Numerical Method for Solving Optimal Control Problem of Stochastic Volterra Integral Equations Using Block Pulse Functions
-
3409
Downloads
-
5256
Views
Authors
M. Saffarzadeh
- Department of Mathematics, Yazd University P.O. Box: 89195-741 Yazd, Iran.
A. Delavarkhalafi
- Department of Mathematics, Yazd University P.O. Box: 89195-741 Yazd, Iran.
Z. Nikoueinezhad
- Department of Mathematics, Yazd University P.O. Box: 89195-741 Yazd, Iran.
Abstract
In this paper, a numerical method for solving a general optimal control of systems is presented. These systems governed by stochastic Volterra integral equations. This method is based on block pulse functions. By using the properties of block pulse functions and associated operational matrices, optimal control problem is converted to an optimization problem and will be solved via mathematical programming techniques. The error estimations and associated theorems have been provided. Finally, some numerical examples are presented to show the validity and efficiency of the proposed method.
Share and Cite
ISRP Style
M. Saffarzadeh, A. Delavarkhalafi, Z. Nikoueinezhad, Numerical Method for Solving Optimal Control Problem of Stochastic Volterra Integral Equations Using Block Pulse Functions, Journal of Mathematics and Computer Science, 11 (2014), no. 1, 22-36
AMA Style
Saffarzadeh M., Delavarkhalafi A., Nikoueinezhad Z., Numerical Method for Solving Optimal Control Problem of Stochastic Volterra Integral Equations Using Block Pulse Functions. J Math Comput SCI-JM. (2014); 11(1):22-36
Chicago/Turabian Style
Saffarzadeh, M., Delavarkhalafi, A., Nikoueinezhad, Z.. "Numerical Method for Solving Optimal Control Problem of Stochastic Volterra Integral Equations Using Block Pulse Functions." Journal of Mathematics and Computer Science, 11, no. 1 (2014): 22-36
Keywords
- Stochastic Volterra integral equations
- Optimal control
- Block pulse functions
- Stochastic operational matrix.
MSC
References
-
[1]
A. W. Heemink, I. D. M. Metzelaar, Data assimilation into a numerical shallow water flow model: A stochastic optimal control approach, J. Marine. Syst., 6 (1995), 145-158.
-
[2]
Y. Shastria, U. Diwekarb, Sustainable ecosystem management using optimal control theory, Part 2 (stochastic systems), J. Theor. Biol. , 241 (2006), 522-532.
-
[3]
A. J. Coldman, J. M. Murray, Optimal control for a stochastic model of cancer chemotherapy, Math. Biosci. , 168 (2000), 187-200.
-
[4]
J. L. Stein, Applications of stochastic optimal control/dynamic programming to international finance and debt crises, Nonlinear. Anal., 63 (2005), e2033-e2041.
-
[5]
J. M. Petersen, M. A. Petersen , Bank management via stochastic optimal control, Automatica, 42 (2006), 1395-1406.
-
[6]
S. U. Acikgoz, U. M. Diwekar, Blood glucose regulation with stochastic optimal control for insulin-dependent diabetic patients, Chem. Eng. Sci. , 65 (2010), 1227-1236.
-
[7]
P. T. Benavides, U. Diweka, Optimal control of biodiesel production in a batch reactor , Part II: Stochastic control, Fuel. , 94 (2012), 218-226.
-
[8]
W. H. Fleming, C. J. Rishel, Deterministic and Stochastic Optimal Control, Springer-Verlag, (1975)
-
[9]
W. H. Fleming, H. M. Soner, Controlled Markov Processes and Viscosity Solutions, Springer, (2006)
-
[10]
R. E. Bellman, Dynamic Programming, Princeton University Press, Princeton, NJ (1957)
-
[11]
R. E. Bellman, S. E. Dreyfus, Applied Dynamic Programming, Princeton University Press, Princeton, NJ ( 1962)
-
[12]
D. E. Kirk , Optimal Control Theory An Introduction, Prentice-Hall , Englewood Cliffs (1970)
-
[13]
A. B. Pantelev, A. C. Bortakovski, T. A. Letova, Some Issues and Examples in Optimal Control, MAI Press , Moscow (in Russian) (1996)
-
[14]
E. R. Pinch, Optimal Control and the Calculus of Variations, Oxford University Press, London (1993)
-
[15]
L. S. Pontryagin, The Mathematical Theory of Optimal Processes, Interscience, John Wiley and Sons (1962)
-
[16]
A. Jajarmi, N. Pariz, S. Effati, A. V. Kamyad, Infinite horizon optimal control for nonlinear interconnected Large-Scale dynamical systems with an application to optimal attitude control, Asian. J. Control. , 15 (2013), 1-12.
