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\(\mathrm{L}^p\)-calculus approach to the random autonomous linear differential equation with discrete delay. (English) Zbl 1417.34198

Summary: In this paper, we provide a full probabilistic study of the random autonomous linear differential equation with discrete delay \(\tau >0\): \[x'(t)=ax(t)+bx(t-\tau), \quad t\ge 0\] with initial condition \(x(t)=g(t)\), \(-\tau \le t\le 0\). The coefficients \(a\) and \(b\) are assumed to be random variables, while the initial condition \(g\)(\(t\)) is taken as a stochastic process. Using \(\mathrm{L}^p\)-calculus, we prove that, under certain conditions, the deterministic solution constructed with the method of steps that involves the delayed exponential function is an \(\mathrm{L}^p\)-solution too. An analysis of \(\mathrm{L}^p\)-convergence when the delay \(\tau \) tends to 0 is also performed in detail.

MSC:

34K50 Stochastic functional-differential equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
65C30 Numerical solutions to stochastic differential and integral equations
34K06 Linear functional-differential equations
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[1] Smith, H.: An Introduction to Delay Differential Equations with Applications to the Life Sciences, Texts in Applied Mathematics. Springer, New York (2011) · Zbl 1227.34001 · doi:10.1007/978-1-4419-7646-8
[2] Driver, Y.: Ordinary and Delay Differential Equations. Applied Mathematical Science Series. Springer, New York (1977) · Zbl 0374.34001 · doi:10.1007/978-1-4684-9467-9
[3] Kuang, Y.: Delay Differential Equations: with Applications in Population Dynamics. Academic Press, Cambridge (2012)
[4] Bocharov, G.A., Rihan, F.A.: Numerical modelling in biosciences using delay differential equations. J. Comput. Appl. Math. 125, 183-199 (2000). https://doi.org/10.1016/S0377-0427(00)00468-4 · Zbl 0969.65124 · doi:10.1016/S0377-0427(00)00468-4
[5] Jackson, M., Chen-Charpentier, B.M.: Modeling plant virus propagation with delays. J. Comput. Appl. Math. 309, 611-621 (2017). https://doi.org/10.1016/j.cam.2016.04.024 · Zbl 1348.92153 · doi:10.1016/j.cam.2016.04.024
[6] Chen-Charpentier, B.M., Diakite, I.: A mathematical model of bone remodeling with delays. J. Comput. Appl. Math. 291, 76-84 (2016). https://doi.org/10.1016/j.cam.2017.01.005 · Zbl 1320.92051 · doi:10.1016/j.cam.2017.01.005
[7] Erneux, T.: Applied Delay Differential Equations, Surveys and Tutorials in the Applied Mathematical Sciences Series. Springer, New York (2009)
[8] Kyrychko, Y.N., Hogan, S.J.: On the Use of delay equations in engineering applications. J. Vib. Control 16(7-8), 943-960 (2017). https://doi.org/10.1177/1077546309341100 · Zbl 1269.70002 · doi:10.1177/1077546309341100
[9] Matsumoto, A., Szidarovszky, F.: Delay Differential Nonlinear Economic Models (in Nonlinear Dynamics in Economics, Finance and the Social Sciences), 195-214. Springer-Verlag, Berlin Heidelberg (2010) · Zbl 1189.37117
[10] Harding, L., Neamtu, M.: A dynamic model of unemployment with migration and delayed policy intervention. Comput. Econ. 51(3), 427-462 (2018). https://doi.org/10.1007/s10614-016-9610-3 · doi:10.1007/s10614-016-9610-3
[11] Oksendal, B.: Stochastic Differential Equations. Springer, New York (1998) · doi:10.1007/978-3-662-03620-4
[12] Shaikhet, L.: Lyapunov Functionals and Stability of Stochastic Functional Differential Equations. Springer, New York (2013) · Zbl 1277.34003 · doi:10.1007/978-3-319-00101-2
[13] Hartung, F., Pituk, M.: Recent Advances in Delay Differential and Differences Equations. Springer-Verlag, Berlin Heidelberg (2014) · Zbl 1297.34002 · doi:10.1007/978-3-319-08251-6
[14] Shaikhet, L.: Stability of equilibrium states of a nonlinear delay differential equation with stochastic perturbations. Int. J. Robust Nonlinear Control 27(6), 915-924 (2016). https://doi.org/10.1002/rnc.3605 · Zbl 1369.93702 · doi:10.1002/rnc.3605
[15] Shaikhet, L.: About some asymptotic properties of solution of stochastic delay differential equation with a logarithmic nonlinearity. Funct. Differ. Equ. 4(1-2), 57-67 (2017) · Zbl 1474.34566
[16] Fridman, E., Shaikhet, L.: Delay-induced stability of vector second-order systems via simple Lyapunov functionals. Automatica 74, 288-296 (2016). https://doi.org/10.1016/j.automatica.2016.07.034 · Zbl 1348.93218 · doi:10.1016/j.automatica.2016.07.034
[17] Benhadri, M., Zeghdoudi, H.: Mean square asymptotic stability in nonlinear stochastic neutral Volterra-Levin equations with Poisson jumps and variable delays. Functiones et Approximatio Commentarii Mathematici 58(2), 157-176 (2018). https://doi.org/10.7169/facm/1657 · Zbl 1395.45014 · doi:10.7169/facm/1657
