0. t We use a sequence of the followin calculations.

5 I.A. DZHALLADOVA: STABILITY ESTIMATES FOR SOLUTIONS (1) Rewrite the riht han sie of inequality (4) in the form e γt M{ x(t) 2 τ,β 1 an substitute the latter into inequality (10). This results in Or equivalently as 1 + c2 M{V [x(t), t + eγt M{[x 2 (t) + x 2 (t τ) t M{V [x(t), t eγt λ min M{[x 2 (t) + x 2 (t τ) c2 +(β γ) { M{V [x(t), t + eγt M{[x 2 (t) + x 2 (t τ). t M{V [x(t), t (β γ)m{v [x(t), t eγt {λ min (S) (β γ)(1 + c 2 ) M{[x 2 (t) + x 2 (t τ). If the equation s coefficients an the functional s parameters are such that λ min (S) > (β γ) (1 + c 2 ) then the followin hols M{V [x(t), t ζm{v [x(t), t, where ζ = β γ. (11) t (2) Rewrite the riht han sie of inequality (4) in the form e γt M{[x 2 (t) + x 2 (t τ) 1 M{V [x(t), t c2 + e γt 1 + c 2 M{ x(t) 2 τ,β an substitute the latter aain into inequality (10). This results in { t M{V [x(t), t λ min(s) c 2 M{V [x(t), t + eγt Or, equivalently, in the inequality 1 + c 2 M{ x(t) 2 τ,β eγt (β γ)m{ x(t) 2 τ,β. t M{V [x(t), t λ min(s) V [x(t), t 1 + c2 { e γt (β γ) λ min(s) 1 + c 2 x(t) 2 τ,β. If the parameter involve are such that the inequality (β γ) > λ min(s[, β, γ) 1 + c 2 hols then we euce t M{V [x(t), t ζm{v [x(t), t, where ζ = λ min(s) 1 + c 2. (12) By simultaneously solvin both ifferential inequalities (11) an (12) we see that M{V [x(t), t V [x(0), 0 e ζt, with ζ = min{β γ, λ min(s) 1 + c 2. By interatin the two-sie inequality (4) for the Liapunov-Krasovskii functional V we obtain e γt M{ x(t) 2 τ,β M{V [x(t), t V [x(0), 0 e ζt { (1 + c 2 )[x 2 (0) + x 2 (τ) + x(0) 2 τ,β e ζt, t 0.

6 66 TWMS J. PURE APPL. MATH., V.4, N.1, 2013 This results in the final estimate for the rate of ecay of solutions iven by the followin inequality [ M{ x(t) 2 τ,β 1 + c 2 ( x(0) + x( τ) ) + x(0) τ,β e θ(,β,γ) t, t 0, where θ(, β, γ) = ζ + γ is iven by formula (9). Suppose that constants, β, γ are such that the assumptions (7) are satisfie with the corresponin rate (8) of ecay of solutions takin place. In the stability situation one now has a flexibility to choose the parameters, β, γ. Consier here a choice for the parameters which can be viewe in a sense as the best one. Accorin to the Silvester criterion the inequalities (7) are sufficient for matrix S to be positive efinite. We will seek for the value of parameter which maximizes the value of of the eterminant 2. This can be viewe as the best case for the positive efiniteness of matrix S[, β, γ since its minimal positive eienvalue is then the larest possible one. A necessary conition for 2 to have a local extreme value in is 2 = { (A γ )(e βτ γc 2 + C) (B + γc 2 ) 2 = = 2e βτ + [(A γ)e βτ + (γc 2 C) = 0. This implies that the corresponin value of is iven by 0 = 1 [ (A γ) + e βτ (γc 2 C). 2 Accorinly, the corresponin matrix S[, β, γ has the form [ 1 [ S = 2 (A γ) + e βτ (γc 2 C) [ B + γc 1 B + γc 2 (A γ) + e βτ (γc 2 C). The quantities σ an 2 are iven by σ = 1 2 (1 + e βτ ) [ (A γ) e βτ (γc 2 C), 2 = 1 4 e βτ [ (A γ) e βτ (γc 2 C) 2 (B + γc) 2. This implies that the smaller solution of the characteristic equation λ 2 σλ + 2 = 0 is iven by λ min = λ min (S[ 0, β, γ) = 1 [σ σ = = 1 { [ (1 + e βτ ) (A γ) e βτ (γc 2 C) 4 (1 e βτ ) 2 [(A γ) e βτ (γc 2 C) (B + γc) 2. Therefore, the above calculation establishes the followin result. Theorem 2.2. Suppose there exist constants β > γ > 0 such that the followin inequality is satisfie A γ e βτ (γc 2 C) > 2e 1 2 βτ B + γc,

