**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**33012

##### Optimal Control of Volterra Integro-Differential Systems Based On Legendre Wavelets and Collocation Method

**Authors:**
Khosrow Maleknejad,
Asyieh Ebrahimzadeh

**Abstract:**

In this paper, the numerical solution of optimal control problem (OCP) for systems governed by Volterra integro-differential (VID) equation is considered. The method is developed by means of the Legendre wavelet approximation and collocation method. The properties of Legendre wavelet together with Gaussian integration method are utilized to reduce the problem to the solution of nonlinear programming one. Some numerical examples are given to confirm the accuracy and ease of implementation of the method.

**Keywords:**
Collocation method,
Legendre wavelet,
optimal control,
Volterra integro-differential equation.

**Digital Object Identifier (DOI):**
doi.org/10.5281/zenodo.1093708

**References:**

[1] E. Tohidi and O. R. N. Samadi, Optimal control of nonlinear Volterra integral equations via Legendre polynomials, IMA J. Math. Control Info, vol. 30, no. 3, pp. 67-83, July 2012.

[2] T. S. Angell, On the optimal control of systems governed by nonlinear Volterra equations, J. Optim. Theory Appl., vol. 19, no. 1, pp. 29-45, 1976.

[3] A. H. Borzabadi, A. Abbasi, and O. S. Fard, Approximate optimal control for a class of nonlinear Volterra integral equation, J. Am. Sci., vol. 6, no. 11, pp. 1017-1021, 2010.

[4] S. A. Belbas, A reduction method for optimal control of Volterra integral equations, Appl. Math. Comput, vol. 197, no. 2, pp. 880-890, April 2008.

[5] G. N Elnegar Optimal control computation for integro-differential aerodynamic equations, Math. Method Appl. Sci., vol. 21, no. 7, pp. 653-664, May. 1998.

[6] S. A. Belbas, A new method for optimal control of Volterra integral equations, Appl. Math. Comput., vol. 189, no. 2, pp. 1902-1915, 2007.

[7] S. A. Belbas, Iterative schemes for optimal control of Volterra integral equations, Nonlinear Anal., vol. 37, no. 1, pp. 57-79, July 1999.

[8] K. Maleknejad and H. Almasieh, Optimal control of Volterra integral equations via triangular functions, Math. Comput. Modeling, vol. 53, no. 9/10, pp. 1902-1909, May 2011.

[9] K. Maleknejad, A. Ebrahimzadeh, The use of rationalized Haar wavelet collocation method for solving optimal control of Volterra integral equation, J. Vib. Control, In Press, Sep. 2013.

[10] M. R. Peygham, M. Hadizadeh, and A. Ebrahimzadeh, Some explicit class of hybrid methods for optimal control of Volterra integral equations, J. Inform. Comput. Sci., vol. 7, no. 4, pp. 253-266, Feb.2012.

[11] N. G. Medhin, Optimal processes governed by integral equations, J. Math. Anal. Applic, vol. 120, no. 1, pp. 1-12, Nov. 1986.

[12] W. H. Schmidt, Volterra integral processes with state constraints, SAMS, vol. 9, pp. 213-224, 1992.

[13] J. T. Betts, Survey of numerical methods for trajectory optimization, J. Guid. Control Dynam., vol. 21, no. 2, pp. 193207, April. 1998.

[14] Y. U. A. Kochetkov and V. P Tomshin Optimal control of deterministic systems described by integro-differential equations, Automat. Remote Control, vol. 39, no. 1, pp. 1-6, 1978.

[15] J. H. Chou and I. R. Horng, Optimal control of deterministic systems described by integro-differential equations via Chebyshev series, J. Dyn. Sys. Meas., Control, vol. 100, no. 4, pp. 345-348, Dec. 1987.

[16] Q. Gong, I. M. Ross, W. Kang and F. Fahroo, Connections between the covector mapping theorem and convergence of pseudospectral methods for optimal control, Comput. Optim. Appl., vol. 41, no. 3, pp. 307-335, Dec. 2008.

[17] D. G. Luenberger and Y. Ye, Linear and Nonlinear Programming 3rd Edition New York: Springer, 2008.

[18] F. Keinert, Wavelets and Multiwavelets (Studies in Advanced Mathematics), Chapman and Hall/CRC, New York, 2003.

[19] S. G. Venkatesh S. K. Ayyaswamy, and S. R. Balachandar, Convergence analysis of Legendre wavelets method for solving Fredholm integral equations, Appl. Math. Sci. vol. 6, no. 46, pp. 2289-2296, 2012.

[20] K. Maleknejad, M. TavassoliKajani, and Y. Mahmoudi, Numerical solution of linear Fredholm and volterra integral equation of the second kind by using Legendre wavelets, Kybernetes, vol. 32, no. 9/10, pp. 1530-1539, 2003.

[21] K. Maleknejad and S. Sohrabi, Numerical solution of Fredholm integral equations of the first kind by using Legendre wavelets, Appl. Math. Comput, vol. 186, no. 1, pp. 836-843, Mar. 2007.

[22] S. A Yousefi, A. lotfi and M. Dehghan, The use of a Legendre multiwavelet collocation method for solving the fractional optimal control problems, J. VIB. CONTROL vol. 17, no. 13, pp. 20592065, Nov. 2011.

[23] M. Razzaghi and S. Yousefi, Legendre wavelets method for constrained optimal control problems, Math. Method Appl. Sci., vol. 25, no. 7, pp. 529-539, May 2002.

[24] M. U. Rehman and R. Ali Khan, The Legendre wavelet method for solving fractional differential equations, Commun. Nonlinear Sci. Numer. Simul,, vol. 16, no. 11, pp. 4163-4173, Nov. 2011.

[25] M. Razzaghi and S. Yousefi, Legendre wavelet direct method for variational problems, Math. Comput. Simulat, vol. 53, no. 3, pp. 185-192, Sep. 2000.

[26] F. Mohammadi and M. M Hosseini, A new Legendre wavelet operational matrix of derivative and its applications in solving the singular ordinary differential equation, J. Franklin I., vol. 348, no. 8, pp. 1787-1796, Oct. 2011.

[27] A. Saadatmandia and M. Dehghan, A new operational matrix for solving fractional-order differential equations, Comput. Math. Appl. vol. 59, no. (3) pp. 13261336, Feb. 2010.

[28] Y. Li and W. Zhao, Haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations, Appl. Math. Comput. vol. 216, no. 8, pp. 2276-2285, June. 2010.

[29] S. A Yousefi, Legendre scaling function for solving generalized emden-fowler equations, International journal of information and systems sciences, vol. 3, no. 2, pp. 243-250, 2007.