Pontryagin s maximum principle

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.). 

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. 

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. 

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 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 ... 

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

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