Study of a stochastic model of the “dangling spider” problem with an infinite previous history and Poisson switchings. (English. Russian original) Zbl 0992.60061
Cybern. Syst. Anal. 36, No. 4, 539-560 (2000); translation from Kibern. Sist. Anal. 2000, No. 4, 79-105 (2000).
The paper deals with a stochastic integro-differential equation defined by an external Gaussian white noise and the integral of a Poisson measure which pictures an infinite previous history. By using Picard’s method, one can prove the existence and the uniqueness of the solution, but to this end, one has to use an alternative to the classical Gronwall’s estimate which does not work here. Then one studies the stability of the solution of this equation: it is shown that the second Lyapunov method applies, and one constructs Lyapunov functionals.
Reviewer: Guy Jumarie (Montréal)
MSC:
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
| 34K50 | Stochastic functional-differential equations |
References:
| [1] | Andreeva, E. A.; Kolmanovskii, V. B.; Shaikhet, L. E., Control of Systems with an Aftereffect (1992), Moscow: Nauka, Moscow · Zbl 0840.93003 |
| [2] | Azbelev, N. V.; Maksimov, V. P.; Rakhmatullina, L. F., Introduction to the Theory of a Functional-Differential Equations (1991), Moscow: Nauka, Moscow · Zbl 0725.34071 |
| [3] | Billingsley, P., Convergence of Probability Measures (1977), Moscow: Nauka, Moscow · Zbl 0172.21201 |
| [4] | Watanabe, S.; Ikeda, N., Stochastic Differential Equations and Diffusion Processes (1986), Moscow: Nauka, Moscow · Zbl 0607.60041 |
| [5] | Gikhman, I. I.; Skorokhod, A. V., Stochastic Differential Equations (1968), Kiev: Naukova Dumka, Kiev · Zbl 0169.48702 |
| [6] | Gikhman, I. I.; Skorokhod, A. V., The Theory of Random Processes (1975), Moscow: Nauka, Moscow · Zbl 0348.60042 |
| [7] | Gikhman, I. I.; Skorokhod, A. V., Stochastic Differential Equations and Their Applications (1982), Kiev: Naukova Dumka, Kiev · Zbl 0557.60041 |
| [8] | Danford, N.; Schwarz, J., The Linear Operators: A General Theory (1962), Moscow: Izd. Inostr. Lit., Moscow |
| [9] | Doob, J. L., Stochastic Processes (1965), Moscow: Izd. Inostr. Lit., Moscow |
| [10] | Dynkin, E. B., Markovian Processes (1963), Moscow: Fizmatgiz, Moscow · Zbl 0132.37701 |
| [11] | Ito, K.; Nisio, M., Stationary solutions of a stochastic differential equation, Matematika: Collection of Translations of Foreign Literature, 11, 5, 117-173 (1967) |
| [12] | Katz, I. Ya.; Krasovskii, N. N., On stability of systems with random parameters, Prikl. Mat. Mekh., 27, 5, 809-823 (1960) · Zbl 0103.36403 |
| [13] | Kolmanovskii, V. B.; Nosov, V. R., Stability and Periodic Conditions of Controlled Systems with Aftereffect (1981), Moscow: Nauka, Moscow |
| [14] | Korolyuk, V. S., Stochastic Models of Systems (1989), Kiev: Naukova Dumka, Kiev · Zbl 0753.60083 |
| [15] | Korolyuk, V. S.; Portenko, N. I.; Skoorokhod, A. V.; Turbin, A. F., Probability Theory and Mathematical Satistics: A Manual (1985), Moscow: Nauka, Moscow · Zbl 0608.60001 |
| [16] | Krasovskii, N. N., Some Problems of the Theory of Stability of Motion (1959), Moscow: Fizmatgiz, Moscow · Zbl 0085.07202 |
| [17] | Krein, M. G.; Rutman, M. A., Linear operators forming an invariant cone in Banach space, Usp. Mat. Nauk., 3, 1, 3-95 (1947) · Zbl 0030.12902 |
| [18] | Kushner, G. J., Stochastic Stability and Control (1989), Moscow: Mir, Moscow |
| [19] | Sverdan, M. L.; Tsarkov, E. F.; Yasinskii, V. K., Stability in Stochastic Simulation of Complex Dynamic Systems (1996), Snyatyn: Nad Prutom, Snyatyn |
| [20] | Sverdan, M. L.; Tsarkov, E. F., Stability of Stochastic Sampled-Data Systems (1994), Riga: RTU, Riga · Zbl 0809.60077 |
| [21] | Skorokhod, A. V., Asymptotic Methods of the Theory of Stochastic Differential Equations (1987), Kiev: Naukova Dumka, Kiev · Zbl 0709.60057 |
| [22] | Khas’minskii, R. Z., Stability of Systems of Differential Equations with Random Disturbances of Their Parameters (1969), Moscow: Nauka, Moscow · Zbl 0214.15903 |
| [23] | Hale, J., The Theory of Functional-Differential Equations (1984), Moscow: Mir, Moscow · Zbl 1092.34500 |
| [24] | Tsar’kov, E. F., Random Disturbances of Differential Functional Equations (1989), Riga: Zinatne, Riga · Zbl 0725.34092 |
| [25] | Tsar’kov, E. F.; Yasinskii, V. K., Quasilinear Stochastic Differential-Functional Equations (1992), Riga: Orientir, Riga |
