Research Article Almost Sure Stability and Stabilization for Hybrid Stochastic Systems with Time-Varying Delays

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1 Mathematcal Problems n Engneerng Volume 212, Artcle ID , 21 pages do:1.1155/212/ Research Artcle Almost Sure Stablty and Stablzaton for Hybrd Stochastc Systems wth Tme-Varyng Delays Hua Yang, 1, 2 Husheng Shu, 3 Xu Kan, 1 and Yan Che 1 1 School of Informaton Scence and Technology, Donghua Unversty, Shangha 251, Chna 2 College of Informaton Scence and Engneerng, Shanx Agrcultural Unversty, Tagu 381, Chna 3 Department of Appled Mathematcs, Donghua Unversty, Shangha 251, Chna Correspondence should be addressed to Husheng Shu, hsshu@dhu.edu.cn Receved 21 June 212; Accepted 1 August 212 Academc Edtor: Bo Shen Copyrght q 212 Hua Yang et al. Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. The problems of almost sure a.s. stablty and a.s. stablzaton are nvestgated for hybrd stochastc systems HSSs wth tme-varyng delays. The dfferent tme-varyng delays n the drft part and n the dffuson part are consdered. Based on nonnegatve semmartngale convergence theorem, Hölder s nequalty, Doob s martngale nequalty, and Chebyshev s nequalty, some suffcent condtons are proposed to guarantee that the underlyng nonlnear hybrd stochastc delay systems HSDSs are almost surely a.s. stable. Wth these condtons, a.s. stablzaton problem for a class of nonlnear HSDSs s addressed through desgnng lnear state feedback controllers, whch are obtaned n terms of the solutons to a set of lnear matrx nequaltes LMIs. Two numercal smulaton examples are gven to show the usefulness of the results derved. 1. Introducton In the past decades, the problems of stablty analyss and stablzaton synthess of stochastc systems have receved sgnfcant attentons, and many results have been reported; see, for example 1 7 and the references theren. Commonly, the above problems can be solved not only n moment sense 8 1 but also n a.s. sense 11, 12. However, n recent years, much nterest has been focused on a.s. stablty problems for stochastc systems; see, for example 8, 13 and the references theren. It s well known that a lot of dynamcal systems have varable structures subject to abrupt changes n ther parameters, whch are usually caused by abrupt phenomena such as component falures or repars, changng subsystem nterconnectons, and abrupt envronmental dsturbances. The HSSs, whch are regarded as the stochastc systems wth

