Simulating and Numerical Solution of Stochastic Differential Systems with Switching Diffusion during the Firing

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1 Jounal of Numeical Analysis an Applie Mathematics Vol., No., 06, pp Simulating an Numeical Solution of Stochastic Diffeential Systems with Switching Diffusion uing the Fiing S. Saabaan, *, H. R. Shaahza Depateman of Mathematics, Imam Hossein Univesity, Tehan, Ian Depateman of Mathematics, Shaif Univesity, Tehan, Ian Abstact Launching the Ramp is one of the pimay means of efence. Rocket stability in lunching has been vey effective in accuate hitting the object which fully epens on the conitions an lunching oscillation. In this pape we consie a stochastic iffeential moel with switching Diffusion fo ocket system an with using the genealization of the Taylo-metho, we will estimate the answe of this poblem. Finally the esults of numeical solution is pesente in Sci-lab. Keywos Slope Rocket Launching, Oscillations, Switching Diffusion, Stochastic Diffeential Systems Receive: July 4, 06 / Accepte: August 5, 06 / Publishe online: August 5, 06 The Authos. Publishe by Ameican Institute of Science. This Open Access aticle is une the CC BY license. Intouction In this stuy, fo etaile esign an efficient lunching system, especially in the ealy non-guie launching of ocket, eview of the oscillation is impotant an necessay. We suppose that the launching evice an the moving ocket fom a complex oscillating system that join togethe a sum of igi boies boun by elastic elements (the vehicle chassis, the tilting platfom an the ockets in the containes []. Suppose the inepenent unknown ynamic vaiables of the ocket-launching evice system motion ae pesente in the fom of the following column vecto []: X6 [ ] T = sϕ yϕz zsγ xγ y () Whee the vehicle chassis tanslation z s, the vehicle chassis otation γ y (the chassis pitch movement), the vehicle chassis otation γ x (the chassis olling movement), the tilting platfom otation ρ z (the gyation movement aoun the vetical axes), the tilting platfom otation ρ y (the pitch movement) an the ocket tanslation s. So, one can obtain the matix fom of the stochastic iffeential equations system that escibes the ocket launching system components motion: Xɺɺ = B. Xɺ + C. X + N. ξ F. ϕ + K. ε Whee 6 6 =(, ) =,6 =,6 is the matix of the velocities coefficients,ẋ 6 x ; 6 6 =(, ) =,6 =,6 is the matix of the unknown vaiables coefficients X. 6 5 =(, ) =,6 =,5 is the matix of the coefficients fo the nonlinea combinations of the unknown vaiables: 5 [ ] T = ɺ x ɺ y ɺ x ɺ y ɺ y ɺ z ɺ y ɺ z ɺ x ɺ y ɺ x ɺ z ɺ y ɺ y ɺ y ɺ zs ɺ xs ɺ ys ɺ ys ɺ z FRy + FRz ξ γ γ γ γ ϕ ϕ ϕ ϕ γ ϕ γ ϕ γ ϕ γ ϕ ɺ γ ɺ γ ɺ ϕ ɺ ϕ µ (3) Fo moe infomation about the components of matices (3), that can be specifie anomly, one can see [, ]. An, the 6 3 =(, ) =,6 =,3 is the matix of the extenal foces that acts on the system: () φ 3 [ gtf ] T = jet (4) * Coesponing autho aess: s.saabaan@yahoo.com (S. Saabaan), hami_ahmaian69@yahoo.com (H. R. Shaahza)

