Optimal Control Analysis

Note that Ji is independent of 6X while J2 is independent of Sy and 6u. Because of this fact, both Ji and J2 must be individually zero. Otherwise, Ji Sy,Su) = —J2 6X), implying the contradiction that Ji depends on and J2 depends on 5y and Su. Therefore, Ji = 0 and J2 = 0 must be satisfied in order to ensure 5J = 0. 

Since 6X is arbitrary in the above equation, J2 = 0 is satisfied by having 

Arising from 6J = 0, both Equations (3.11) and (3.12) are the necessary conditions for the optimum of J. On the basis of the Lagrange Multiplier Rule, these two equations are also the necessary conditions for the constrained optimum of I. Both optima are still subject to the given initial condition, i. e.. Equation (3.6). The equivalence between the two optima will be shown later in Section 4.3.3. 

We now proceed to obtain workable equations that will ensure the satisfaction of Equation (3.12), and consequently, be necessary along with Equation (3.11) for the optimum of /. 

Note that J3 does not depend on Su, and J4 does not depend on 6y. Therefore, for the above equation to hold, both integrals must be individually zero. Otherwise, Jz 5y) = —Ji 8u), implying that J3 depends on 5u and J4 depends on 6y, which is contradictory. Thus, 

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In other words, functionals are more likely to have variations than differentials. Thus, most of the time, we will use the variation of functionals in optimal control analysis. Frechet and Gateaux differentials will be invoked only when their typical properties are needed. 

Extending the optimal control analysis (Section 3.2.2, p. 59) to the generalized problem with multiple states and controls, and defining the Hamiltonian as... 

The optimal control analysis developed above is also applicable when a control function is piecewise continuous with a finite number of jump discontinuities, as shown in Figure 3.4. Observe that a jump discontinuity suddenly changes y or the slope of the state variable. The result is a corner in the graph of y t) at the time of discontinuity where y is not defined. Geometrically, there is no unique slope of the curve at a corner. One can draw an infinite number of tangents there. 

Let us understand the import of the above result in light of the optimal control analysis we have done so far. The augmented functional for this problem is... 

HSmalainen, R.P. (1992) Optimal control analysis of the breathing pattern by using the WBREPAT computer software, in Control of Breathing and its Modelling Perspective (eds Y. Honda, Y. Miyamoto, K. Konno and J.G. Widdicombe), Plenum Press, New York, pp. 393-6. 

Khludneva E. Yu. (1990a) Optimal control of external forces in contact problems for a viscoelastic plate. In Algebra and Math. Analysis. Novosibirsk, 8-14 (in Russian). 

Metabolic control analysis (MCA) assigns a flux control coefficient (FCC) to each step in the pathway and considers the sum of the coefficients. Competing pathway components may have negative FCCs. To measure FCCs, a variety of experimental techniques including radio isotopomers and pulse chase experiments are necessary in a tissue culture system. Perturbation of the system, for example, with over-expression of various genes can be applied iteratively to understand and optimize product accumulation. 

A reported application of canonical analysis involved a novel combination of the canonical form of the regression equation with a computer-aided grid search technique to optimize controlled drug release from a pellet system prepared by extrusion and spheronization [28,29]. Formulation factors were used as independent variables, and in vitro dissolution was the main response, or dependent variable. Both a minimum and a maximum drug release rate was predicted and verified by preparation and testing of the predicted formulations. Excellent agreement between the predicted values and the actual values was evident for the four-component pellet system in this study. 

It may be useful to point out a few topics that go beyond a first course in control. With certain processes, we cannot take data continuously, but rather in certain selected slow intervals (c.f. titration in freshmen chemistry). These are called sampled-data systems. With computers, the analysis evolves into a new area of its own—discrete-time or digital control systems. Here, differential equations and Laplace transform do not work anymore. The mathematical techniques to handle discrete-time systems are difference equations and z-transform. Furthermore, there are multivariable and state space control, which we will encounter a brief introduction. Beyond the introductory level are optimal control, nonlinear control, adaptive control, stochastic control, and fuzzy logic control. Do not lose the perspective that control is an immense field. Classical control appears insignificant, but we have to start some where and onward we crawl. 

The simulation control includes the methods of generating price simulation scenarios either manually, equally distributed or using stochastic distribution approaches such as normal distribution. In addition, the number of simulation scenarios e g. 50 is defined. The optimization control covers preprocessing and postprocessing phases steering the optimization model. The optimization model is then iteratively solved for a simulated price scenario and optimization results including feasibility of the model are captured separately after iteration. Simulation results are then available for analysis. 

Coupled columns packed with different stationary phases can be used to optimize the analysis time (71, 75). In this approach the different columns are connected in a series or in parallel. liie sample mixture is first fractioned on a relatively short column. Subsequently the fractions of the partially separated mixture are separated on other columns containing the same or other stationary phases in order to obtain the individual components. Columns differing in length (number of theoretical plates), adsorptive strength or phase ratio (magnitude of specific surface area), and selectivity (nature of the stationary phase) can be employed, whereas, the eluent composition remains unchanged. Identification of the individual sample components via coupled column technique requires a careful optimization of each column and precise control of each switching step. 

S. P. Shah and S. A. Rice. Controlling quantum wavepacket motion in reduced-dimensional spaces reaction path analysis in optimal control of HCN isomerization. Faraday Trans., 113 319-331(1999). 

The full development of these mechanism analysis tools has not been achieved at this time, but some of the basic features have been set fbrth[18]. The key concept is to introduce further modulation into the optimal control field e(f) to reveal the dominant quantum pathways. This task calls for special care, as the control field already is temporally modulated in a delicate fashion to manipulate the quantum system. A way around this difficulty is feasible by introducing a new pseudo time-like variable s such that e(r) — In practice, s is a... 

Suppose the reactive polyatomic molecule of interest can undergo uni-molecular reaction to form several products, and we imagine carrying out a constrained reaction path analysis for each of the product channels. To carry out the analysis of a particular constrained reaction path, Zhao and Rice adopted a system-bath model [74] in which the reaction path coordinate delines the system and all other coordinates constitute the bath. The use of this representation permits the elimination of the bath coordinates, which then increases the efficiency of calculation of the optimal control field for motion along the reaction coordinate. 

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... 

Below we show how the energy-optimal control of chaos can be solved via a statistical analysis of fluctuational trajectories of a chaotic system in the presence of small random perturbations. This approach is based on an analogy between the variational formulations of both problems [165] the problem of the energy-optimal control of chaos and the problem of stability of a weakly randomly perturbed chaotic attractor. One of the key points of the approach is the identification of the optimal control function as an optimal fluctuational force [165],... 

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. 

However, using a method proposed [60,62,95,112] for experimental analysis of the Hamiltonian flow in an extended phase space of the fluctuating system, we can exploit the analogy between the Wentzel-Freidlin and Pontryagin Hamiltonians arising in the analysis of fluctuations, and the energy-optimal control problem in a nonlinear oscillator. To see how this can be done, let us consider the fluctuational dynamics of the nonlinear oscillator (35). 

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