Optimal control theory objectives

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. 

The formalism used to calculate the pulse shape that maximizes J is optimal control theory. This formalism can be considered to be an extension of the calculus of variations to the case where the constraints include differential equations. In general, the constraints expressed in the form of differential equations express the restriction that the amplitude must always satisfy the Schrodinger equation. In addition, there can be a variety of other constraints, such as a restriction on the total energy in the pulse or on the shape of the pulse. These constraints are accounted for by the method of Lagrange multipliers, which modify the objective functional (4.6) and thereby permit the calculation of the unconstrained maximum of the modified objective functional. When the only constraints are satisfaction of the Schrodinger equation and limitation of the pulse energy, the modified objective functional can be written in the form... 

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. 

Optimal control theory aims to maximize or minimize certain transition probabilities, called objectives, such as the production of a specified wave function at a specified time tf, given a wave function F(t0) at time f0. The general principles of OCT are best understood via a case study due to Rice and coworkers [104, 119], illustrated in Figure 4.2, in which the objective is to concentrate the wave function in one of the exit channels of a bifurcating chemical reaction ... 

Therefore, the controller is a linear time-invariant controller, and no online optimization is needed. Linear control theory, for which there is a vast literature, can equivalently be used in the analysis or design of unconstrained MPC (Garcia and Morari, 1982). A similar result can be obtained for several MPC variants, as long as the objective function in Eq. (4). remains a quadratic function of Uoptfe+ -iife and the process model in Eq. (22) remains linear in Uoptfe+f-ife. Incidentally, notice that the appearance of the measured process output y[ ] in Eq. (22) introduces the measurement information needed for MPC to be a feedback controller. This is in the spirit of classical hnear optimal control theory, in which the controlled... 

Rawlings and Muske (1993) have shown that this idea can be extended to unstable processes. In addition to guaranteeing stability, their approach provides a computationally efficient method of on-line implementation. Their idea is to start with a finite control (decision) horizon but an infinite prediction (objective function) horizon, i.e., m < < and p = , and then use the principle of optimality and results from optimal control theory to substitute the infinite prediction horizon objective by a finite prediction horizon objective plus a terminal penalty term of the form... 

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. 

Optimal control theory is a formalism created to be similar in structure to Hamiltonian mechanics. One describes the evolution of a system in terms of some set of state variables and a control function (or functions) which also depends on those variables or related ones. The problem is stated in terms of a goal whose degree of achievement is measured by a performance index or objective function that we can write as a time integral of a time-dependent function L, relating the state variables x(t) and the control variables u(t) ... 

Let II II denote the Euclidean norm and define = gk+i gk- Table I provides a chronological list of some choices for the CG update parameter. If the objective function is a strongly convex quadratic, then in theory, with an exact line search, all seven choices for the update parameter in Table I are equivalent. For a nonquadratic objective functional J (the ordinary situation in optimal control calculations), each choice for the update parameter leads to a different performance. A detailed discussion of the various CG methods is beyond the scope of this chapter. The reader is referred to Ref. [194] for a survey of CG methods. Here we only mention briefly that despite the strong convergence theory that has been developed for the Fletcher-Reeves, [195],... 

Optimal control problems involving multiple integrals are constrained by partial differential equations. A general theory similar to the Pontryagin s minimum principle is not available to handle these problems. To find the necessary conditions for the minimum in these problems, we assume that the variations of the involved integrals are weakly continuous and find the equations that eliminate the variation of the augmented objective functional. 

The approach used here to develop the control structure for the modules is a hybrid of the formal mathematical approach [14,26] and the heuristic-based approach [7,19,48]. The portion of the mathematical approach used involves the application of optimization (steady-state) theory to identify active control constraints for each module. This optimization uses an objective function, O, that attempts to minimize the loss in profit (see Equation (3)) for a given module. 

One object of separation theory is to predict H in terms of controllable experimental parameters such as flow velocity, field strength, and support particle diameter. This theory is important in optimizing separations. Details will be provided in later chapters. 

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