Optimal control theory derivation

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]. 

Optimal control theory A method for determining the optimum laser field used to maximize a desired product of a chemical reaction. The optimum field is derived by maximizing the objective function, which is the sum of the expectation value of the target operator at a given time and the cost penalty function for the laser field, under the constraint that quantum states of the reactants satisfy the Schrodinger equation. 

Recently, a semiclassical formulation of the optimal control theory has been derived [23, 24] by combining the conjugate gradient search method... 

Optimal Control for Chemical Engineers gives a detailed treatment of optimal control theory that enables you to formulate and solve optimal control problems. With a strong emphasis on problem solving, the book provides all the necessary mathematical analyses and derivations of important results. 

Since optimal control theory was applied in the history matching problem, an adjoint system of equations similar to the state system of equations (equations 1 to 3) was derived. In the adjoint system of equations, Q was substituted for the state primary dependent variable, P. The adjoint system of equations was solved backward in time. The final adjoint equation, the final condition and the associated boundary conditions are shown in equations (4), (5) and (6), respectively. 

The chapter is arranged in this order. The first section presents the three models representing the functioning of our ecosystem (in an increasing order of complexity). We present a state-of-the-art approach in finding controllability of these three systems based on a complex systems approach in the second section. The controllabUity defined in terms of number of nodes or decision variables is then used to select important decision variables for the three systems respectively in the third section. The derivation of technological and socio-economic poHcies based on optimal control theory is presented in the same section. A summary of the chapter is presented at the end. 

Doshi R, Diwekar U, Benavides P, Yenkie K, Cabezas H. Maximizing sustainability of ecosystem model through socio-economic policies derived from multivariable optimal control theory. Clean Technol Environ Policy 2015 17 1573. 

The identification of relations between statics and dynamics became a constituting part in the explanation of unity of the laws of mechanics in (Lagrange, 1788). Deriving the equations of trajectories from the equation of state (1) turned out to be possible owing to the assumptions made about observance of the relativity principle of Galileo and the third law of Newton and, hence, about representability of any trajectory in the form of a continuous sequence of equilibrium states. From the representability, in turn, follow the most important properties of the Lagrange motion curves existence of the functions of states (independent of attainability path) at each point possibility to describe the curves by autonomous differential equations that have the form x = f x) dependence of the optimal configuration of any part of the curve upon its initial point only. These properties correspond to the extreme principles of the optimal control theory. 

Optimal control theory was a useful tool for finding the pathway, in terms of the time dependence of temperature T and volume V, that yields the optimum power or the optimum efficiency for a given fixed cycle time. We shall only outline the method and show some of the results here the full derivation is available [6]. 

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. 

This derivation shows that retention time is dependant on three factors temperature, energies of intermolecular interactions and flow rate. Temperature and flow rate are controlled by the user. Energies of intermolecular interactions are controlled by stationary phase choice. This theory is also the basis for the popular software programs that are available for computer-assisted method development and optimization [4,5,6,7]. More detailed descriptions of the theory behind retention times can be found in the appropriate chapters in the texts listed in the bibliography. 

Schramm et al. (2001) have presented a model-based control approach for direct control of the product purities of SMB processes. Based on wave theory, relationships between the front movements and the flow rates of the equivalent TMB process were derived. Using these relationships, a simple control concept with two PI controllers was proposed. This concept is very easy to implement however, it does not address the issue of optimizing the operating regime in the presence of disturbances or model mismatch. 

A different theory of local control has been derived from the viewpoint of global optimization, applied to finite time intervals [58-60]. This approach can also be applied within a classical context, and local control fields from classical dynamics have been used in quantum problems [61]. In parallel, Rabitz and coworkers developed a method termed tracking control, in which Ehrenfest s equations [26] for an observable is used to derive an explicit expression for the electric field that forces the system dynamics to reproduce a predefined temporal evolution of the control observable [62, 63]. In its original form, however, this method can lead to singularities in the fields, a problem circumvented by several extensions to this basic idea [64-68]. Within the context of ground-state vibration, a procedure similar to tracking control has been proposed in Ref. 69. In addition to the examples already mentioned, the different local control schemes have found many applications in molecular physics, like population control [55], wavepacket control [53, 54, 56], control within a dissipative environment [59, 70], and selective vibrational excitation or dissociation [64, 71]. Further examples include isomerization control [58, 60, 72], control of predissociation [73], or enantiomer control [74, 75]. 

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