2 2 Mathematcal Problems n Engneerng Markovan swtchng n ths paper, have been used to model the prevous phenomena; see, for example and the references theren. The HSSs combne a part of the state x t that takes values n R n contnuously and another part of the state r t that s a Markov chan takng dscrete values n a fnte space S {1, 2,...,N}. One of the mportant ssues n the study of HSSs s the analyss of stablty. In partcular, t s not necessary for the stable HSSs to requre every subsystem to be stable; n other words, even all the subsystems are unstable; as the result of Markovan swtchng, the HSSs may be stable. These reveal that the Markovan jumps play an mportant role n the stablty analyss of HSSs. Therefore, n the past few decades, a great deal of lterature has appeared on the topc of stablty analyss and stablzaton synthess of HSSs; see, for example 2, 13, 14, 19, 2. On the other hand, tme delays are frequently encountered n a varety of dynamc systems, such as nuclear reactors, chemcal engneerng systems, bologcal systems, and populaton dynamcs models. They are often a source of nstablty and poor performance of systems. So the problems of stablty analyss and stablzaton synthess of HSDSs have been of great mportance and nterest. The classcal efforts can be classfed nto two categores, namely, moment sense crtera, see, for example 21 23, and a.s. sense crtera, see, for example 24, 25. Among the exstng results, n 25, based on the technques proposed n 26 whch were developed va the results of 11, a.s. stablty and stablzaton of HSDSs were studed. In 24, the a.s. stablty analyss problem for a general class of HSDSs was derved from extendng the results n 25 to HSSs wth mode-dependent nterval delays. However, to the author s best knowledge, when the dfferent tme-varyng delays n the drft part and n the dffuson part are consdered, the a.s. stablty analyss and stablzaton synthess problems for nonlnear HSDSs have not been adequately addressed and reman an nterestng and challengng research topc. Ths stuaton motvates the present study. In ths paper, we are concerned wth a.s. stablty analyss and stablzaton synthess problems for HSDSs. The purpose of stablty s to develop condtons such that the underlyng systems are a.s. stable. Followng the same dea as n dealng wth the stablty problem, lnear state feedback controllers are desgned such that the specal nonlnear or lnear closed-loop systems are a.s. stable. The explct expressons for the desred state feedback controllers are gven by means of the solutons to a set of LMIs. Two numercal smulaton examples are exploted to verfy the effectveness of the theoretcal results. The man contrbuton of ths paper s manly twofold: 1 the dfferent tme-varyng delays n the drft part and n the dffuson part are consdered for nonlnear HSDSs; 2 for a class of nonlnear HSDSs, the stablzaton synthess problem s nvestgated n the a.s. sense. Ths paper s organzed as follows. In Secton 2, we formulate some prelmnares. In Secton 3, we nvestgate the a.s. stablty for the hybrd stochastc systems wth tmevaryng delays. In Secton 4, the results of Secton 3 are then appled to establsh a suffcent crteron for the stablzaton. In Secton 5, two examples are dscussed for llustraton. Fnally, conclusons are drawn n Secton 6. Notaton 1. The notaton used here s farly standard unless otherwse specfed. R n and R n m denote, respectvely, the n dmensonal Eucldean space and the set of all n m real matrces, and let R,. Ω, F, {F t } t, P be a complete probablty space wth a natural fltraton {F t } t satsfyng the usual condtons.e., t s rght contnuous, and F contans all P-null sets. Ifx, y are real numbers, then x y stands for the maxmum of x and y, andx y the mnmum of x and y. M T represents the transpose of the matrx M. λ max M and λ mn M denote the largest and smallest egenvalue of M, respectvely. denotes the Eucldean norm n R n. E{ } stands for the mathematcal expectaton. P{ } means the probablty. C τ, ; R n

3 Mathematcal Problems n Engneerng 3 denotes the famly of all contnuous R n -valued functon ϕ on τ, wth the norm ϕ sup{ ϕ θ : τ θ }. C b F τ, ; R n beng the famly of all F -measurable bounded C τ, ; R n -value random varables ξ {ξ θ : τ θ }. L 1 R ; R denotes the famly of functons λ : R R such that λ t dt <. 2. Problem Formulaton In ths paper, let r t,t be a rght-contnuous Markov chan on the probablty space takng values n a fnte state space S {1, 2,...,N} wth generator Γ γ j N N gven by P { r t Δ j r t } { γj Δ o Δ f / j, 1 γ Δ o Δ f j, 2.1 where Δ > andγ j s the transton rate from mode to mode j f / j whle γ j / γ j. Assume that the Markov chan r s ndependent of the Brownan moton B. It s known that almost all sample paths of r are rght-contnuous step functons wth a fnte number of smple jumps n any fnte subnterval of R :,. Let us consder a class of stochastc systems wth tme-varyng delays: dx t f x t,x t τ 1 t,t,r t dt g x t,x t τ 2 t,t,r t db t 2.2 wth ntal data x {x θ : τ θ } ξ C b F τ, ; R n and r r S, where τ max{τ 1,τ 2 }, τ 1 and τ 2 are postve constant and τ 1 t and τ 2 t are nonnegatve dfferental functons whch denote the tme-varyng delays and satsfy τ 1 t τ 1, τ 1 t d τ1 < 1, τ 2 t τ 2, τ 2 t d τ2 < The nonlnear functons f : R n R n R S R n and g : R n R n R S R n m satsfy the local Lpschtz condton n x, y, z ; that s, for any K>, there s L K > such that f ( x, y, t, ) f ( x, y, t, ) g x, z, t, g x, z, t, L K ( x x y y z z ), 2.4 for all x y z x y z K, t and S, and moreover, sup t, S { f,,t, g,,t, : t, S} K wth some nonnegatve number K. Remark 2.1. It should be ponted out that the systems 2.2 can be seen as the specalzaton of multple tme-varyng delays systems whch are of the form dx t f x t,x t τ 1 t,x t τ 2 t,t,r t dt g x t,x t τ 1 t,x t τ 2 t,t,r t db t. 2.5