2 4 S. Saabaan an H. R. Shaahza: Simulating an Numeical Solution of Stochastic Diffeential Systems with Switching Diffusion uing the Fiing The vecto (4) is use to expess the influence of the extenal foces on the motion system. In this vecto, the fist tem coespons to the weight foce, the secon tem coespons to the ocket thust an the last tem to the ocket jet foce []. In this equation K=(k i,j ) i=,6j=,6 is the noise matix of coefficients that is because of the uncetain of all conitions. In this wok, ε 6 x ae assume as the nomal inepenent vaiables [3] Stochastic iffeential equations (SDEs) povie a founation fo many banches of applie sciences. Inee, they ensue an aequate esciption fo systems subjecte to anom istubances. SDEs epesent a majo object of eseach in moen contol theoy. The significance of such equations is also explaine by thei close connection to equations aising in mathematical physics. A well-known (yet, pomising) appoach to numeical integation of Ito SDEs pocees fom the stochastic analogs of Taylo expansion. Theoetically, this fomula allows esigning methos with abitay lage oes of accuacy (une appopiate assumptions egaing the coefficients of combine equations). An impotant istinguishing featue of the stochastic analogs of Taylo expansion use to solve Ito SDEs lies in the following. These analogs incopoate the so-calle epeate stochastic integals (of the Ito o Statonovich type) being complex functional of the components of the n-imensional Wiene pocess [3]. In Ito s theoy, Hemite polynomials seve as analogs of stana egees of the Wiene pocess. On the othe pat, the exponential supe matingale uplicates the stana exponent of a stochastic integal. In the sense of the mean-squae convegence citeion, the poblem of joint numeical simulation of sets of epeate Ito stochastic integals appeas complicate, both theoetically an pactically. Recently, eseaches have focuse thei attention on mathematical moels in the fom of SDEs with Makovian jumps of the iffusion component. They ae calle switching iffusion moels. Such mathematical objects escibe complex systems subjecte to abupt changes in thei stuctue an paametes (ue to possible failues, isupt ata communication an impacts of an extenal envionment). Switching iffusion moels gain gowing populaity in moen contol theoy an infomation theoy. As a ule, investigatos teat switching iffusions govene by finite-state Makov chains (Makovian switching). Duing compute simulation of such systems, it is necessay to solve numeically SDEs with Makovian switching. Thee exist publications eicate to contollability, stability an stabilization of such systems. Howeve, the issues of numeical solution of SDEs with Makovian switching have been insufficiently exploe. The famous Eule metho was consiee in [3] fo numeical solution of stochastic iffeential equations with Makovian switching; in aition, poofs fo some special esults wee emonstate. This pape veifies the applicability of numeical schemes base on stochastic analogs of Taylo expansion to appoximate solutions of such equations. The est of the pape is oganize as follows: In the Section, the genealize of the Taylo-metho is expesse fo solving the stochastic iffeential equation with switching iffusion. This metho is applie to the poblem fo solving the stochastic iffeential equation fo launching of ocket in Section 3. The numeical esults ae shown in the Section 4.. Review on the Solution of the Stochastic Diffeential Equation with Switching Diffusion Let (Ω, F, P) be a pobabilistic space, Ft (t0 t t0 + T) epesent a non-eceasing family σ compising sub algebas of F, an ((t), Ft), =,...,, inicate inepenent Wiene pocesses. Consie the Ito stochastic iffeential equation: X = a( t, X) t + σ ( t, X) ( t) (5) Whee X, a, an σ mean vectos of length n. The functions a(t, x) an σ ( t, X ) ae given an continuous une t [t0, t0 + T], x _n. Moeove They meet the Lipschitz conition fo all t [t0, t0 + T], x _n, y _n: a( t, x) a( t, y) + σ( t, x) σ( t, y) K x y (6) In the sequel, we aopt the following notation: x efines the Eucliean nom of the vecto x; xy yiels the scala pouct of the vectos x an y; K is a positive constant; X t,x(t) o simply X(t) specifies the solution of Eq. (5) Define the following one-step appoximation t,x(t + h), t0 t t + h t0 + T, which is geneate epening on x, t, h, an {(ϑ) (t),..., (ϑ) (t): t ϑ t + h}: X t, x( t + h) = x + f( t, x, h; i( ϑ) i( t)), i =,...,, t ϑ t + h (7) Assume that (Ω, F, P) is a pobabilistic space, Ft (t0 t t0 + T) epesents a non-eceasing family compising σ-sub

