Population based algorithms for approximate optimal distributed control of wave equations

Iranian Journal of Numerical Analysis and Optimization

مقاله 2، دوره 4، شماره 2 - شماره پیاپی 6، بهمن 2014، صفحه 31-41   

نوع مقاله: Research Article

شناسه دیجیتال (DOI): 10.22067/ijnao.v4i2.40230

نویسندگان

A. H. Borzabadi^* ؛ S. Mirassadi؛ M. Heidari

School of Mathematics and Computer Science, Damghan University, Damghan, Iran.

چکیده

In this paper, a novel hybrid iterative scheme to find approximate optimal distributed control governed by wave equations is considered. A partition of the time-control space is considered and the discrete form of the problem is converted to a quasi assignment problem. Then a population based algorithm, with a finite difference method, is applied to extract approximate optimal distributed control as a piecewise linear function. A convergence analysis is proposed for discretized form of the original problem. Numerical computations are given to show the profciency of the proposed algorithm and the obtained results applying two popular evolutionary algorithms, genetic and particle swarm optimization algorithms.

کلیدواژه‌ها

Optimal control problem؛ Evolutionary algorithm؛ finite difference method؛ Wave equation

مراجع

1. Alavi, S.A., Kamyad, A.V. and Farahi, M.H. The optimal control of an inhomogeneous wave problem with internal control and their numerical solution, Bulletin of the Iranian Mathematical society 23(2) (1997) 9-36.

2. Borzabadi, A.H. and Mehne, H. H. Ant colony optimization for optimal control problems, Journal of Information and Computing Science, 4(4)

3. Borzabadi, A.H. and Heidari, M. Comparison of some evolutionary algorithms for approximate solutions of optimal control problems, Australian Journal of Basic and Applied Sciences, 4(8) (2010) 3366-3382.

4. Borzabadi, A.H. and Heidari, M. Evolutionary algorithms for approximate optimal control of the heat equation with thermal sources, Journal of Mathematical Modelling and Algorithms, DOI: 10.1007/s10852-011-9166-0.

5. Farahi, M.H., Rubio, J.E. and Wilson, D.A. The Optimal control of the linear wave equation, International Journal of control, 63 (1996) 833-848.

6. Farahi, M.H., Rubio, J.E. and Wilson, D.A. The global control of a nonlinear wave equation, International Journal of Control 65(1) (1996) 1-15.

7. Fard, O.S. and Borzabadi, A.H. Optimal control problem, quasiassignment problem and genetic algorithm, Enformatika, Transaction on Engin. Compu. and Tech., 19 (2007) 422 - 424.

8. Gerdts, M., Greif, G. and Pesch, H.J. Numerical optimal control of the wave equation: Optimal boundary control of a string to rest in finite time, Proceedings 5th Mathmod Vienna, February (2006).

9. Glowinski, R., Lee, C.H. and Lions, J.L. A numerical approach to the exact boundary controllability of the wave equation (I) Dirichlet controls: description of the numerical methods, Japan J. Appl. Math., 7 (1990) 1-76.

10. Goldwyn, R.M., Sriram, K.P. and Graham, M.H. Time optimal controlof a linear hyperbolic system, International Journal of Control, 12 (1970)645-656.

11. Gugat, M., Keimer, A. and Leugering, G. Optimal distributed control ofthe wave equation subject to state constraints, ZAMM . Z. Angew. Math. Mech. 89(6) (2009) 420-444.

12. Gugat, M. Penalty techniques for state constrained optimal control problems with the wave equation, SIAM J. Control Optim. 48 (2009) 3026-3051.

13. Hasanov, K.K. and Gasumov, T.M. Minimal energy control for the wave equation with non-classical boundary condition, Appl. Comput. Math.,9(1) (2010) 47-56.

14. Kim, M. and Erzberger, H. On the design of optimum distributed parameter system with boundary control function, IEEE Transactions on Automatic Control, 12(1) (1967) 22-28.

15. Lions, J.L. Optimal control of systems governed by partial differential

equations, Springer, Berlin, 1971.