Augmented Functional

In the present problem defined by Equation (3.4)-(3.6), we adjoin the state equation constraint to / using a Lagrange multiplier A and obtain the augmented functional 

For the given initial condition, i.e., Equation (3.6), the optimization of J is then equivalent to the optimization of I constrained by the state equation. Observe how the integrand / in Equation (3.7) is formed by multiplying A by G, which consists of all terms of the state equation constraint moved to the right-hand side. We will follow this approach, which will later on enable us to introduce a useful mnemonic function called the Hamiltonian. 

The Lagrange multiplier A is also known as the adjoint or costate variable. It is an undetermined function of an independent variable, which is t in the present problem. Both A and the optimal u are determined by the necessary condition for the optimum of J. The subsequent analysis expands the necessary condition, which is terse as such, into a set of workable equations or necessary conditions to be satisfied at the optimum. 

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The penalty term of an augmented Lagrangian method is designed to add positive curvature so that the Hessian of the augmented function is positive-definite. 

Optimization of augmenting functions for the description of electron affinities, weak interactions, or core-valence correlation effects. 

IV) Functions, called augmenting functions (AF), which are needed to describe small residual distortions of the valence charge distribution on formation of the molecule. 

Early studies demonstrated a role for CRH in augmenting function of immune cells, such as natural killer cells (Carr et al 1990). The localization and the synthesis of CRH in the immune system and in immune cells were initially suggested by finding mRNA for CRH in leukocytes (Stephanou et al., 1990) and significant concentrations of CRH in areas of inflammation (KaraUs et al., 1991). This suggested CRH was produced locally that is, in the periphery in areas of inflammation, probably in part by immune cells. Further studies supported these ideas. In Lew/N rats, which are deficient in hypothalamic CRH responses to inflammatory stimuli, there are high levels of CRH in inflamed joints (Crofford et al., 1992). These data suggested that CRH plays the role of an autocrine or paracrine inflammatory factor. The obvious paradox is that CRH plays a powerful antiinflammatory role as the primary CNS activator of the HPA axis, yet in local peripheral sites it has just the opposite effect. Thus, CRH expression and function with respect to inflammation is site specific. 

A more enlightening example is that in which not only is the constraint of normalization of the probability distribution imposed, but it is assumed that we know the average value of some quantity, for example, E) = EiPi Ei). In this case, we again consider an augmented function, which involves two Lagrange multipliers, one tied to each constraint, and given by... 

Note that once again the first constraint imposes normalization of the distribution. Maximizing this augmented function leads to the conclusion that pj =. ... 

It now remains to define the tail N icr) of the augmented muffin-tin orbital in a suitable form. Here we recall that the original spherical Bessel and Neumann functions obey the expansion theorem (5.14), and it is therefore natural to require that the augmented functions and also satisfy this theorem. Hence, we are led to the definition... 

One application of partial derivatives is in the search for minimum and maximum values of a function. An extremum (minimum or maximum) of a function in a region is found either at a boundary of the region or at a point where all of the partial derivatives vanish. A constrained maximum or minimum is found by the method of Lagrange, in which a particular augmented function is maximized or minimized. 

To proceed, we need to apply the Lagrange Multiplier Rule, the details of which will be provided later in Chapter 4. According to this rule, the above constrained problem is equivalent to the problem of finding the control T t) that maximizes the following augmented functional ... 

At this point it is sufficient to learn how the augmented functional J is formed by adjoining the constraint [Equation (2.22)] to the original functional / after bringing aU constraint terms to the right-hand side and multiplying them by an undetermined multipher A. 

Derive the augmented functional for the optimal periodic control problem in Section 1.3.5 (p. 11). Find the variation of that functional. 

This is the simple batch reactor control problem (see Example 2.10, p. 45) for which the augmented functional is... 

Using the Lagrange Multiplier Rule, the augmented functional is... 

This rule is based on the Lagrange Multiplier Theorem. Consider the augmented functional... 

Now the minimum of J implies that the state equation is already satisfied for the given initial condition r/(0) = 0. The augmented functional is given by... 

Hence from the serial application of the John Multiplier Theorem, [similar to that in Section 4.3.3.1 (p. 100)], the final augmented functional is given by... 

The equivalent problem is to minimize J given by Equation (4.15) on p. 105 subject to the initial conditions and the algebraic constraints. It is assumed that the preconditions for the minimum of J (see p. 105) are satisfied. The augmented functional for this modified problem is given by... 

Show that the augmented functional formed by adjoining even the initial condition of the state equation leads to the same necessary conditions for the optimum. 

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... 

Based on the Lagrange Multiplier Rule (see Section 4.3.3.2, p. 103), the above problem is equivalent to minimizing the augmented functional... 

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