Optimal control theory

A number of other early experiments confirming the Tannor-Rice scenario are discussed in detail in the monograph of Rice and Zhao [105], i 

In what follows we outline the general principles of OCT. Specific cases ate-described in Chapter 13. 

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It turns out that there is another branch of mathematics, closely related to tire calculus of variations, although historically the two fields grew up somewhat separately, known as optimal control theory (OCT). Although the boundary between these two fields is somewhat blurred, in practice one may view optimal control theory as the application of the calculus of variations to problems with differential equation constraints. OCT is used in chemical, electrical, and aeronautical engineering where the differential equation constraints may be chemical kinetic equations, electrical circuit equations, the Navier-Stokes equations for air flow, or Newton s equations. In our case, the differential equation constraint is the TDSE in the presence of the control, which is the electric field interacting with the dipole (pemianent or transition dipole moment) of the molecule [53, 54, 55 and 56]. From the point of view of control theory, this application presents many new features relative to conventional applications perhaps most interesting mathematically is the admission of a complex state variable and a complex control conceptually, the application of control teclmiques to steer the microscopic equations of motion is both a novel and potentially very important new direction. 

It was felt that a nonisothermal policy might have considerable advantages in minimizing the reaction time compared to die optimal isothermal policy. Modem optimal control theory (Sage and White (1977)), was employed to minimize the reaction time. The mathematical development is presented below. 

The above formulation is a well posed problem in optimal control theory and its solution can be obtained by the application of Pontryagin s Minimum Principle (Sage and White (1977)). 

G. 147 Swan, Applications of Optimal Control Theory in Biomedicine (1984)... 

V. F. Krotov, Global Methods in Optimal Control Theory (1996)... 

OPTIMAL CONTROL THEORY FOR MANIPULATING MOLECULAR PROCESSES... 

Appendix F. Convergence of the Iterative Solution of the Optimal Control Theory Equations... 

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 early work in the optimal control theory design of laser helds to achieve desired transformations, the optimal control equations were solved directly, without constraints other than those imposed implicitly by the inclusion of a penalty term on the laser huence [see Eq. (1)]. This inevitably led to laser helds that suddenly increased from very small to large values near the start of the laser pulse. However, physically realistic laser helds should tum-on and -off smoothly. Therefore, during the optimization the held is not allowed to vary freely but is rather expressed in the form [60] ... 

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. 

OPTIMAL CONTROL THEORY equation can be written in the form ... 

One of the difficulties with optimal control theory is in identifying the underlying physical mechanism, or mechanisms, leading to control. Methods [2, 7, 9, 14, 26-29], that utilize a small number of interfering pathways reveal the mechanism by construction. On the other hand, while there have been many successful experimental and theoretical demonstrations of control based on OCT, there has been little analytical work to reveal the mechanism behind the complicated optimal pulses. In addition to reducing the complexity of the pulses, the many methods for imposing explicit restrictions on the pulses, see Section II.B, can also be used to dictate the mechanisms that will be operative. However, in this section we discuss some of the analytic approaches that have been used to understand the mechanisms of optimal control or to analytically design optimal pulses. Note that we will not discuss numerical methods that have been used to analyze control mechanisms [145-150]. 

This chapter has provided a brief overview of the application of optimal control theory to the control of molecular processes. It has addressed only the theoretical aspects and approaches to the topic and has not covered the many successful experimental applications [33, 37, 164-183], arising especially from the closed-loop approach of Rabitz [32]. The basic formulae have been presented and carefully derived in Section II and Appendix A, respectively. The theory required for application to photodissociation and unimolecular dissociation processes is also discussed in Section II, while the new equations needed in this connection are derived in Appendix B. An exciting related area of coherent control which has not been treated in this review is that of the control of bimolecular chemical reactions, in which both initial and final states are continuum scattering states [7, 14, 27-29, 184-188]. 

APPENDIX F CONVERGENCE OF THE ITERATIVE SOLUTION OF THE OPTIMAL CONTROL THEORY EQUATIONS... 

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