Pontryagin, maximum principle

In the original derivation of the maximum principle, Pontryagin (1986) chose the final costate along the normal p, i. e., away from the set of final states. 

Variational calculus, Dreyfus (1962), may be employed to obtain a set of differential equations with certain boundary condition properties, known as the Euler-Lagrange equations. The maximum principle of Pontryagin (1962) can also be applied to provide the same boundary conditions by using a Hamiltonian function. 

Depending on the numerical techniques available for solving optimal control or optimisation problems the model reformulation or development of simplified version of the original model was always the first step. In the Sixties and Seventies simplified models represented by a set of Ordinary Differential Equations (ODEs) were developed. The explicit Euler or Runge-Kutta methods (Huckaba and Danly, 1960 Domenech and Enjalbert, 1981) were used to integrate the model equations and the Pontryagin s Maximum Principle was used to obtain optimal operation policies (Coward, 1967 Robinson, 1969, 1970 etc.). 

The Maximum Principle of Pontryagin (Pontryagin et al., 1964) may be stated briefly as follows. For a given set of ODEs ... 

Table 5.2. Summary of Minimum Time Problem Using Pontryagin s Maximum Principle...

Diwekar (1992, 1995) has extensively used Pontryagin s Maximum Principle for solving all types of optimisation problems (section 5.2) using the short-cut model presented in Chapter 4. Refer to the original references for example problems. 

Pontryagin s Maximum Principle to pose the dynamic optimisation problems as NLP problems. The problems were then solved by SQP based or other efficient methods. 

The problem of choosing whether and when to recycle each off-cut and the size of the cut is a difficult one. Liles (1966) considered dynamic programming approach and Luyben (1988) considered repetitive simulation approach to tackle this problem. Mayur et al. (1970) and Christensen and Jorgensen (1987) tackled it as a dynamic optimisation problem using Pontryagin s Maximum Principle applied to very simplified column models as mentioned in Chapters 4 and 5. 

To find the optimal temperature profile, the Pontryagin maximum principle can be used. However, in this case, the Pontryagin maximum principle results in a very simple optimization criterion, that is at each point along the length of the reactor find the temperature that maximizes the net rate of reaction at this point . Proof of the fact that the Pontryagin maximum principle in this case gives this simple criterion is given in the next section. 

The maximization problem is to maximize (7.35) subjected to constraints (7.32-7.34). The derivation of the Pontryagin maximum principle for the system and the optimality conditions are given in Appendix D, for one internal collocation point. The derivation of the optimality conditions for N internal collocation points is given by Elnashaie and El-Rifaie (1978). 

Poisoning of catalyst 42, 398, 400 Pontryagin maximum principle 276, 286-290 Pore diffusion 54 Porosity 19, 96, 300 Porous catalyst pellet 15, 139, 192-260, 436-448 structure 15 Prandtl number 299 Pre-exponential (frequency) factor 62,... 

Optimization for Reversible Reactions Pontryagin Maximum Principle and the Simple Optimality Criterion for Exothermic Reversible Reactions (The Single Reaction Pseudo-homogeneous Model)... 

Lynn ei ai (1970) obtain the optimal temperature profiles for a tubular reactor with axial dispersion (Lynn et ai, 1970). Weighted residual techniques were used to solve the state and adjoint differential equations which result from the application of Pontryagin s maximum principle to the optimal control problem. 

Pontryagin maximum principle under the constraint < T < The maximum principle uses a temperature selection for every axial position in a tubular reactor for which the Hamiltonian is maximal with respect to T and remains constant along the temperature profile. 

One of the most profound results of applied mathematics, Pontryagin s minimum principle provides the necessary conditions for the minimum of an optimal control problem. The elegance of the principle lies in the simplicity of its application to a vast variety of optimal control problems. Boltyanskii et al. (1956) developed the principle originally as a maximum principle requiring the Hamiltonian to be maximized at the minimum. 

The va/Mg-based approach significantly improves the effectiveness of procedures of controlling chemical reactions. Optimal control on the basis of the value method is widely used with Pontryagin s Maximum Principle, while simultaneously calculating the dynamics of the value contributions of individual steps and species in a reaction kinetic model. At the same time, other methods of optimal control are briefly summarized for a) calculus of variation, b) dynamic programming, and c) nonlinear mathematical programming. 

Rudick, AP. Nuclear reactors and Pontryagin s maximum principle [in Russian], Atomizdat, Moscow, 1971. 

The method of determining the initial concentration of the inhibitor that results in maximum inhibition effect is based on Pontryagin s maximum principle. For this case the kinetic equation and the appropriate Hamiltonian are written as... 

Indirect or variational approaches are based on Pontryagin s maximum principle [8], in which the first-order optimality conditions are derived by applying calculus of variations. For problems without inequality constraints, the optimality conditions can be written as a set of DAEs and solved as a two-point boundary value problem. If there are inequality path constraints, additional optimality conditions are required, and the determination of entry and exit points for active constraints along the integration horizon renders a combinatorial problem, which is generally hard to solve. There are several developments and implementations of indirect methods, including [9] and [10]. 

Chapter 4 applies variational calculus to problems that include control variables as well as state variables. Optimal control strategies are developed that extremize precise performance criteria. Necessary conditions for optimization are shown to be conveniently expressed in terms of a mathematical function called the Hamiltonian. Pontryagin s maximum principle is developed for systems that have control constraints. Process applications of optimal control are presented. 

The rigorous optimization could be performed with several mathematical techniques — see Beveridge and Schechter [1970], and for a concise discussion of the Pontryagin maximum principle, see Ray and Szekely [1973] also see Aris [1961] for specific chemical reactor examples. 

Optimal control theory known as the Continuous Maximum Principle due to Pontryagin etal. (1962), which may be stated as... 

The results discussed earlier for concentration dependent deactivation kinetics were based on Pontryagin s maximum principle (e.g., Koppel 1972) as extended to pseudo-steady state systems (Sirazetdinov and Degtyarev 1967). Here, a weak but more general maximum principle is considered due to Ogunye and Ray (1971). Let the concentrations (conversions) and temperatures be denoted by state variables X and the control variables such as temperatures (which are not state variables) by U and the catalytic activities by A. In general, under the pseudo-steady state assumption one has ... 

The minimization of the cost criterion can be accomplished using the maximum principle introduced by Pontryagin and the principle of optimality introduced by Bellman. The procedure results in a system of differential equations as follows ... 

Application of rigorous optimization methods, such as Pontryagin s maximum principle or the calculus of variations... 

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