[18] Nouri, K., Ranjbar, H.: Improved Euler-Maruyama method for numerical solution of the Itô stochastic differential systems by composite previous-current-step idea. Mediterr. J. Math. 15, 140 (2018). https://doi.org/10.1007/s00009-018-1187-8 · Zbl 1393.60063 · doi:10.1007/s00009-018-1187-8
[19] Santonja, F., Shaikhet, L.: Probabilistic stability analysis of social obesity epidemic by a delayed stochastic model. Nonlinear Anal. Real World Appl. 17, 114-125 (2014). https://doi.org/10.1016/j.nonrwa.2013.10.010 · Zbl 1292.93141 · doi:10.1016/j.nonrwa.2013.10.010
[20] Santonja, F., Shaikhet, L.: Analysing social epidemics by delayed stochastic models. Discret. Dyn. Nat. Soc. 2012, 13 (2012). https://doi.org/10.1155/2012/530472. (Article ID 530472) · Zbl 1248.91088 · doi:10.1155/2012/530472
[21] Liu, L., Caraballo, T.: Analysis of a stochastic 2D-Navier-Stokes model with infinite delay. J. Dyn. Differ. Equ. pp 1-26 (2018). https://doi.org/10.1007/s10884-018-9703-x · Zbl 1427.35182
[22] Caraballo, T., Colucci, R., Guerrini, L.: On a predator prey model with nonlinear harvesting and distributed delay. Commun. Pure Appl. Anal. 17(6), 2703-2727 (2018). https://doi.org/10.3934/cpaa.2018128 · Zbl 1394.92104 · doi:10.3934/cpaa.2018128
[23] Smith, RC, Uncertainty Quantification (2014), Philadelphia · Zbl 1284.65019
[24] Soong, T.T.: Random Differential Equations in Science and Engineering. Academic Press, New York (1973) · Zbl 0348.60081
[25] Nouri, K., Ranjbar, H.: Mean square convergence of the numerical solution of random differential equations. Mediterr. J. Math. 12(3), 1123-1140 (2015). https://doi.org/10.1007/s00009-014-0452-8 · Zbl 1338.60174 · doi:10.1007/s00009-014-0452-8
[26] Zhou, T.: A stochastic collocation method for delay differential equations with random input. Adv. Appl. Math. Mech. 6(4), 403-418 (2014). https://doi.org/10.4208/aamm.2012.m38 · Zbl 1305.65010 · doi:10.4208/aamm.2012.m38
[27] Shi, W., Zhang, C.: Generalized polynomial chaos for nonlinear random delay differential equations. Appl. Numer. Math. 115, 16-31 (2017). https://doi.org/10.1016/j.apnum.2016.12.004 · Zbl 1358.65010 · doi:10.1016/j.apnum.2016.12.004
[28] Lupulescu, V., Abbas, U.: Fuzzy delay differential equations. Fuzzy Optim. Decis. Mak. 11(1), 91-111 (2012). https://doi.org/10.1007/s10700-011-9112-7 · Zbl 1254.34006 · doi:10.1007/s10700-011-9112-7
[29] Liu, S., Debbouche, A., Wang, J.R.: Fuzzy delay differential equations. On the iterative learning control for stochastic impulsive differential equations with randomly varying trial lengths. J. Comput. Appl. Math. 312, 47-57 (2017). https://doi.org/10.1016/j.cam.2015.10.028 · Zbl 1351.65005 · doi:10.1016/j.cam.2015.10.028
[30] Krapivsky, P.L., Luck, J.L., Mallick, K.: On stochastic differential equations with random delay. J. Stat. Mech. Theory Exp. (2011). https://doi.org/10.1088/1742-5468/2011/10/P10008 · doi:10.1088/1742-5468/2011/10/P10008
[31] Garrido-Atienza, M.J., Ogrowsky, A., Schmalfuss, B.: Random differential equations with random delays. Stoch. Dyn. 11(2-3), 369-388 (2011). https://doi.org/10.1142/S0219493711003358 · Zbl 1270.34182 · doi:10.1142/S0219493711003358
[32] Khusainov, D.Y., Ivanov, A.F., Kovarzh, I.V.: Solution of one heat equation with delay. Nonlinear Oscil. 12, 260-282 (2009). https://doi.org/10.1007/s11072-009-0075-3 · Zbl 1277.35193 · doi:10.1007/s11072-009-0075-3
[33] Asl, F.M., Ulsoy, A.G.: Analysis of a system of linear delay differential equations. J. Dyn. Syst. Meas. Control 125, 215-223 (2003). https://doi.org/10.1115/1.1568121 · doi:10.1115/1.1568121
[34] Kyrychko, Y.N., Hogan, S.J.: On the use of delay equations in engineering applications. J. Vib. Control 16(7-8), 943-960 (2010). https://doi.org/10.1177/1077546309341100 · Zbl 1269.70002 · doi:10.1177/1077546309341100
[35] Villafuerte, L., Braumann, C.A., Cortés, J.C., Jódar, L.: Random differential operational calculus: theory and applications. Comput. Math. Appl. 59(1), 115-125 (2010). https://doi.org/10.1016/j.camwa.2009.08.061 · Zbl 1189.60126 · doi:10.1016/j.camwa.2009.08.061
[36] Strand, J.L.: Random ordinary differential equations. J. Diff. Equ. 7(3), 538-553 (1970). https://doi.org/10.1016/0022-0396(70)90100-2 · Zbl 0231.34051 · doi:10.1016/0022-0396(70)90100-2
[37] Khusainov, D.Y., Pokojovy, M.: Solving the linear 1d thermoelasticity equations with pure delay. Int. J. Math. Math. Sci. 2015, 1-11 (2015). https://doi.org/10.1155/2015/479267 · Zbl 1476.35264 · doi:10.1155/2015/479267
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