-
[17]
B. Kafash, A. Delavarkhalafi, S. M. Karbassi, Application of variational iteration method for Hamilton–Jacobi–Bellman equations, Appl. Math. Modell., 37 (2013), 3917–3928.
-
[18]
H. M. Jaddu, Direct solution of nonlinear optimal control problems using quasilinearization and Chebyshev polynomials, J. Franklin Inst., 39 (2002), 479-498.
-
[19]
B. Kafash, A. Delavarkhalafi, S. M. Karbassi, Application of Chebyshev polynomials to derive efficient algorithms for the solution of optimal control problems, Sci. Iran., 19 (2012), 795-805.
-
[20]
K. Maleknejad, H. Almasieh, Optimal control of Volterra integral equations via triangular functions, Math. Comput. Modell., 53 (2011), 1902–1909.
-
[21]
H. R. Erfanian, M. H. Noori Skandari, Optimal control of an HIV model, The Journal of Mathematics and Computer Science , 2 (2011), 650-658.
-
[22]
E. Hesameddini, A. Fakharzadeh Jahromi, M. Soleimanivareki, H. Alimorad, Differential transformation method for solving a class of nonlinear optimal control problems , The Journal of Mathematics and Computer Science , 5 (2012), 67-74.
-
[23]
C. Myers, Stochastic Control, Sciyo, Croatia (2010)
-
[24]
B. Øksendal, T. Zhang, Optimal control with partial information for stochastic Volterra equations, Int. J. Stoch. Anal. Art. ID 329185, (2010), 25 pp.
-
[25]
S. Ji, X. Y. ZHOU, A maximum principle for stochastic optimal control with terminal state constraints and its applications , Commun. Info. Sys. , 6 (2006), 321-338.
-
[26]
Z. Wu, A general maximum principle for optimal control of forward-backward stochastic systems, Automatica, 49 (2013), 1473-1480.
-
[27]
S. Bonaccorsi, F. Confortola, E. Matrogiacomo, Optimal control for stochastic Volterra equations with completely monotone kernels, Siam. J. Control. Optim. , 50 (2012), 748-789.
-
[28]
N. Kuchkina, L. Shaikhet, Optimal control of Volterra Type stochastic difference equations , Comput. Math. Appl. , 36 (1998), 251-259.
-
[29]
H. J. Kushner, Numerical methods for stochastic control problems in finance, Lefschetz Center for Dynamical Systems and Center for Control Sciences, Division of Applied Mathematics, Brown University (1995)
-
[30]
C. Munk, Numerical methods for continuous-time, continuous-state stochastic control problems, Publications from department of management , 97, No. 11 (1997)
-
[31]
C. Munk , The Markov chain approximation approach for numerical solution of stochastic control problems: experiences from Merton’s problem, Appl. Math. Comput. , 136 (2003), 47-77.
-
[32]
W. Chavanasporn, C. O. Ewald, A numerical method for solving stochastic optimal control problems with linear control, Comput. Econ., 39 (2012), 429-446.
-
[33]
K. Maleknejad, M. Khodabin, M. Rostami, Numerical solution of stochastic Volterra integral equations by a stochastic operational matrix based on block pulse functions, Math. Comput. Modell., 55 (2012), 791-800.
-
[34]
G. P. Rao, Piecewise constant orthogonal functions and their application to systems and control, Springer, Berlin (1983)
-
[35]
Z. H. Jiang, W. Schaufelberger, Block pulse functions and their applications in control systems, Springer-Verlag, (1992)
-
[36]
K. Maleknejad, M. Khodabin, M. Rostami, A numerical method for solving m-dimensional stochastic Itô-Volterra integral equations by stochastic operational matrix, Comput. Math. Appl. , 63 (2012), 133-143.
-
[37]
M. Khodabin, K. Maleknejad, M. Rostami, M. Nouri , Numerical approach for solving stochastic Volterra-Fredholm integral equations by stochastic operational matrix, Comput. Math. Appl. , 64 (2012), 1903-1913.
-
[38]
D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, Siam Rev. , 43 (2001), 525-546.
-
[39]
J. Engwerda, LQ Dynamic Optimization and Diffrential Games, John Wileyn and Sons LTD, (2005)