eesti teaduste
akadeemia kirjastus
Proceedings of the Estonian Academy of Sciences. Physics. Mathematics
Evolution strategies in optimization problems; 299-309

Pedro A. F. Cruz, Delfim F. M. Torres

Evolution strategies are inspired in biology and form part of a larger research field known as evolutionary algorithms. Those strategies perform a random search in the space of admissible functions, aiming to optimize some given objective function. We show that simple evolution strategies are a useful tool in optimal control, permitting one to obtain, in an efficient way, good approximations to the solutions of some recent and challenging optimal control problems.


1. Beyer, H.-G. and Schwefel, H.-P. Evolution strategies – a comprehensive introduction. Nat. Comput., 2002, 1, 3–52.

2. Auger, A. Convergence results for the (1, λ)-SA-ES using the theory of Φ-irreducible Markov chains. Theor. Comput. Sci., 2005, 334, 35–69.

3. Arnặutu, V. and Neittaanmäki, P. Optimal Control from Theory to Computer Programs. Kluwer Acad. Publ., Dordrecht, 2003.

4. Szarkowicz, D. S. Investigating the brachistochrone with multistage Monte Carlo method. Int. J. Systems Sci., 1995, 26, 233–243.

5. Porter, B. and Merzougui, T. Evolutionary synthesis of optimal control policies for manufacturing systems. In Emerging Technologies and Factory Automation Proceedings, 1997. ETFA ’97, 6th International Conference on 9–12 Sept. 1997, 304–309.

6. Raimúndez Álvarez, J. C. Port controller Hamiltonian synthesis using evolution strategies. In Dynamics, Bifurcations, and Control (Kloster Irsee, 2001). Lecture Notes in Control and Inform. Sci., 2002, 273, 159–172.

7. Raimúndez, C. Robust stabilization for the nonlinear benchmark problem (TORA) using neural nets and evolution strategies. In Nonlinear Control in the Year 2000, Vol. 2 (Paris). Lecture Notes in Control and Inform. Sci., 2001, 259, 301–313.

8. Beielstein, T., Ewald, C.-P. and Markon, S. Optimal elevator group control by evolution strategies. Lecture Notes Comput. Sci., 2003, 1963–1974.

9. Aktan, B., Greenwood, G. W. and Shor, M. H. Using optimal control principles to adapt evolution strategies. In 2006 IEEE Congress on Evolutionary Computation, Vancouver, Canada, July 16–21. 2006, 291–296.

10. Tikhomirov, V. M. Stories About Maxima and Minima. Amer. Math. Soc., Providence, RI, 1990 (translated from the 1986 Russian original by Abe Shenitzer).

11. Plakhov, A. Yu. and Torres, D. F. M. Newton’s aerodynamic problem in media of chaotically moving particles. Sb. Math., 2005, 196, 885–933.

12. Sussmann, H. J. and Willems, J. C. The brachistochrone problem and modern control theory. In Contemporary Trends in Nonlinear Geometric Control Theory and Its Applications (México City, 2000). World Sci. Publ., River Edge, NJ, 2002, 113–166.

13. Ramm, A. G. Inequalities for brachistochrone. Math. Inequal. Appl., 1999, 2, 135–140.

14. Torres, D. F. M. and Plakhov, A. Yu. Optimal control of Newton-type problems of minimal resistance. Rend. Semin. Mat. Univ. Politec. Torino, 2006, 64, 79–95.

15. Plakhov, A. Yu. Billiards in unbounded domains that reverse the direction of motion of the particles. Uspekhi Mat. Nauk, 2006, 61, 183–184.

16. R (Development Core Team). R: a Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria, 2005. ISBN 3-900051-07-0.

Back to Issue

Back issues