4 4 Mathematcal Problems n Engneerng But t s easy to see that the results n ths paper can be appled to the systems 2.5 by the smlar assumpton n 2.4. Let C 2,1 R n R S; R denote the famly of all nonnegatve functons V x, t, on R n R S that are twce contnuously dfferentable n x and once n t.ifv C 2,1 R n R S; R, defne an operator L assocated wth 2.2 from R n R n R n R S to R by LV ( x, y, z, t, ) V t x, t, V x x, t, f ( x, y, t, ) 1 ] [g 2 trace T x, z, t, V xx x, t, g x, z, t, N γ j V ( x, t, j ). j Remark 2.2. LV s thought as a sngle notaton and s defned on R n R n R n R S whle V s defned on R n τ, S. Defnton 2.3. The system 2.2 s sad to be a.s. stable f for all ξ C b F τ, ; R n and r S ( ) P lm t Man Results Theorem 3.1. Assume that there exst nonnegatve functons V C 2,1 R n R S; R, λ L 1 R ; R, ω 1,ω 2,ω 3 C R n ; R such that LV ( x, y, z, t, ) λ t k 1 ω 1 x k 2 ω 2 ( y ) k3 ω 3 z, ( x, y, z, t, ) R n R n R n R S, 3.1 ω 1 x >ω 2 x ω 3 x, x /, 3.2 lm nf x t, S V x, t,, 3.3 where k 1,k 2 and k 3 are postve numbers satsfyng k 1 max{k 2 / 1 d τ1,k 3 / 1 d τ2 }. Then system 2.2 s almost surely stable. To prove ths theorem, let us present the followng lemmas. Lemma 3.2 see 24, 25. If V C 2,1 R n R S; R, then for any t, the generalzed Itô s formula s gven as dv x t,t,r t LV x t,x t τ 1 t,x t τ 2 t,t,r t dt V x x t,t,r t g x t,x t τ 2 t,t,r t db t V x t,t,r t l r t,α V x t,t,r t μ dt, dα, R 3.4 where functon l, and martngale measure μ, are defned as, for example, 2.6 and 2.7 n [25].

5 Mathematcal Problems n Engneerng 5 Lemma 3.3 see 27. Let A 1 t and A 2 t be two contnuous adapted ncreasng processes on t wth A 1 A 2 a.s., let M t be a real-valued contnuous local martngale wth M a.s., and let ζ be a nonnegatve F -measurable random varable such that Eζ <. Denote X t ζ A 1 t A 2 t M t for all t.ifx t s nonnegatve, then { } { } { } lm A 1 t < lm X t < lm A 2 t < t t t a.s., 3.5 where C D a.s. means P C D c. In partcular, f lm t A 1 t < a.s., then, lm X t <, lm t t A 2 t <, < lm t M t < a.s That s, all of the three processes X t,a 2 t, and M t converge to fnte random varables wth probablty one. Lemma 3.4 see 25. Under the condtons of Theorem 3.1, for any ntal data {x θ : τ θ } ξ C b F τ, ; R n and r S, 2.2 has a unque global soluton. Proof. Fx any ntal data ξ, r,andletβ be the bound for ξ. For each nteger k β, defne f k ( x, y, t, ) ( x k y ) k f x, x y y, t,, 3.7 where we set x k/ x x when x. Defne g k x, z, t, smlarly. By 2.4, we can observe that f k and g k satsfy the global Lpschtz condton and the lnear growth condton. By the known exstence-and-unqueness theorem, there exsts a unque global soluton x k t on t τ, to the equaton dx k t f k x k t,x k t τ 1 t,t,r t dt g k x k t,x k t τ 2 t,t,r t db t 3.8 wth ntal data {x k θ : τ θ } ξ and r r. Defne the stoppng tme σ k nf{t : x k t k}, 3.9 where we set nf as usual. It s easy to show that x k t x k 1 t f t σ k, whch mples that σ k s ncreasng n k. Lettng σ lm k σ k, the property above also enables us to defne x t for t τ, σ as x t x k t f τ t σ k.