3 Jounal of Numeical Analysis an Applie Mathematics Vol., No., 06, pp algebas of F, an ( ), =,...,, inicate inepenent Wiene pocesses. Let M = {,...,m} be a finite set. Consie the following stochastic iffeential equation with Makovian switching: X ( t ) = a ( β( t ), X ( t )) t + σ ( β( t ), X ( t )) ( t ) (8) The tansient function of such pocess is efine by a set of functions P(t, u, l) = pul(t); they fom the stochastic matix P(t) of tansition ates (pul(t) 0, l pul(t) = ). The values pul act as the pobabilities of tansition fom u to l uing the peio h (une the conition that the pocess β(t) leaves state u uing this peio).[4] The evolution of the whole pocess β(t) is escibe by the values qul, ql. The latte fom the tansition intensity matix m m P( t) P(0) Q( t) = ( qul( t)) R, Q( t) = lim (9) t 0 t Fo each t, we have qul(t) 0 une quu=-qu, = 0, fo each u M []. Suppose that the functions a(β(t), x(t)) an σ(β(t), x(t)) ae efine, enjoy the continuity popety fo t [t0, t0 +T],!, as well as satisfy the Lipschitz conition fo all t [t0, t0 +T],!, "!, # a( u, x) a( u, y) + σ( u, x) σ( u, y) K x y (0) In aition, these functions meet the constaint. a( u, x) + σ( u, x) K( + x) (boune velocity of components with espect to x). Une the state conitions, Eq. (8) amits a unique continuous solution Xu,x on the inteval t 0 fo each initial conition [8]. Set the numeical solution poblem fo Eq. (9). In othe wos, it is necessay to appoximate its solution on the above time inteval. In this fomulation, the tansition intensity matix Q tuns out inepenent of x (but epens only on t). Thus, the poblem can be euce to the wellknown solution poblem fo the stochastic iffeential Eq. (5) on a set of anom subintevals [0, t), [t, t + t),..., whee tk esignate anom instants of switching in the Makov chain (they can be efine a pioi). Recall that the next switching instant tk an the state β(tk) ae anom vaiables whose istibution epens only on (β(tk), tk). Theefoe, one can geneate a pioi the set (β(tk), tk) of the ight-han sies of Eq. () an solve the latte though famous methos. The appoximation accuacy (which epens on step choice within sepaate subintevals) becomes eachable on the whole time inteval. Howeve, anothe solution poceue is applicable to (8), as well. In the case of the stochastic iffeential Eq. (5), E. Platen popose simple eivations (involving meely Ito s fomula) to expan the solution Xt,x(t + h) with espect to powes of h an integals epening on the incements (ϑ) (t), whee t ϑ t +h, =,...,. In the eteministic setting, this expansion epouces Taylo s fomula fo Xt,x(t+h) with espect to powes of h in a neighbohoo of the point (t, x) [4]. In what follows, we employ Platen s expansion (see [4] fo etails) to solve Eq. (8). Fo this, substitute the function f(t, x) by the function f(β(t), x(t)) with the switching component. Such appoach was aopte in [4,8]; the cite authos pesente the coesponing lemmas an theoems in a special case (Eule s scheme stuie in these woks epesents a paticula case of the scheme. Howeve, in the moel of state-epenent switching, numeical solution leas to the following. Switching instants an post-switching states ae also estimate appoximately. Hee a seies of questions aise concening pope intepetation of the iffeence between the appoximate solution an its exact countepat. Pobably, one shoul intouce a cetain measue fo the eviation of the appoximate istibution of switching instants fom thei exact istibution. An amissible way is to moify such measue (making all jumps Poisson jumps); subsequently, the poblem acquies anothe fom. It is necessay to assess (5)the eviation of such measue fom the exact one an() the iffeence between the estimate mean values of the functionals an the exact values. 3. Solving the Stochastic Diffeential Equation with Switching Diffusion fo Launching of Rocket At fist, to solve the equation, system () has been change to the fom of (5) X = a( t, X) t + σ ( t, X) ( t) In top equation, the coefficients must satisfy the LIPSCHITZ popety. Since the coefficients of the equation ae linea, they apply in this conition an the genealize of the Taylometho with switching Diffusion can be use, one can see. In aition Makov- moel can be examine by the genealize of the Taylo-metho with switching iffusion. So all moes of pojectiles an ockets fom beginning of fiing to ening ae use fo Makov states.

4 6 S. Saabaan an H. R. Shaahza: Simulating an Numeical Solution of Stochastic Diffeential Systems with Switching Diffusion uing the Fiing Accoing to this, the system () is euce to fist oe stochastic iffeential equation. ν =ɺ s () s ν =ɺ () zs zs =ɺ ϕ (3) ϕ y y =ɺ ϕ (4) ϕ z z =ɺ γ (5) γ x x algebas of F, an ( ), =,...,, inicate inepenent Wiene pocesses. Let M = {,...,m} be a finite set. Consie the following stochastic iffeential equation with Makovian switching: X( t) = a( β( t), X( t)) t + σ ( β( t), X( t)) ( t) (0) To fin solution to Eq. (0), we popose the following numeical scheme base on the one-step appoximation (with the oot-mean-squae oe of accuacy of 3/ ): =ɺ γ (6) γ y y Using those new vaiables ()-(6), the unknown vaiables vecto can be pesente as follows []: X = [ ν sϕ ϕ ν z γ γ sϕ yϕz zsγ xγ y] (7) y z s x y Using the notations ()-(6) an the vecto (7), as well as the equation (), we obtain the new matix fom of the fist oe iffeential equations, which escibes the motion of the ocket-launching evice system: Xɺ = P. X + Q. ξ + R. ϕ + S. ε (8) Whee, T X = X + a( β, X ) h + σ( β, X ) tk+ tk tk tk tk tk k (, ) (, ){( ) + σ βt X k t σ β k t X k t k k h } + a ( β, X ) σ( β, X ) Z tk tk tk tk k + ( a ( βt, X k t ) a ( β k t, X k tk ) + σ β β + ( a( β, X ) σ ( β, X ) ( t, X ) (, )) k t a k t X k t h k tk tk tk tk + σ ( βt, X ) (, )){ k t σ β k t X k t k k h Zk} + σ ( βt, X k t )( σ ( β k t, X k tk ) σ ( βtk, Xtk ) + ( σ ( βt, X )) ) { ( ) } k t k k h k 3 () P Q R S B6 6 C6 6 = ; I6 6 O6 6 N6 5 = ; O6 5 F6 3 = ; O6 3 K6 6 = O6 6 (9) 4. Numeical Results In this section, expansion of the Taylo-metho with switching Diffusion is teste fo solving the system (8). All numeical esults caie out with Sci-lab. The appoximation of the components of X ae shown in figual fom. (The iffeential of oscillations paametes of ocket-launching evice) Which O 6 6, O 6 5 an O 6 3 ae zeos matices an as mentione befoe othe blocks ae the anom matices that thei elements ae anom values imposing the launching evice uing the fiing. Hee, we solve the matix system (8) by applying aial basis functions. The 6 scala equations ae necessay to calculate the 6 unknown vaiables that escibe the movement of the ocket-launching evice system uing fiing ( s, ϕ y, ϕ z, z s, γ x, γ y ) while the othe scala equations allow to compute the evolutions of the iffeentials of 6 main unknown vaiables efine with ()- (6) []. At fist, the fomula is assume to solve the above equations, Assume that (Ω, F, P) is a pobabilistic space, Ft (t 0 t t 0 + T) epesents a non-eceasing family compising σ-sub Fig.. X changes ove time.

