Optimal Control versus Optimization

It is easy to perceive from the above example that optimal control involves optimization of an objective functional subject to the equations of change in a system and additional constraints, if any. Because of this fact, optimal control is also known as dynamic or trajectory optimization. 

A control function used in optimal control comprises a number of values, one for each value of the independent variable. That number is infinity if at least a part of the function is continuous. Thus, in the previous example of the batch reactor, the control T t) is a set of optimization parameters 

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Figure 7.9 The initially guessed and optimal controls versus time for Example 7.3...

Because of very high dielectric constants k > 20, 000), lead-based relaxor ferroelectrics, Pb(B, B2)02, where B is typically a low valence cation and B2 is a high valence cation, have been iavestigated for multilayer capacitor appHcations. Relaxor ferroelectrics are dielectric materials that display frequency dependent dielectric constant versus temperature behavior near the Curie transition. Dielectric properties result from the compositional disorder ia the B and B2 cation distribution and the associated dipolar and ferroelectric polarization mechanisms. Close control of the processiag conditions is requited for property optimization. Capacitor compositions are often based on lead magnesium niobate (PMN), Pb(Mg2 3Nb2 3)02, and lead ziac niobate (PZN), Pb(Zn 3Nb2 3)03. 

Figure 5.164. Tank temperature versus time for two values of Kc (1.5 and 2.0), with XI = 10000. The changes at T=10 and T=20 are programmed step changes in the inlet water flow rate. Oscillations and offset are caused by sub-optimal controller tuning.

Fig. 10 Blood level versus time profile simulations following (A) a single dose representing 100 units of a drug from a rapidly releasing dosage (B) Three divided ddoses of 33 units each from the same rapidly releasing product and (C) a single 100 unit dose from an optimized controlled-release dosage form. A hypothetical effective level (80 units) and toxic level (160 units) are depicted. The dosing units are typically in mg and the blood level concentration units in pg or ng.

The issue of parallel versus sequential synthesis using multimode or monomode cavities, respectively, deserves special comment. While the parallel set-up allows for a considerably higher throughput achievable in the relatively short timeframe of a microwave-enhanced chemical reaction, the individual control over each reaction vessel in terms of reaction temperature/pressure is limited. In the parallel mode, all reaction vessels are exposed to the same irradiation conditions. In order to ensure similar temperatures in each vessel, the same volume of the identical solvent should be used in each reaction vessel because of the dielectric properties involved [86]. As an alternative to parallel processing, the automated sequential synthesis of libraries can be a viable strategy if small focused libraries (20-200 compounds) need to be prepared. Irradiating each individual reaction vessel separately gives better control over the reaction parameters and allows for the rapid optimization of reaction conditions. For the preparation of relatively small libraries, where delicate chemistries are to be performed, the sequential format may be preferable. This is discussed in more detail in Chapter 5. 

Our analysis of multiple, field-driven qubits that are coupled to partly correlated or independent baths or undergo locally varying random dephasing allows one to come up with an optimal choice between global and local control, based on the observation that the maximal suppression of decoherence is not necessarily the best one. Instead, we demand an optimal phase relation between different but synchronous local modulations of each particle. The merits of local versus global modulations have been shown to be essentially twofold ... 

An enantioselective intermolecular Michael addition of aldehydes (138) to enones (139), catalysed by imidazolidinones (140), has been reported. Chemoselectivity (Michael addition versus aldol) can be controlled through judicious choice of hydrogen bond-donating co-catalysts. The optimal imidazolidinone-hydrogen bond donor pair affords Michael addition products in <90% ee. Furthermore, the enamine intermediate was isolated and characterized and its efficacy as a nucleophile in the observed Michael addition reactions was demonstrated.172... 

The previous section assumed that product composition (or product flow) requirements are fixed. In this very common situation, the optimum design minimizes the costs of achieving these requirements. Often, product specs are not fixed, but depend on economics. Even when a product must obey a "less than" purity spec, better purity may fetch a better price. The better price may justify additional investment in equipment and/or a higher operating cost. Here, a design must optimize product purity value versus distillation cost. This optimization is also important in an operating column and is commonly performed by on-line computer control. It is outlined below, and discussed in detail elsewhere (1,2). 

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