6 6 Mathematcal Problems n Engneerng It s clear that x t s a unque soluton of 2.2 for t τ, σ. To complete the proof, we only need to show P{σ } 1. By Lemma 3.2, we have that for any t>, EV x k t σ k,t σ k,r t σ k EV x k,,r σk E L k V x k s,x k s τ 1 s,x k s τ 2 s,s,r s ds, 3.1 where operator L k V s defned smlarly as LV was defned by 2.6. By the defntons of f k and g k,f s t σ k, we hence observe that L k V x k s,x k s τ 1 s,x k s τ 2 s,s,r s LV x k s,x k s τ 1 s,x k s τ 2 s,s,r s By the condtons of 3.1 and 3.2, we derve that EV x k t σ k,t σ k,r t σ k V ξ,,r E λ s ds V ξ,,r E E τ2 t ( k3 τ 2 1 d τ2 k 1 ω 1 x s k 2 ω 2 x s τ 1 s k 3 ω 3 x s τ 2 s ds k 1 ω 1 x s ds E ) ω 3 s ds λ s ds τ1 t ( k2 τ 1 1 d τ1 ) ω 2 s ds 3.12 V ξ,,r E k 1 ω 2 ξ θ ω 3 ξ θ dθ τ E k 1 ω 1 s ω 2 s ω 3 s ds V ξ,,r E τ λ s ds k 1 ω 2 ξ θ ω 3 ξ θ dθ λ s ds. On the other hand, EV x k t σ k,t σ k,r t σ k V x k t σ k,t σ k,r t σ k dp {σ k t} P{σ k t} nf x k,t, S V x, t,. 3.13

7 Mathematcal Problems n Engneerng 7 Ths yelds P{σ k t} V ξ,,r E τ k 1 ω 2 ξ θ ω 3 ξ θ dθ λ s ds nf x k,t, S V x, t, Lettng k and usng 3.3, weobtanp σ t. Snce t s arbtrary, we must have P σ 1. The proof s therefore complete. Let us now begn to prove our man result. Proof. Let ω x ω 1 x ω 2 x ω 3 x for all x R n. Inequalty 3.2 mples ω x > whenever x /. Fx any ntal value ξ and any ntal state r, and for smplcty wrte x t; ξ, r x t. By Lemma 3.2 and condton 3.1, we have V x t,t,r t V ξ,,r LV x s,x s τ 1 s,x s τ 2 s,s,r s ds V x x s,s,r s g x s,x s τ 2 s,s,r s db s R V ξ,,r V x s,s,r l r s,α V x s,s,r s μ ds, dα λ s ds k 1 ω 1 x s k 2 ω 2 x s τ 1 s k 3 ω 3 x s τ 2 s ds V x x s,s,r s g x s,x s τ 2 s,s,r s db s R V x s,s,r l r s,α V x s,s,r s μ ds, dα V ξ,,r λ s ds k 1 ω 2 x s ω 3 x s ds τ k 1 ω x s ds R V x x s,s,r s g x s,x s τ 2 s,s,r s db s V x s,s,r l r s,α V x s,s,r s μ ds, dα. 3.15

8 8 Mathematcal Problems n Engneerng Snce λ s ds <, applyng Lemma 3.3 we obtan that lm t ω x s ds ω x s ds < a.s., 3.16 lm t sup V x t,t,r t < a.s Defne β : R that R as β r nf x r, t<, S V x, t,. Then, t s obvous to see from 3.17 sup t< β x t sup V x t,t,r t < a.s t< On the other hand, by 3.3 we have sup t< x t < a.s.. It s easy to fnd an nteger k such that ξ <k a.s. because of ξ C b F τ, ; R n. Furthermore, for any nteger k>k, we can defne the stoppng tme ρ k nf{t : x t k}, 3.19 where nf as usual. Clearly, ρ k a.s. as k. Moreover, for any gven ε>, there s k ε k such that P{ρ k < } ε for any k k ε. It s straghtforward to see from 3.16 that lm t nf ω x t a.s.; then we clam that lm t ω x t a.s The rest of the proof s carred out by contradcton. That s, assumng that 3.2 s false, we have { } P lm sup ω x t > > t Furthermore, there exst ε > andε>ε 1 > such that P ( σ 2j < : j Z ) ε, 3.22 where Z s a set of natural numbers and {σ j } j 1 are a sequence of stoppng tmes defned by σ 1 nf{t :ω x t 2ε 1 }, σ 2j nf { t σ 2j 1 : ω x t ε 1 }, j 1, 2,..., 3.23 σ 2j 1 nf { t σ 2j : ω x t 2ε 1 }, j 1, 2,...

