Minima Optima Sufficient conditions

On one rather technical point, the optimum theory does differ from classical calculus of variations. This is in regard to the Weierstrass- or end condition, which is not satisfied by the rigorous optimum control theory. The reason for this nicety in the sufficient conditions for an optimum (and the mathematics will not be pursued further) is that optimum control theory is based on the utilitarian search for a greatest or a least value of a functional rather than a maximum or a minimum. These latter terms have precise mathematical meaning in relation to their turning values, whereas all we may require in application is the least value of a cost functional. The example of a finite straight line may make this point clearer, for such a line has neither maximum nor minimum and yet will have, at its ends, greatest and least values. 

Examples illustrating what can go wrong if the constraint gradients are dependent at x can be found in Luenberger (1984). It is important to remember that all local maxima and minima of an NLP satisfy the first-order necessary conditions if the constraint gradients at each such optimum are independent. Also, because these conditions are necessary but not, in general, sufficient, a solution of Equations (8.17)-(8.18) need not be a minimum or a maximum at all. It can be a saddle or inflection point. This is exactly what happens in the unconstrained case, where there are no constraint functions hj = 0. Then conditions (8.17)-(8.18) become... 

For the development of new reaction sequences for solid-phase synthesis, the optimum conditions for each step must be identified. This includes determination of the minimum reaction time and temperature, and the minimum amounts of reagents required to obtain sufficiently pure products with a representative selection of different reagents. If reaction times or amounts of reagents are excessive, library production will be costly and inefficient. 

Mayur et al. (1970) formulated a two level dynamic optimisation problem to obtain optimal amount and composition of the off-cut recycle for the quasi-steady state operation which would minimise the overall distillation time for the whole cycle. For a particular choice of the amount of off-cut and its composition (Rl, xRI) (Figure 8.1) they obtained a solution for the two distillation tasks which minimises the distillation time of the individual tasks by selecting an optimal reflux policy. The optimum reflux ratio policy is described by a function rft) during Task 1 when a mixed charge (BC, xBC) is separated into a distillate (Dl, x DI) and a residue (Bl, xBi), followed by a function r2(t) during Task 2, when the residue is separated into an off-cut (Rl, xR2) and a bottom product (B2, x B2)- Both r2(t)and r2(t) are chosen to minimise the time for the respective task. However, these conditions are not sufficient to completely define the operation, because Rl and xRI can take many feasible values. Therefore the authors used a sequential simplex method to obtain the optimal values of Rl and xR which minimise the overall distillation time. The authors showed for one example that the inclusion of a recycled off-cut reduced the batch time by 5% compared to the minimum time for a distillation without recycled off-cut. 

For energy exchange equipment Supply sufficient excess of heat transfer area in reboilers, condensers, cooling jackets, and heat removal systems for reactors to be able to handle the anticipated upsets and dynamic changes. Sometimes extra area is needed in overhead condensers to subcool the condensate to prevent flashing in the downstream control valves. Too frequently, overzealous engineers size the optimum heat exchangers based on an economic minimum based on steady-state conditions and produce uncontrollable systems. 

In practice, it is often not possible to calculate the selectivity for this type of process from first principles. Experimental studies have to be done under various conditions, leading to an empiricd optimum on a small scale. It has been shown in section 5.2.2 that the selectivity can be sensitive to the meso-mixe(hiess, and that these can both decrease on larger scales. It may be practical to study the reaction first in a well mixed semi-batch reactor, preferably equipped with an effective turbine impeller, or propeller. The feed tube should end in the exit flow of the impeller. With this equipment one measures the influence of the feed rate on the selectivity. It may be that a sufficiently high selectivity is obtained when the feeding time has a certain minimum value. The specific energy dissipation should be measured or calculated. 

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