Extremal Shift Rule and Viability Property for Mean Field-Type Control Systems

Yurii Averboukh (), Antonio Marigonda () and Marc Quincampoix ()
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Yurii Averboukh: Krasovskii Institute of Mathematics and Mechanics
Antonio Marigonda: University of Verona
Marc Quincampoix: Univ Brest

Journal of Optimization Theory and Applications, 2021, vol. 189, issue 1, No 11, 244-270

Abstract: Abstract We investigate when a mean field-type control system can fulfill a given constraint. Namely, given a closed set of probability measures on the torus, starting from any initial probability measure belonging to this set, does there exist a solution to the mean field control system remaining in it for any time? This property—the so-called viability property—is equivalently characterized through a property involving normals to the given set of probability measures. We prove that, if the Hamiltonian is nonpositive at any normal distribution to the given set, then the feedback strategy realizing the extremal shift rule provides the approximate viability. This implies the usual viability property. Conversely, the Hamiltonian is nonpositive at any normal distribution if the given set is viable. Our approach enables us to derive generalized feedback laws which ensure the trajectory to fulfill the constraint. This generalized feedback called here extremely shift rule is inspired by constructive motions developed by Krasovskii and Subbotin for differential games.

Keywords: Mean field-type control; Viability; Proximal normal distribution; Extremal shift; 49N35; 93B52; 93C10; 93A14; 28A33 (search for similar items in EconPapers)
Date: 2021
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10957-021-01832-z

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