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Minimal operation time of energy devices. (English) Zbl 1090.80003

The author raises an interesting question concerning the minimal switch-off time for an energy supplier, with an energy supply rate \(E'( t) \), which has to transfer a givenamount of energy to a system. In his first main theorem, the author assumes that \(E'( t) \), is continuous on \(\left[ 0,+\infty \right) \) and piecewise monotone with three parts, starting in an increasing way from 0 at \(t=0\), then with a plateau between \(t_{0}\) and \(t_{1}\geq t_{0}\), finally decreasing after \( t_{1}\). He characterizes the minimal switch-off time using the anti-derivatives of the monotone functions building \(E^{\prime }\left( t\right) \). The authors completes his paper with two examples: one with exponential functions, one with affine functions. In both examples, he explicitly computes the switch-off time. The paper ends with a study concerning the case where \(E'( t) \) is supposed to be the solution of a first-order differential equation with noise on subintervals.

MSC:

80M50 Optimization problems in thermodynamics and heat transfer
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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