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Mourot A, Fehrentz T, Le Feuvre Y, Smith CM, Herold C, Dalkara D, Nagy F, Trauner D, Kramer RH (2012) Nature Methods 9:396-402.

(80) For (delta le t le omega ), we know V ( t ) le biglVert V ( delta ) bigrVert _{tau} e^{ ( a_{5} + b_{5} )t},quad delta le t le omega.

From the fact that η is nondecreasing we conclude (seta(t)le seta(t)) for all (0le t le s).

The nondecreasing property of μ yields (smu(t)le smu(t)) for all (0le t le s).

If (p(t)le -7), then, it is highly negative.

Constraints of these state variables are expressed in equations as: 0 le nu_{text{AC}} (t) le 1 (10) 0 le nu_{text{MT}} (t) le 1 (11) 0 le nu_{text{ACh}} (t) le 1 (12).

This means that (X t) = tilde{X}(t)) for (t_{0} le t le T).

The level of negative inventory at time (t), where (0 le t le t_{1}).

Under this situation, three cases such as ( t_{1} le T ), ( T le t_{1} le T + t_{2} ) and ( T + t_{2} le t_{1} ) are to be considered.

When (delta le t le omega ), we can conclude by (39) and Lemma 3 V ( t ) le biglVert V ( delta ) bigrVert _{tau} e^{ ( a_{2} + b_{2} )t},quad delta le t le omega.

Solution region: Sigma = { 0 le S le infty, 0 le t le T }.

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