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Journal Articles SIAM Journal on Control and Optimization Year : 2013

Chaperoning State Evolutions by Variable Durations


McKendrick partial di erential equations in demography provide the state of a population depending on chronological time T and age D, and involve two time evolutions : the chronological one t 2 [0; T] 7! t 0 and the calendar age t 2 [T D; T] 7! t (T D) 0, both with constant velocity equal to 1. The calendar age evolution chaperons, so to speak, the evolution t 2 [T D; T] 7! x(t) of the state of a system. Some physical, biological and economic problems motivate the introduction of variable durations with variable velocities (representing the fluidity of time) o ffering mathematical metaphors of a subjective fleeting specious time passing more or less slowly. Variable durations are no longer prescribed, but chosen among available ones and regulated : the joint evolution of the variable evolution and the state is assumed to be governed by a di erential inclusion (or control system) and provides, as a by product, the unknown temporal windows on which they evolve together. In economics, for instance, the state is a commodity, its velocity a transaction. If a cost function is de ned on the fluidity of the variable evolution and the transaction of commodities, their cumulated cost over the fluidity-transaction pairs could be minimized. Slowing down the fluidity, which widens the investment period (from the milliseconds of high-frequency markets to the centuries of cathedral building) by inventing a shareholder value tax and decelerating transactions (by implementing the Tobin tax) could be an objective of salubrious nancial and corporate management. This was the real motivation of this study, which fi nds applications in other fields, where duration, this particular meaning of time, should have a value.
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hal-00800156 , version 1 (13-03-2013)



Jean-Pierre Aubin. Chaperoning State Evolutions by Variable Durations. SIAM Journal on Control and Optimization, 2013, 51 (2), pp.2081-2101. ⟨10.1137/120879853⟩. ⟨hal-00800156⟩
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