Variables, adjoint state

Results from this model were verified by Neupauer and Wilson [51] using the adjoint method. In this method, the forward governing equation, with concentration as the dependent variable, is replaced by the adjoint equation, with the adjoint state as the dependent variable. They showed that backward-in-time location and travel time probabilities are adjoint states of the forward-in-time resident concentration. In this and the follow-up paper, Neupauer and Wilson [51,52] presented the adjoint method as a formal framework for obtaining the backward-in-time probabilities for multidimensional problems and more complex domain geometries. 

Application of the maximum principle to the above problem involves the addition of nx adjoint variables zt (one adjoint variable per state variable), nx adjoint equations, and a Hamiltonian, which satisfies the following relations ... 

These methods are efficient for problems with initial-value ODE models without state variable and final time constraints. Here solutions have been reported that require from several dozen to several hundred model (and adjoint equation) evaluations (Jones and Finch, 1984). Moreover, any additional constraints in this problem require a search for their appropriate multiplier values (Bryson and Ho, 1975). Usually, this imposes an additional outer loop in the solution algorithm, which can easily require a prohibitive number of model evaluations, even for small systems. Consequently, control vector iteration methods are effective only when limited to the simplest optimal control problems. 

The accelerated gradient method is used because of its advantages especially when the control is constrained. The system and its adjoint equations are coupled hyperbolic partial differential equations. They can be solved numerically using the method of characteristics (Lapidus, 1962b Chang and Bankoff, 1969). This method is used with the fourth order Runge-Kutta method (with variable step size to ensure accuracy of the integration) to solve the state and adjoint equations. 

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 open-loop strategy implies that each players control is only a function of time, Ui = Ui t). A feedback strategy implies that each players control is also a function of state variables, ui = Ui t Xi t) Xj(t)). As in the static games, NE is obtained as a fixed point of the best response mapping by simultaneously solving a system of first-order optimality conditions for the players. Recall that to find the optimal control we first need to form a Hamiltonian. If we were to solve two individual non-competitive optimization problems, the Hamiltonians would be Hi = fi XiQi, i = 1,2, where Xi t) is an adjoint multiplier. However, with two players we also have to account for the state variable of the opponent so that the Hamiltonian becomes... 

The optimal control problem represents one of the most difficult optimization problems as it involves determination of optimal variables, which are vectors. There are three methods to solve these problems, namely, calculus of variation, which results in second-order differential equations, maximum principle, which adds adjoint variables and adjoint equations, and dynamic programming, which involves partial differential equations. For details of these methods, please refer to [23]. If we can discretize the whole system or use the model as a black box, then we can use NLP techniques. However, this results in discontinuous profiles. Since we need to manipulate the techno-socio-economic poHcy, we can consider the intermediate and integrated model for this purpose as it includes economics in the sustainabiHty models. As stated earlier, when we study the increase in per capita consumption, the system becomes unsustainable. Here we present the derivation of techno-socio-economic poHcies using optimal control appHed to the two models. 

If the adjoint function satisfies these equations and boundary conditions, Lis a stationary expression, insensitive to small errors in the density, whose numerical value will yield C. Inspection shows that the Lagrangian has a certain symmetry such that, if N satisfies its equation and boundary conditions, then the Lagrangian is stationary to errors in iV (stationary, in fact, to large errors, since /o and M are not functions of the costate variable). In practice, both equations are perturbed by a change in the control variable and simultaneous errors are made in both functions. For small control perturbations, we anticipate small perturbations in the state and costate variables and that the resulting expression is in error in the cost function only through terms involving the product of small errors. We write... 

Assume that the Lagrangian has been established and the adjoint equation and boundary conditions imposed as before. Suppose some perturbation in the control is considered, small but arbitrary within the limits set by the constraints on controls. We would, in general, need to consider not the partial derivative of a functional with respect to the perturbation in control variable, but rather the total differential that included the indirect effect of the control perturbation on the state and control variables ... 

The adjoint operator is defined by a commutation relation that is to be valid for any state variable satisfying the boundary conditions of the problem and of the possible Lagrange multipliers through the relation... 

Equation (7.7.8) is called the Fundamental Property of the Adjoint System. It is a relationship between the initial state and adjoint variables and their values at the final conditions. The right-hand side is only a function of the adjoint variables, /, and the known forcing vector, /, of the system equations. The adjoint variables p t) can be calculated using equation (7.7.2) and arbitrary initial values. 

This equation can either be integrated forward in time with the state equations or backward in time w ith the adjoint equations. The choice of the backward in time integration is more efficient since we do not have to store the values of p2- Instead y can be computed simultaneously while computing the adjoint variables. 

By application of optimal control theory, the singular feed rate (Fg) can be expressed as a nonlinear feedback expression involving state and adjoint variables (Modak, Lim, Tayeb, 1986). Depending on the process kinetics, this feedback law maintains the substrate concentration constant or allows its variation in a predetermined manner. If the... 

Example 5.5 Solution of the Optimal Temperature Profile for Penicillin Fermentation. Apply the orthogonal collocation method to solve the two-point boundary-value problem arising from the application of the maximum principle ofPontryagin to a batch penicillin fermentation. Obtain the solution of this problem, and show the profdes of the state variables, the adjoint variables, and the optimal temperature. The equations that describe the state of the system in a batch penicillin fermentation, developed by Constantinides et al.(6], are ... 

Eqs. (1 )-(6) form a two-point boundary-value problem. Apply the orthogonal collocation method to obtain the solution of this problem, and show the profiles of the state variables, the adjoint variables, and the optimal temperature. 

For each control variable, the sensitivity method involves an additional Hnear IVP to be solved. Thus for a large number of parameters, this method is not reasonable. Another method is more feasible which avoids calculating the sensitivity of the state directly, the so-called adjoint method which enables to compute the sensitivities in one shot. 

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