7 I.A. DZHALLADOVA: STABILITY ESTIMATES FOR SOLUTIONS where constants A, B, an C are efine by (5). Then the zero solution of equation (1) is asymptotically stable in the square mean. Moreover, every solution x(t) satisfies the followin converence estimate for all t 0 M { x(t) 2 [ 1 + c 2 τ,β ( x(0) + x( τ) ) + x(0) τ,β e θ(β,γ)t, where θ(β, γ) an λ min are iven by { λ min θ(β, γ) = min β, 1 + c 2 + γ, λ min = 1 { (1 + e βτ )[(A γ) e βτ (γc 2 C) 4 (1 + e βτ ) 2 [(A γ) e βτ (γc 2 C) (B + γc) 2. References [1 Boukas, El-Kabir, (2002), Deterministic an Stochastic Time Delay Systems, Birkhauser, 432p. [2 Diekmann, O., Van Gils, S., Veruyn Lunel, S., Walther, H.O., (1995), Delay Equations: Functional-, Complex-, an Nonlinear Analysis, Spriner-Verla, New York, 552p. [3 Dzhallaova, I.A., (1997), Investiation of the stabilization of a matehamtical moel of a ynamic system uner ranom perturbations in the resonance moe, Ukrainian Math. J., 49(9), pp (in Russian). [4 Dzhallaova, I.A., (1998), Investiation of a system of linear ifefrential equations with ranom coefficients, Ukrainian Math. J., 50(8), pp (in Russian). [5 Honjiu Yan, Yuanqin Xia, (2012), Stability of Markovian jump systems over networks via elta operator approach, Circuits Syst Sinal Process, 31, pp [6 Erneux, T., (2009), A Book on Applie Functional Differential Equations, Birkhauser. [7 Gikhman I.I., Skorokho A.V., (1968), Stochastic Differential Equations, Kiev, Naukova Dumka Publ. Co., 354p. (in Russian). [8 Glass, L., Mackey, M.C., 1(988), From Clocks to Chaos, The Rhythms of Life, Prinston University Press. [9 Hale, J.K., Veruyn Lunel, S.M., (1993), Introuction to Functional Differential Equations, Spriner Applie Mathematical Sciences, 99. [10 Khasminskii, R.Z., (1969), Stability of Systems of Difefrential Equations uner Stochastic Perturbations of Parameters, Moscow, Science Publ. Co., 368p. (in Russian). [11 Khusainov, D.Y., Bychkov, A.S., (1992), Stability of neutral stochastic systems with small eviation of arument, Differential Equations, 28(12), pp (in Russian). [12 Kolmanovskii, V., Myshkis, A., (1999), Introuction to the Theory an Applications of Functional Differential Equations, Ser.: Mathematics an Its Applications, 463, Kluwer Acaemic Publishers, 648p. [13 Korenevskii, D.G., (1989), Stability of Dynamical Systems uner Ranom Perturbations of Parameters, Alebraic Criteria, Kiev, Naukova Dumka Publ. Co., 208p. (in Russian). [14 Korenevskii, D.G., (1992), Stability of Solutions in Deterministic an Stochastic Differential-Difference Equations, Alebraic Criteria, Kiev, Naukova Dumka Publ. Co., 148p. (in Russian). [15 Krasovskii, N.N., (1959), Some Problems in the Stability Theory of Motion, Moscow, Fiz.- Tek. Publishin Co. (in Russian). [16 Kuan, Y., (2003), Delay Differential Equations with Applications in Population Dynamics, Acaemic Press Inc., Series: Mathematics in Science an Enineerin, 191, 398p. [17 Kushner, G.J., (1969), Stochastic Stability an Control, Moscow, Worl Publ. Co., 200p. [18 Razumikhin, B.S., (1988), Stability in Heriatary Systems, Moscow, Science Publ. Co., 112p. (in Russian). [19 Sveran, M.L., Tsarkov, E.F., Yasinskii, V.K., (1996), Stabilty in Stochastic Moelin of Complex Dynamical Systems, Svyatyn, Na Prutom Publ. Co., 448 p. (in Ukrainian). [20 Tsarkov, E.F., Yasinskii, V.K., (1992), Quasi-Linear Stochastc Functional Differential Equations, Ria, Orientir Publ. Co., 322p. (in Russian).

8 68 TWMS J. PURE APPL. MATH., V.4, N.1, 2013 Iraa Dzhallaova was born in Azerbaijan Republic. She rauate from Faculty of Mechanics an Mathematics of Baku State University in She ot Ph.D. an Doctor of Sciences erees in 1994 an 2009 corresponently in Kiev National University by name T.G. Shevchenko. Since 2010 year she is a Professor an Chief of Department of Hiher Mathematics at Kiev National Economical University, Kiev, Ukraine. She publishe more than 140 research articles an supervise 5 Ph.D. stuents. Her research interests inclue stochastic ynamical systems, mathematical moels in economics an others.