# American Institute of Mathematical Sciences

January  2018, 23(1): 473-485. doi: 10.3934/dcdsb.2018032

## Solution to a stochastic pursuit model using moment equations

 1 University of Białystok, Faculty of Mathematics and Informatics, K. Ciołkowskiego 1M, 15-245 Białystok, Poland 2 Kiev National Economics V. Hetman University Faculty of Information System and Technology, Kyiv 03068, Peremogy 54/1, Ukraine 3 Palacký University Olomouc, Faculty of Education, Žižkovo nám. 5, Olomouc, Czech Republic 4 Brno University of Technology, Faculty of Civil Engineering, Department of Mathematics and Descriptive Geometry, Veveří 331/95,602 00 Brno, Czech Republic

* Corresponding author: Miroslava Růžičková

Received  September 2016 Published  January 2018

The paper investigates the navigation problem of following a moving target, using a mathematical model described by a system of differential equations with random parameters. The differential equations, which employ controls for following the target, are solved by a new approach using moment equations. Simulations are presented to test effectiveness of the approach.

Citation: Miroslava Růžičková, Irada Dzhalladova, Jitka Laitochová, Josef Diblík. Solution to a stochastic pursuit model using moment equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 473-485. doi: 10.3934/dcdsb.2018032
##### References:
 [1] A. G. Aleksandrov, Optimal and Adaptive Systems, Nauka, Moscow, 1989. [2] J. Diblík, I. Dzhalladova, M. Michalková and M. Růžičková, Modeling of applied problems by stochastic systems and their analysis using the moment equations, Adv. Difference Equ., 2013 (2013), 12 pp. doi: 10.1186/1687-1847-2013-152. Google Scholar [3] J. Diblík, I. Dzhalladova, M. Michalková and M. Růžičková, Moment equations in modeling a stable foreign currency exchange market in conditions of uncertainty Abstr. Appl. Anal. 2013 (2013), Art. ID 172847, 11 pp. doi: 10.1155/2013/172847. Google Scholar [4] I. A. Dzhalladova, Optimization of Stochastic System, KNEU, Kiev, 2005. [5] I. Dzhalladova, M. Růžičková, M. Štoudková and M. Růžičková, Stability of the zero solution of nonlinear differential equations under the influence of white noise, Adv. Difference Equ., 2015 (2015), 11 pp. doi: 10.1186/s13662-015-0482-y. Google Scholar [6] A. I. Egorov, Optimal Control with Linear Systems, KNEU, Kiev, 1988. [7] H. Gao and C. Sun, Well-posedness of stochastic primitive equations with multiplicative noise in three dimensions, Discrete Continuous Dynam. Systems -B, 21 (2016), 3053-3073. doi: 10.3934/dcdsb.2016087. Google Scholar [8] C. G. Guo, C. X. Guo, S. Ahmed and F. X. Liu, Moment stability for nonlinear stochastic growth kinetics of breast cancer stem cells with time-delays, Discrete Continuous Dynam. Systems -B, 21 (2016), 2473-2489. doi: 10.3934/dcdsb.2016056. Google Scholar [9] J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes, Springer-Verlag, New York, 1987. doi: 10.1007/978-3-662-02514-7. [10] M. Růžičková and I. Dzhalladova, The optimization of solutions of the dynamic systems with random structure Abstr. Appl. Anal. 2011 (2011), Art. ID 486714, 18 pp. doi: 10.1155/2011/486714. Google Scholar [11] M. Šagát, Stochastic Differential Equations and their Applications Mgr. thesis University of Žilina, Žilina -2814420132009,2013.Google Scholar [12] K. G. Valeev and I. Dzhalladova, Optimization of Random Processes, KNEU, Kiev, 2006.

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##### References:
 [1] A. G. Aleksandrov, Optimal and Adaptive Systems, Nauka, Moscow, 1989. [2] J. Diblík, I. Dzhalladova, M. Michalková and M. Růžičková, Modeling of applied problems by stochastic systems and their analysis using the moment equations, Adv. Difference Equ., 2013 (2013), 12 pp. doi: 10.1186/1687-1847-2013-152. Google Scholar [3] J. Diblík, I. Dzhalladova, M. Michalková and M. Růžičková, Moment equations in modeling a stable foreign currency exchange market in conditions of uncertainty Abstr. Appl. Anal. 2013 (2013), Art. ID 172847, 11 pp. doi: 10.1155/2013/172847. Google Scholar [4] I. A. Dzhalladova, Optimization of Stochastic System, KNEU, Kiev, 2005. [5] I. Dzhalladova, M. Růžičková, M. Štoudková and M. Růžičková, Stability of the zero solution of nonlinear differential equations under the influence of white noise, Adv. Difference Equ., 2015 (2015), 11 pp. doi: 10.1186/s13662-015-0482-y. Google Scholar [6] A. I. Egorov, Optimal Control with Linear Systems, KNEU, Kiev, 1988. [7] H. Gao and C. Sun, Well-posedness of stochastic primitive equations with multiplicative noise in three dimensions, Discrete Continuous Dynam. Systems -B, 21 (2016), 3053-3073. doi: 10.3934/dcdsb.2016087. Google Scholar [8] C. G. Guo, C. X. Guo, S. Ahmed and F. X. Liu, Moment stability for nonlinear stochastic growth kinetics of breast cancer stem cells with time-delays, Discrete Continuous Dynam. Systems -B, 21 (2016), 2473-2489. doi: 10.3934/dcdsb.2016056. Google Scholar [9] J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes, Springer-Verlag, New York, 1987. doi: 10.1007/978-3-662-02514-7. [10] M. Růžičková and I. Dzhalladova, The optimization of solutions of the dynamic systems with random structure Abstr. Appl. Anal. 2011 (2011), Art. ID 486714, 18 pp. doi: 10.1155/2011/486714. Google Scholar [11] M. Šagát, Stochastic Differential Equations and their Applications Mgr. thesis University of Žilina, Žilina -2814420132009,2013.Google Scholar [12] K. G. Valeev and I. Dzhalladova, Optimization of Random Processes, KNEU, Kiev, 2006.
Distribution of forces acting on the missile
The mean value of the process s(t) with parameters λ and p: λ = 0.01; p = 0.1, 0.2, ..., 1; s(0) = 0, 4, 8, ..., 200
The mean value of the process s(t) with parameters λ and p: λ = 0.01, 0.02, ..., 0.2; p = 0.1, 0.2, ..., 1; s(0) = 0, 4, 8, ..., 200
The mean value of the process $s(t)$ with parameters $\lambda$ and $p$: $\lambda=0.01, 0.011, 0.012, \ldots, 0.21$; $p=0, 0.01, \ldots, 1$; $s(0)=0$
The mean value of the process $s(t)$ with parameters $\lambda$ and $p$: $\lambda=0.01, 0.011, 0.012, \ldots, 0.21$; $p=0, 0.01, \ldots, 1$; $s(0)=0, 20, \ldots, 100$
The mean value $E_{1}^{(1)}\{s( t )\}$ of the stochastic process as the solution to (12) and the mean value of two simulations of the stochastic process
The mean value $E_{1}^{(1)}\{s( t )\}$ of the stochastic process as the solution to(12) and the mean value of one hundred simulations of the stochastic process
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