5 Jounal of Numeical Analysis an Applie Mathematics Vol., No., 06, pp. 3-9 Fig. 5. X5 changes ove time. Fig.. X changes ove time. Fig. 6. X6 changes ove time. Fig. 3. X3 changes ove time. Fig. 4. X4 changes ove time. Fig. 7. X7 changes ove time. 7

6 8 S. Saabaan an H. R. Shaahza: Simulating an Numeical Solution of Stochastic Diffeential Systems with Switching Diffusion uing the Fiing Fig.. X changes ove time. Fig. 8. X8 changes ove time. Fig.. X changes ove time. Fig. 9. X9 changes ove time. Fig. 0. X0 changes ove time. Notations an Symbols X Inepenent unknown ynamic vaiables of the ocket-launching evice system motion zs Vehicle chassis tanslation γy Chassis pitch movement γx Chassis olling movement ϕz Gyation movement aoun the vetical axes ϕy Pitch movement s Rocket tanslation ξ Matix of the coefficients fo the nonlinea φ Extenal foces that acts on the system ε Shape paamete

7 Jounal of Numeical Analysis an Applie Mathematics Vol., No., 06, pp Eucliean istance X C Set of cente ν s Deivative of s ν zs Deivative of Z s ϕ y Deivative of ϕ y γ x Deivative of γ x γ y Deivative of γ y Refeences [] P. Somoiag, C. Moloveanu, Numeical eseach on the stability of launching evices uing fiing, Defence Technology 9, (03), [] P. Somoiag, F. Moau, D. Safta, C. Moloveanu, A mathematical moel fo the motion of a ocket-launching evice system on a heavy vehicle, Militay technical acaemy, Romania, (0). [3] Jagaeep Thota Benan J. O Toole Mohame B. Tabia Optimization of shock esponse within a militay vehicle space fame (0): [4] Azamas Polytechnic Institute of Alekseev Nizhni Novgoo State,Technical Univesity,Numeical Solution Algoithms fo Stochastic Diffeential Systems with Switching Diffusion (0). [5] R. Schaback, Impove eo bouns fo scattee ata intepolation by aial basis functions, Math Comput, 68, (999), [6] Kuznetsov, D. F., Stokhasticheskie iffeentsial nye uavneniya: teoiya i paktika chislennogo esheniya (Stochastic Diffeential Equations: Theoy an Pactice of Numeical Solution), St. Petesbug: Politekh. Univ., 007. [7] Kuznetsov, D. F., Stochastic Diffeential Equations: Theoy an Pactice of Numeical Solution, Diffe. Uavn. Pots. Upavlen., 008, no.. [8] Yin, G., Mao, X., Yuan, C., an Cao, D., Appoximation Methos fo Hybi Diffusion Systems with State-epenent Switching Pocesses: Numeical Algoithms an Existence an Uniqueness of Solutions, SIAM J. Math. Anal., 00, vol. 4, no. 6, pp [9] N. Katz, H. Mukai, H. Schattle, M. Zhang, 6 an M. Xu, Solution of a Diffeential Game Fomulation of Militay Ai Opeations by the Metho of Chaacteistics, (005). [0] Lee B, Saitou K Thee-imensional assembly synthesis fo obust imensional integity base on scew theoy. J Mech (006): [] Yuan, C. an Lygeos, J., Stochastic Makovian Switching Hybi Pocesses, Poject IST ,COLUMBUS, Design of Embee Contolles fo Safety Citical Systems, Cambige: Univ. of Cambige, 004.