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.   Google Scholar
[2]

J. DiblíkI. DzhalladovaM. 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.   Google Scholar
[5]

I. DzhalladovaM. RůžičkováM. 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.   Google Scholar
[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. GuoC. X. GuoS. 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.  Google Scholar
[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.   Google Scholar

show all references

References:
[1] A. G. Aleksandrov, Optimal and Adaptive Systems, Nauka, Moscow, 1989.   Google Scholar
[2]

J. DiblíkI. DzhalladovaM. 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.   Google Scholar
[5]

I. DzhalladovaM. RůžičkováM. 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.   Google Scholar
[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. GuoC. X. GuoS. 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.  Google Scholar
[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.   Google Scholar
Figure 1.  Distribution of forces acting on the missile
Figure 2.  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
Figure 3.  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
Figure 4.  The mean value of the process $s(t)$ with parameters $\lambda$ and $p$: $\lambda=0.01, 0.011, 0.012, \ldots, 0.21$; $p=0, 0.01, \ldots, 1$; $s(0)=0$
Figure 5.  The mean value of the process $s(t)$ with parameters $\lambda$ and $p$: $\lambda=0.01, 0.011, 0.012, \ldots, 0.21$; $p=0, 0.01, \ldots, 1$; $s(0)=0, 20, \ldots, 100$
Figure 6.  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
Figure 7.  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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