9 Mathematcal Problems n Engneerng 9 By the local Lpschtz condton 2.4, for any gven k>, there exsts L k > such that f ( x, y, t, ) g x, z, t, Lk, 3.24 for all x y z k, t and S. For any j Z, lett < σ 2j σ 2j 1 ;byhölder s nequalty and Doob s martngale nequalty, we compute { ( E I {σ2j <ρ k } sup x σ2j 1 t ) x ( } ) σ 2j 1 2 t T σ2j 1 t E I {σ 2j <ρ k } sup f x s,x s τ 1 s,s,r s ds t T 2E I {σ 2j <ρ k } sup t T { 8E I {σ2j <ρ k } sup t T σ 2j 1 σ2j 1 t σ 2j 1 σ2j 1 t σ 2j 1 σ2j 1 t σ 2j 1 2 g x s,x s τ 2 s,s,r s db s 2 f x s,x s τ 1 s,s,r s ds g x s,x s τ 2 s,s,r s 2 ds } L 2 kt T 4, where I A s the ndcator of set A. Snce ω x s contnuous n R n, t must be unformly contnuous n the closed ball S k {x R n : x k}. For any gven b>, we can choose c b > such that ω x ω y <b whenever x, y S k and x y <c b. Furthermore, let us choose ε ε 3, k k ε, b ε By nequalty 3.25 and Chebyshev s nequalty, we have P ({ ( { }) {σ2j } ( ( ρ k σ 2j P <ρ k sup ω x σ2j 1 t )) ω ( x ( }) )) σ 2j 1 ε1 t T P ({ } { ρ k σ 2j σ2j }) P ({ } { ρ k σ 2j σ2j < }) ( { {σ2j } <ρ k P sup t T x ( σ 2j 1 t ) x ( }) ) σ 2j 1 c ε L2 T T 4 k 1 2ε. cε 2 1

10 1 Mathematcal Problems n Engneerng Meanwhle, we can also choose T T ε, ε 1,k suffcently small for 2L 2 T T 4 k ε cε 2 1 And then, 3.27 and 3.28 yeld P ({ σ 2j <ρ k } Ωj ) ε, 3.29 where Ω j {sup t T ω x σ 2j 1 t ω x σ 2j 1 <ε 1 }. In the followng, we can obtan from 3.16 and 3.29 that > E ω x t dt [ σ2j ] E I {σ2j <ρ k } ω x t dt j 1 σ 2j 1 [ ( ) ] ε 1 E I {σ2j <ρ k } σ2j σ 2j 1 j 1 Tε 1 P ({ } ) σ 2j <ρ k Ωj j 1 1 ε j 1Tε 1 Tε ε 1. 3 j Ths s a contradcton. So there s an Ω Ω wth P Ω 1 such that lm ω x t, ω, sup t x t, ω <, ω Ω t< Fnally, any fxed ω Ω, {x t, ω } t s bounded n R n. By Bolzano-Weerstrass theorem, there s an ncreasng sequence{t } 1 such that {x t, ω } 1 converges to some z R n wth z <. Snce ω x > whenever x /, we must have ω x fand only f x. Ths mples that the soluton of 2.2 s a.s. stable, and the proof s therefore completed. Remark 3.5. The technques proposed n Theorem 3.1 can be used to deal wth the a.s. stablty problem for other HSDSs, such as the ones n 25. In a very specal case when τ 1 t τ 2 t τ for all t and S, tseasytoseethat τ 1 t τ 2 t, and Theorem 3.1 s exactly Theorem 2.1 n 25. Smlarly, Theorem 2.2 n 25 can be generalzed to system 2.2 as a LaSalle-type theorem see 24, 26 for HSSs wth multple tme-varyng delays.