| [26] | Yasinskaya, L. I.; Yasinskii, V. K., On asymptotical behavior of solutions of stochastic differential equations with variable delay and with Poisson perturbations, Stochastic Systems and Their Applications, 107-116 (1990), Kiev: Inst. Mat. Akad. Nauk Ukr. SSR, Kiev |
| [27] | Yasinskaya, L. I., Almost sure stability of a trivial solution of a system with an aftereffect and with discrete random disturbances, Approximate Methods of Investigation of Nonlinear Vibrations, 180-188 (1983), Kiev: Inst. Mat. Akad. Nauk Ukr. SSR, Kiev · Zbl 0554.60064 |
| [28] | Yasinskii, I. V.; Yasinskaya, L. I., On global stability of solutions of stochastic functional differential equations, Visnyk Kyiv. Univ., Mat. Mekh., 127-133 (1995), Kiev: Issue 38, Lybid’, Kiev |
| [29] | Yasinskii, I. V.; Stoyanov, J.; Yasinskii, V. K., Stability of Stochastic Differential-Functional Equations with Infinite Aftereffect (1993), Sofia: Izd. Inst. Matern. Bolgar. AN, Sofia |
| [30] | Chang, M. H.; Ladde, G. S.; Lik, P. T., Stability of stochastic functional differential equations, J. Math. Phys., 15, 9, 1471-1478 (1984) · Zbl 0285.60045 |
| [31] | Chow, P. L., Stability of nonlinear stochastic evolution equations, J. Math. Anal. Appl., 89, 400-419 (1982) · Zbl 0496.60059 · doi:10.1016/0022-247X(82)90110-X |
| [32] | Coleman, B. D.; Mizel, V. J., Norms and semigroups in the theory of fading nemory, Arch. Rational Mech. Anal., 23, 87-123 (1966) · Zbl 0146.46104 · doi:10.1007/BF00251727 |
| [33] | Coleman, B. D.; Owen, D. R., On the initial value problem for a class of functional differential equations, Arch. Rational Mech. Anal., 55, 275-299 (1974) · Zbl 0293.34094 · doi:10.1007/BF00250436 |
| [34] | Ichikawa, A., Absolute stability of stochastic evolution equations, Stochastic, 11, 143-158 (1983) · Zbl 0531.93065 |
| [35] | Mizel, V. J.; Trutzer, V., Stochastic hereditary equations: existence and asymptotic stability, J. Integral Equations, 7, 1-72 (1984) · Zbl 0539.60052 |
| [36] | Trutzer, V., Existence and Asymptotic Stability for Solutions to Stochastic Hereditary Equations: Diss. Ph.D. (1982), Pittsburgh: Carnegie Mellon Univ, Pittsburgh |
| [37] | I. V. Yasinskii,“On strong solutions of stochastic differential-functional equations with infinite aftereffect,” in: Proc. of the Latvian Probability Seminar, Pt.1 (1992), pp. 189-214. |
| [38] | I. V. Yasinskii,“Stability of differential-functional equations with infinite aftereffect and Markovian parameters,” Proc. of Intern. Conf. to the Memory of Acad. M. P. Kravchuk (Ukraine, Kiev-Lutsk, September, 22-28) [in Ukrainian], Kiev (1992). |
| [39] | I. V. Yasinskii M. L. Sverdan,“Continuous dependence of solutions of stochastic differential-functional equations on the initial data,” Proc. of Intern. Conf. to the Memory of Acad. M. P. Kravchuk (Ukraine, Kiev-Lutsk, September, 22-28) [in Ukrainian], Kiev (1992). |
| [40] | I. V. Yasinskii,“The comparison theorem for solutions of stochastic differential-functional equations with discrete trajectories,” Proc. of the Conf. on Nonlinear Problems of Differential Equations and Mathematical Physics, 2nd Bogolyubov Readings (Kiev, September 14-18, 1992) [in Russian], Kiev (1992). |
| [41] | I. V. Yasinskii,“Exponential stability of solutions of stochastic differential-functional equations,” 4th Colloquium on Diff. Eqs., Bulgaria, August 18-22, Plovdiv (1993). |
| [42] | I. V. Yasinskii and L. I. Yasinskaya, “On the inverse theorem to the Lyapunov method for stochastic differential-functional equations with the Poisson disturbances,” All-Ukrainian Scientific Conference (Drogobych, January 25-27) [in Ukrainian], Kiev (1994). |
| [43] | Yasinskii, I. V.; Gotynchan, G. I., A semigroup approach to investigation of stability of linear stochastic differential-functional equations, Intern. Math. Conf. to G. Gan’s Memory (Chernivtsi, Ukraine, October, 10-15) (1994), Chernivtsi: Ruta, Chernivtsi |
| [44] | Yasinskii, I. V.; Sverdan, M. L., Stability of stochastic systems with infinite aftereffect under Markovian disturbances, Intern. Math. Conf. to G. Gan’s Memory (1994), Chernivtsi: Ruta, Chernivtsi |
| [45] | Yasinskii, I. V., Characteristics of stochastic differential-functional equations with infinite sequel, Proc of the Latvian Probability Seminar, 116-132 (1994), Riga: Techn. Univ., Riga |
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