Optimal control theory approximation

Optimal control theory, as discussed in Sections II-IV, involves the algorithmic design of laser pulses to achieve a specified control objective. However, through the application of certain approximations, analytic methods can be formulated and then utilized within the optimal control theory framework to predict and interpret the laser fields required. These analytic approaches will be discussed in Section VI. 

In this section, we provide some examples of optimal control theory calculations using the ENBO approximation. The reader is referred to Ref. [42], from where aU the examples are taken, for further details. 

Messina et al. consider a system with two electronic states g) and e). The system is partitioned into a subset of degrees of freedom that are to be controlled, labeled Z, and a background subset of degrees of freedom, labeled x the dynamics of the Z subset, which is to be controlled, is treated exactly, whereas the dynamics of the x subset is described with the time-dependent Hartree approximation. The formulation of the calculation is similar to the weak-response optimal control theory analysis of Wilson et al. described in Section IV [28-32], The solution of the time-dependent Schrodinger equation for this system can be represented in the form... 

If the loop interactions are not severe, then each single-loop controller can be designed using the techniques described earlier in this section. However, the presence of strong interactions requires that the controllers be detuned to reduce oscillations. There are multivariable control techniques, such as optimal control, that provide frameworks in which to handle the interactions between various inputs and outputs. Optimal control methods also allow one to deal directly with nonlinear system dynamics, rather than the linear model approximation required for the techniques discussed in this section. A thorough presentation of optimal control theory is given by Bryson and Ho (1975). Optimal control is discussed further in Section 9.5. 

Gluck et al. (1996) adapted optimal control theory (OCT) to the damper placement problem. OCT is used to minimise the performance objective by optimising the location of linear passive devices. Since passive dampers cannot provide feedback in terms of optimal control gains, three approaches (response spectrum approach, single mode approach, and truncation approach) are proposed to remove the off-diagonal state interactions within the gain matrix and allow approximation of floor damping coefficients. Combination of these methods with OCT and passive devices achieves an equivalent effect compared to active control. 

Equation (32) is the working equation that is used in the optimal control applications using the ENBO approximation. Further details of the theory are given in Ref. [41]. 

We emphasize that the question of stability of a CA under small random perturbations is in itself an important unsolved problem in the theory of fluctuations [92-94] and the difficulties in solving it are similar to those mentioned above. Thus it is unclear at first glance how an analogy between these two unsolved problems could be of any help. However, as already noted above, the new method for statistical analysis of fluctuational trajectories [60,62,95,112] based on the prehistory probability distribution allows direct experimental insight into the almost deterministic dynamics of fluctuations in the limit of small noise intensity. Using this techique, it turns out to be possible to verify experimentally the existence of a unique solution, to identify the boundary condition on a CA, and to find an accurate approximation of the optimal control function. 

Model-based approaches allow fast derivative computation by relying on a process model, yet only approximate derivatives are obtained. In self-optimizing control [12,21], the idea is to use a plant model to select linear combinations of outputs, the tracking of which results in optimal performance, also in the presence of uncertainty in other words, these linear combinations of outputs approximate the process derivatives. Also, a way of calculating the gradient based on the theory of neighbouring extremals has been presented in [13] however, an important limitation of this approach is that it provides only a first-order approximation and that the accuracy of the derivatives depends strongly on the reliability of the plant model. 

If we add a solution of species A to a solution of species B, eddies of solution A in solution B are created. As a first approximation, these eddies can be considered as spherical drops with constant mean radius R. The lifetime of such an eddy can be estimated to be 0.01-1 s. The radius R depends on the intensity of the turbulence created by mixing and may be controlled, for example, by mechanical stirring. From the theory of turbulence, one can estimate the minimum mean size of such elements of liquid. For the common solvents water, methanol, and ethanol, the mean minimum radius R of the eddies in optimal turbulence is approximately 10 to 10 cm. 

Milnadpran is a rather newer SNRI licensed as an antidepressant in France. It is associated with clear-cut efficacy judged on the placebo-controlled studies [Lecrubier et al. 1996 Macher et al. 1989]. Milnacipran inhibits the reuptake of both noradrenaline and serotonin (Moret et al. 1985]. It has a relatively short half-life and is given optimally in a dose of 50 mg twice daily. The proportion of reuptake inhibition between serotonin and noradrenaline is approximately equal with this antidepressant, and so one would expect that milnacipran would be more effective than SSRIs, assuming the theory is correct that two actions are better than one. 

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