11 Mathematcal Problems n Engneerng Almost Sure Stablzaton of Nonlnear HSDSs Consder the followng nonlnear HSDSs: dx t [ A r t x t A d r t x t τ 1 t f x t,x t τ 1 t,t,r t B u r t u t ] dt g x t,x t τ 2 t,t,r t db t, 4.1 where B u r t are known constant matrces wth approprate dmensons and B t represents a scalar Brownan moton Wener process on Ω, F, {F t } t, P that s ndependent of Markov chan r t and satsfes: E{dB t }, { E db t 2} dt, 4.2 f and g are both functons from R n R n R S to R n whch satsfy local Lpschtz condton and the followng assumptons: f x t,x t τ1 t,t,r t 2 x T t F 1 r t x t x T t τ 1 t F 2 r t x t τ 1 t, g x t,x t τ2 t,t,r t x T t G 1 r t x t x T t τ 2 t G 2 r t x t τ 2 t, where, for each r t j S, A r t, A d r t are known constant matrces wth approprate dmensons, and F r t R n n, G r t R n n 1, 2 are postve defnte matrces. In the sequel, we denote the matrx assocated wth the th mode by Γ Γ r t, 4.4 where the matrx Γ could be A, A d,b u,f 1,F 2,G 1,G 2,G,orG d. As the gven HSDSs 4.1 s nonlnear, we here consder the resultng systems can be stablzed only by lnear state feedback controller whch s of the form u t K r t x t, 4.5 where K r t are controller parameters to be desgned. Under control law 4.5, the closed-loop system can be gven as follow: dx t [ A r t x t A d r t x t τ 1 t f x t,x t τ 1 t,t,r t B u r t K r t x t ] dt 4.6 g x t,x t τ 2 t,t,r t db t.

12 12 Mathematcal Problems n Engneerng The stablzaton problem s therefore to desgn matrces K r t for the closed-loop system 4.6 to be a.s. stable. In order to guarantee the solvablty of K r t, the followng theorem s gven. Theorem 4.1. If there exst sequences of scalars ε 1 >, ε 2 >, δ >, postve defnte matrces X > and matrces Y such that the followng LMIs M 1 M 2 M 4 M 3 <, j S, 4.7 M 5 X δ I 4.8 hold, where M 1 A X X A T B u Y Y T BT u ε 1A d A T d ε 2I γ X, M 2 X,X,X,X,X, ( ) M 3 dag ε 2 F 1 1,c 1ε 2j F 1 2j,δ G 1 1,c 2δ G 1 2j,c 1ε 1j I, M 4 [ γ 1 X,..., γ 1 X, γ 1 X,..., γ N X ], 4.9 M 5 dag X 1,...,X 1,X 1,...,X N, c 1 1 d τ1, c 2 1 d τ2, then the controlled system 4.6 s a.s. stable and the state feedback controller determned by u t K x t, K Y X 1, S. 4.1 Proof. Let P X 1 and V x, x T P x t τ 1 t xt s Q 1 x s ds t τ 2 t xt s Q 2 x s ds. The operator LV : R n R n R n S R has the form LV ( x, y, z, ) x T Q 1 x 1 τ 1 t y T Q 1 y x T Q 2 x 1 τ 2 t z T Q 2 z 2x T P ( A x A d y f ( x, y, ) B u K x ) g T x, z, P g x, z, N γ j x T P j x j 1

13 Mathematcal Problems n Engneerng 13 x T Q 1 Q 2 P A A T P P B u K B u K T P ε 1 P A d A T d P ε 2 P 2 N γ j P j ε 1 2 F 1 δ 1 j 1 G 1 x y T[ ] ε 1 1 I ε 1 2 F 2 1 d τ1 Q 1 y z T[ δ 1 G 2 1 d τ2 Q 2 ]z So LV ( x, y, z, ) ω 1 x 1 d τ1 ω 2 ( y ) 1 dτ2 ω 3 z, 4.12 where ω 1 x x T [ Q 1 Q 2 P A A T P P B u K B u K T P ε 1 P A d A T d P N ε 2 P 2 ε 1 2 F 1 δ 1 G 1 γ k P k ]x, k ω 2 x x T[ c 1 1 ε 1 1 I c 1 1 ε 1 2 F 2 Q 1 ]x, ω 3 x x T[ c 1 2 δ 1 G 2 Q 2 ]x. By assumpton 1, t s easy to see that we can choose Q 1 and Q 2 such that ω 2 x, ω 3 x for all x R n, S. Notng that P X 1 and Y K X, we can pre- and postmultply 4.7 by dag P,...,P, and usng Schur complements, we can obtan Φ j <, 4.14 where Φ j P A A T P P B u K B u K T P ε 1 P A d A T d P ε 2 P 2 δ 1 N ε 1 2 F 1 γ k P k c 1 I c 1 2j F 2j c 1 2 δ jg 2j. k 1 1 ε 1 1j 1 ε 1 G Ths mples ω 1 x >ω 2j x ω 3j x, x / Let ω 1 x mn S ω 1 x,ω 2 x max S ω 2 x,andω 3 x max S ω 3 x.

