American Institute of Mathematical Sciences

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2013, 2013(special): 129-137. doi: 10.3934/proc.2013.2013.129

Numerical optimal unbounded control with a singular integro-differential equation as a constraint

Shihchung Chiang 1,

1. 

Department of Applied Statistics, Chung Hua University, Hsinchu, Taiwan

Received  August 2012 Revised  March 2013 Published  November 2013

This study presents a discussion of numerical methods for optimal control using an integro-differential equation of singular kernel as a constraint. The proposed scheme attempts to set the objective to minimize the gap between optimal state and target function for certain period of time. By assuming that control is unbounded, this study proposes a method of feedback correction that makes correction each step for optimal control. These corrections are proportional to the corresponding state-target distance until a certain accuracy criterion is satisfied. There are several advantages to this method, including user-decided accuracy, user-decided number of iterations, and time saving. This study presents a comparison of the numerical results with the results of other methods [4],[7].
Keywords: objective function, Integro-differential equations, optimal state., optimal control.
Mathematics Subject Classification: Primary: 34K28, 49J30; Secondary: 45E1.
Citation: Shihchung Chiang. Numerical optimal unbounded control with a singular integro-differential equation as a constraint. Conference Publications, 2013, 2013 (special) : 129-137. doi: 10.3934/proc.2013.2013.129
References:
[1]

J. A. Burns, E. M. Cliff and T. L. Herdman, A state-space model for an aeroelastic system, "22nd IEEE Conference on Decision and Control", (1983), pp. 1074-1077.

[2]

J. A. Burns, T. L. Herdman, Harlan W. Stech, Linear functional differential equations as semigroups on product spaces, Siam J. Math. Anal., Vol.14, No. 1, (January 1983), pp. 98-116.

[3]

Hsin-Hao Chen and S. Chiang, A specific procedure for analytic solutions to a class of singular integral equations, Chung Hua Journal of Computational Science, No. 2, (March 2012), pp. 7-17.

[4]

S. Chiang, Numerical optimal issues to a class of neutral singular integro-differential equations, Chung Hua Journal of Computational Science, No. 2, (March 2012), pp. 7-17.

[5]

S. Chiang, Notes on the solution of a class of singular integral equations, Chung Hua Journal of Science and Engineering, Vol. 3, No. 4, (2005), pp. 89-95.

[6]

S. Chiang, On the numerical solution of a class of singular integro-differential equations, Chung Hua Journal of Science and Engineering, Vol. 4, No. 3, (2006), pp. 43-48.

[7]

S. Chiang and T. L. Herdman, Revised numerical methods on the optimal control problem for a class of singular integral equations, Mathematics in Engineering, Science and Aerospace, Vol. 4, No. 2, (2013), pp. 171-187.

[8]

Chien-Chi Yu and S. Chiang, On the numerical optimal controls for a class of integro-differential equations of neutral type, Chung Hua Journal of Computational Science, No. 1, (2011), pp. 1-7.

[9]

F. Kappel and K. P. Zhang, Equivalence of functional equations of neutral type and abstract Cauchy problems, Monatsh Math., Vol. 101, (1986), pp. 115-133.

show all references

References:
[1]

J. A. Burns, E. M. Cliff and T. L. Herdman, A state-space model for an aeroelastic system, "22nd IEEE Conference on Decision and Control", (1983), pp. 1074-1077.

[2]

J. A. Burns, T. L. Herdman, Harlan W. Stech, Linear functional differential equations as semigroups on product spaces, Siam J. Math. Anal., Vol.14, No. 1, (January 1983), pp. 98-116.

[3]

Hsin-Hao Chen and S. Chiang, A specific procedure for analytic solutions to a class of singular integral equations, Chung Hua Journal of Computational Science, No. 2, (March 2012), pp. 7-17.

[4]

S. Chiang, Numerical optimal issues to a class of neutral singular integro-differential equations, Chung Hua Journal of Computational Science, No. 2, (March 2012), pp. 7-17.

[5]

S. Chiang, Notes on the solution of a class of singular integral equations, Chung Hua Journal of Science and Engineering, Vol. 3, No. 4, (2005), pp. 89-95.

[6]

S. Chiang, On the numerical solution of a class of singular integro-differential equations, Chung Hua Journal of Science and Engineering, Vol. 4, No. 3, (2006), pp. 43-48.

[7]

S. Chiang and T. L. Herdman, Revised numerical methods on the optimal control problem for a class of singular integral equations, Mathematics in Engineering, Science and Aerospace, Vol. 4, No. 2, (2013), pp. 171-187.

[8]

Chien-Chi Yu and S. Chiang, On the numerical optimal controls for a class of integro-differential equations of neutral type, Chung Hua Journal of Computational Science, No. 1, (2011), pp. 1-7.

[9]

F. Kappel and K. P. Zhang, Equivalence of functional equations of neutral type and abstract Cauchy problems, Monatsh Math., Vol. 101, (1986), pp. 115-133.

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