Xenon restraint

VI. EXAMPLE WITH STATE RESTRAINTS A. Xenon Restraint Model... 

The problems that occur when one tries to estimate affinity in terms of component terms do not arise when perturbation methods are used with simulations in order to compute potentials of mean force or free energies for molecular transformations simulations use a simple physical force field and thereby implicitly include all component terms discussed earlier. We have used the molecular transformation approach to compute binding affinities from these first principles [14]. The basic approach had been introduced in early work, in which we studied the affinity of xenon for myoglobin [11]. The procedure was to gradually decrease the interactions between xenon atom and protein, and compute the free energy change by standard perturbation methods, cf. (10). An (issential component is to impose a restraint on the... 

Indeed, in Fig. 2, it is seen that only those trajectories coming from initial conditions close to the target curve lie below the xenon boundary, X, and could therefore admit a startup from zero flux. Other trajectories, calling for operation at nonzero flux, could not be realized. This is typical of the problem where restraints on the state variables are to be taken into consideration. 

Of these three conditions derived by Pontryagin to treat the problem of state restraints, the most significant is that of Eq. (52), the time dependence of the additional Lagrange multiplier. The use of this condition is illustrated in the following section in the context of the xenon shutdown problem. 

When state restraints were neglected in the xenon shutdown problem, as in Sect. Ill, several of the optimum trajectories in the phase space exceeded the acceptable xenon concentration, X. If these trajectories required operation other than at zero power, they would not be realistic or acceptable. Thus, we seek to introduce the restraint that at no time in the control period should the optimum trajectory exceed X. ... 

For free end time problems, H (and H ) will be zero on the boundary of restraint, along which the rate of change of the xenon concentration vanishes. We therefore have the solution for the iodine adjoint function by considering... 

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