Abstract
It is shown that, if a parametrized fämily of extremals F can be stratified in a way compatible with the flow map generated by F, then those trajectories of the family which realize the minimal values of the cost in F are indeed optimal in comparison with all trajectories which lie in the region R covered by the trajectories of F. It is not assumed that F is a field covering the state space injectively. As illustration, an optimal synthesis is constructed for a system where the flow of extremals exhibits a simple cusp singularity.
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Ledzewicz, U., Nowakowski, A. & Schättler, H. Stratifiable Families of Extremals and Sufficient Conditions for Optimality in Optimal Control Problems. Journal of Optimization Theory and Applications 122, 345–370 (2004). https://doi.org/10.1023/B:JOTA.0000042525.50701.9a
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DOI: https://doi.org/10.1023/B:JOTA.0000042525.50701.9a