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Necessary Optimality Conditions for Implicit Control Systems with Applications to Control of Differential Algebraic Equations

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Abstract

In this paper we derive necessary optimality conditions for optimal control problems with nonlinear and nonsmooth implicit control systems. Implicit control systems have wide applications including differential algebraic equations (DAEs). The challenge in the study of implicit control system lies in that the system may be truly implicit, i.e., the Jacobian matrix of the constraint mapping may be singular. Our necessary optimality conditions hold under the so-called weak basic constraint qualification plus the calmness of a perturbed constraint mapping. Such constraint qualifications allow for singularity of the Jacobian and hence are suitable for implicit systems. Specifying these results to control of semi-explicit DAEs we obtain necessary optimality conditions for control of semi-explicit DAEs with index higher than one.

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Acknowledgments

We thank the anonymous reviewers of this paper for valuable comments that helped us to improve the presentation of the manuscript.

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Authors and Affiliations

  1. School of Mathematical Sciences, Xiamen University, Xiamen, 361005, Fujian, China

    An Li

  2. Department of Mathematics and Statistics, University of Victoria, Victoria, B.C., V8W 2Y2, Canada

    Jane J. Ye

Authors
  1. An Li
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  2. Jane J. Ye
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Corresponding author

Correspondence to Jane J. Ye.

Additional information

Dedicated to the memory of Jonathan Michael Borwein

The research of the first author was partially supported by the National Natural Science Foundation of China (Grant No. 11671335), the Natural Science Foundation of Fujian Province, China (Grant No. 2016J01033) and the Fundamental Research Funds for the Central Universities (Grant No. 20720160036).

The research of the second author was supported by NSERC.

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Li, A., Ye, J.J. Necessary Optimality Conditions for Implicit Control Systems with Applications to Control of Differential Algebraic Equations. Set-Valued Var. Anal 26, 179–203 (2018). https://doi.org/10.1007/s11228-017-0444-5

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  • DOI: https://doi.org/10.1007/s11228-017-0444-5

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