14 14 Mathematcal Problems n Engneerng Clearly ω 1 x >ω 2 x ω 3 x, x / Moreover, by 4.24 we further obtan LV ( x, y, z, ) ω 1 x 1 d τ1 ω 2 ( y ) 1 dτ2 ω 3 z The requred asserton now follows from Theorem 3.1. If the systems 4.6 reduces to lnear HSDSs of the form dx t A r t x t A d r t x t τ 1 t B u r t K r t x t dt G r t x t G d r t x t τ 2 t db t, 4.19 where A r t,a d r t,b u r t,g r t, andg d r t are known constant matrces wth approprate dmensons. Then, the followng corollary follows drectly from Theorem 4.1. Corollary 4.2. If there exst sequences of scalars ε 1 >, ε 2 >, postve defnte matrces X > and matrces Y such that the followng LMIs M 1 M 2 M 4 M 3 <, j S 4.2 M 5 hold, where M 1 A X X A T B u Y Y T BT u ε 1A d A T d γ X, [ M 2 2X G T,X j,x j, ] 2X T j G dj, M 3 dag ( X,c 1 ε 1j I,ε 2j I,X j ), M 4 [ γ 1 X,..., γ 1 X, γ 1 X,..., γ N X ], 4.21 M 5 dag X 1,...,X 1,X 1,...,X N, c 1 1 d τ1, c 2 1 d τ2, then the controlled system 4.19 s a.s. stable and the state feedback controller determned by u t K x t, K Y X 1, S. 4.22

15 Mathematcal Problems n Engneerng 15 Proof. Let P X 1 and V x, x T P x t τ 1 t xt s Q 1 x s ds t τ 2 t xt s Q 2 x s ds. The operator LV : R n R n R n S R has the form LV ( x, y, z, ) x T Q 1 x 1 τ 1 t y T Q 1 y x T Q 2 x 1 τ 2 t z T Q 2 z 2x T [ P A x A d y B u K x ] G x G d z T P G x G d z N γ k x T P k x k 1 x T [Q 1 Q 2 P A A T P P B u K B u K T P 4.23 ε 1 P A d A T d P N γ k P k 2G T P G ]x k 1 y T[ ε 1 1 I 1 d τ 1 Q 1 ] y z T[ ε 1 2 I 2GT d P G d 1 d τ2 Q 2 ]z. So LV ( x, y, z, ) ω 1 x 1 d τ1 ω 2 ( y ) 1 dτ2 ω 3 z, 4.24 where ω 1 x x T [ Q 1 Q 2 P A A T P P B u K B u K T P ε 1 P A d A T d P N γ k P k 2G T P G ]x, k 1 ω 2 x x T[ ] c 1 1 ε 1 1 I Q 1 x, 4.25 ω 3 x x T[ ε 1 2 I 2c 1 2 GT d P G d Q 2 ]x. It s easy to see that we can choose Q 1 and Q 2 such that ω 2 x, ω 3 x for all x R n, S. Notng that P X 1 and Y K X, we can pre- and postmultply 4.7 by dag P,...,P, and usng Schur complements, we can obtan Φ j <, 4.26

