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. 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.

[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.

[4] I. A. Dzhalladova, Optimization of Stochastic System, KNEU, Kiev, 2005.
[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.

[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.

[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.

[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.

[11]

M. Šagát, Stochastic Differential Equations and their Applications Mgr. thesis University of Žilina, Žilina -2814420132009,2013.

[12] K. G. Valeev and I. Dzhalladova, Optimization of Random Processes, KNEU, Kiev, 2006.

show all references

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

[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.

[4] I. A. Dzhalladova, Optimization of Stochastic System, KNEU, Kiev, 2005.
[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.

[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.

[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.

[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.

[11]

M. Šagát, Stochastic Differential Equations and their Applications Mgr. thesis University of Žilina, Žilina -2814420132009,2013.

[12] K. G. Valeev and I. Dzhalladova, Optimization of Random Processes, KNEU, Kiev, 2006.
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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