Journal of Optimization Theory and Applications

, Volume 163, Issue 3, pp 957–988 | Cite as

Switching Time and Parameter Optimization in Nonlinear Switched Systems with Multiple Time-Delays

  • Chongyang Liu
  • Ryan Loxton
  • Kok Lay TeoEmail author


In this paper, we consider a dynamic optimization problem involving a general switched system that evolves by switching between several subsystems of nonlinear delay-differential equations. The optimization variables in this system consist of: (1) the times at which the subsystem switches occur; and (2) a set of system parameters that influence the subsystem dynamics. We first establish the existence of the partial derivatives of the system state with respect to both the switching times and the system parameters. Then, on the basis of this result, we show that the gradient of the cost function can be computed by solving the state system forward in time followed by a costate system backward in time. This gradient computation procedure can be combined with any gradient-based optimization method to determine the optimal switching times and parameters. We propose an effective optimization algorithm based on this idea. Finally, we consider three numerical examples, one involving the 1,3-propanediol fed-batch production process, to illustrate the effectiveness and applicability of the proposed algorithm.


Switched system Time-delay system Switching times  Nonlinear optimization 

Mathematics Subject Classification

49M07 49M37 65K10 



The first author is supported by the Natural Science Foundation for the Youth of China (Grant Nos. 11201267 and 11001153), the Tian Yuan Special Funds of the National Natural Science Foundation of China (Grant No. 11126077), and the Shandong Province Natural Science Foundation of China (Grant Nos. ZR2010AQ016 and ZR2011AL003). The second author is supported by the Natural Science Foundation of China (Grant No. 11350110208) and the Australian Research Council (Discovery Grant DP110100083). The third author is supported by the Australian Research Council (Discovery Grant DP110100083).


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.School of Mathematics and Information ScienceShandong Institute of Business and TechnologyYantaiChina
  2. 2.Department of Mathematics and StatisticsCurtin UniversityPerthAustralia
  3. 3.Institute of Cyber-Systems and ControlZhejiang UniversityHangzhouChina

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