Minimum energy control problem

This is an identical equation for the minimum energy path [9,10] or the so-called steepest descent path. The first implementation of an algorithm to compute minimum energy paths based on the above formula was the LUP (Locally Updated Planes) method [9] that did not include the constraint on the displacement size. This is formally correct since different parameterizations of the path are possible, but may lead to numerical problems in which the distances between the intermediates grow without control. This was adjusted to produce more stable algorithms by the Nudge Elastic Band approach [11] and later by the String method [10]. 

For comparison, we now study the classical control problem and determine the field from Eq. (31), where the expectation values are replaced by the values of the classical coordinates and momenta, respectively. This yields the classical trajectory shown in Fig. 16, which is superimposed on the potential energy contours. Here, a perfect transfer is found where the particle stops in the minimum of the target potential well. A comparison with the trajectory derived from the quantum calculation (Fig. 14) shows that the classical orbit follows the quantum orbit closely until the reaction barrier is passed. At later times, due to the missing dispersion in the classical treatment, deviations are found. 

To illustrate the concept, consider a single distillation column with distillate and bottoms products. To produce these products while using the minimum amount of energy, the compositions of both products should be controlled at their specifications. Figure 8.13u shows a dual composition control system. The disadvantages of this structure arc (1) two composition analyzers are required, (2) the instrumentation is more complex, and (3) there may be dynamic interaction problems since the two loops are interacting. This system may be difficult to design and to tune. 

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