16 16 Mathematcal Problems n Engneerng where Φ j P A A T P P B u K B u K T P ε 1 P A d A T d P N γ k P k 2G T P G c 1 k 1 1 ε 1 1j I ε 1I 2c 1 2 GT dj P jg dj. 2j 4.27 Ths mples ω 1 x >ω 2j x ω 3j x, x / Let ω 1 x mn S ω 1 x,ω 2 x max S ω 2 x,andω 3 x max S ω 3 x. Clearly ω 1 x >ω 2 x ω 3 x, x / Moreover, by 4.24 we further obtan LV ( x, y, z, ) ω 1 x 1 d τ1 ω 2 ( y ) 1 dτ2 ω 3 z. 4.3 The requred asserton now follows from Theorem Examples In ths secton we wll provde two examples to llustrate our results. In the followng examples we assume that B t s a scalar Brownan moton, γ t s a rght-contnuous Markov chan ndependent of B t and takng values n S {1, 2}, and the step sze Δ.1. By usng the YALMIP toolbox, smulatons results are shown n Fgures 1 3. Fgure 1 gves a porton of state γ t of Example 5.1 for clear dsplay. Fgure 2 smulates the numercal results for Example 5.1. The smulaton results have llustrated our theoretcal analyss. Followng from Theorem 4.1, the smulaton results for Example 5.2 can be founded n Fgure 3, whch verfy our desred results. Example 5.1. Let Γ ( ( ) ).8.8 γ j Consder scalar nonlnear HSDSs: dx t f x t,t,r t dt g x t,x t τ 2 t,t,r t db t, 5.2

17 Mathematcal Problems n Engneerng t Fgure 1: The state γ t of Example X(t) Fgure 2: The state evoluton of Example 5.1. t where f x, t, x, g x, z, t, 1 5 x z 3, 3 f x, t, x, t g x, z, t, 2 5 x 3 cos t 5 5 z 4 3 sn t, 5.3 τ 2 t.3.3sn t.

18 18 Mathematcal Problems n Engneerng X(t) The controlled states The uncontrolled states Fgure 3: The state evoluton of Example 5.2. t To examne the stablty of system 5.2, we consder a Lyapunov functon canddate V : R S R as V x, x 2 for 1, 2. Then we have LV x, z, t, 1 1x 6/5 4z 6/5, LV x, z, t, 2 3x 3 1 t 6x 6/ z6/ By the elementary nequalty α c β 1 c cα 1 c β for all α, β, and c 1, we see that nequalty ( ) 3x 6 6/5 ( ( κ 3 1 t 5 κx6/5 6 5 ) 5 1 t 2 ) 1/6 κx 6/5 κ 1 1 t holds for any κ>, where κ 1 κ/5 5. From nequaltes , we have LV x, z, t, κ 1 1 t 2 6 κ x6/5 4z 6/5, 5.6 for all t and S. Byτ 2 t.3.3sn t, tseasytoseethatd τ2 t < 1/3; then, we choose constant κ such that <κ< 2 6d τ2 / 1 d τ2, and hence condtons of Theorem 3.1 are satsfed.

19 Mathematcal Problems n Engneerng 19 Example 5.2. Let Γ ( ( ) ).6.6 γ j Consder scalar nonlnear closed-loop HSDSs: dx t [ f x t,x t τ 1 t,t,r t B r t K r t x t ] dt g x t,x t τ 2 t,t,r t db t 5.8 wth f ( x, y, t, 1 ) x 1 2 y 2x 3 g x, z, t, 1 x cos t f ( x, y, t, 2 ) 2x y g x, z, t, 2 2x sn t y sn t, 2 x 1 z 3 z 1 2, x 3 x 2 y 3 2 ( ) 2, y 1 x 3 2 x 1 z 3 2 z 2, τ 1 t.1.1sn t, τ 2 t.2.2sn 2t, B 1 2, B 2 3, A 1 1, A 2 2, A d1 1/2, A d2 1, F 11 8, F 12 G 11 2, G 12 F 21 F 22 2, G 21 1/2, G By Theorem 4.1 we can fnd the feasble soluton K 1 3, K 2 2 for the a.s. stablty. 6. Conclusons In ths paper, we have nvestgated the a.s. stablty analyss and stablzaton synthess problems for nonlnear HSDSs. Some suffcent condtons are gven to guarantee the resultng systems to be a.s. stable. Under these condtons, a.s. stablzaton problem for a class of nonlnear HSDSs s solved n terms of the solutons to a set of LMIs. Fnally, the results of ths paper have been demonstrated by two numercal smulaton examples. Acknowledgment Ths work s supported n part by the Natonal Natural Scence Foundaton of P.R. Chna no References 1 J. Hu, Z. Wang, H. Gao, and L. K. Stergoulas, Robust sldng mode control for dscrete stochastc systems wth mxed tme delays, randomly occurrng uncertantes, and randomly occurrng nonlneartes, IEEE Transactons on Industral Electroncs, vol. 59, no. 7, pp , 212.

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