Geodesic
Optimization in Mean and Phase Transitions in Controlled Dynamical Systems
Funkcionalʹnyj analiz i ego priloženiâ, Tome 36 (2002) no. 2, pp. 1-11

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The time mean of a smooth objective function along a phase trajectory of a controlled dynamical system is maximized. The simplest singularities of the dependence of the optimal mean value on the parameter in generic one-parameter families of controlled systems of this kind are listed. It turns out that the most common generic stable singularity is the discontinuity of the first or second derivative of the optimal mean value with respect to the parameter.
Keywords: time mean, generic singularities, variational problems, mixed strategies, Harnack theorem, Sturm theorem. 
@article{FAA_2002_36_2_a0,
     author = {V. I. Arnol'd},
     title = {Optimization in {Mean} and {Phase} {Transitions} in {Controlled} {Dynamical} {Systems}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {1--11},
     publisher = {mathdoc},
     volume = {36},
     number = {2},
     year = {2002},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2002_36_2_a0/}
}
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V. I. Arnol'd. Optimization in Mean and Phase Transitions in Controlled Dynamical Systems. Funkcionalʹnyj analiz i ego priloženiâ, Tome 36 (2002) no. 2, pp. 1-11. http://geodesic.mathdoc.fr/item/FAA_2002_36_2_a0/