Pontryagin’s method

Pontryagin s method can be applied to a number of different cases, but only the form needed for comparison will be given here. Consider an n-dimensional state space vector x = (jc. .. "), with r control variables u = (w1,... ur), related by... 

Studies in optimization-VII The application of Pontryagin s methods to the control of batch and tubular reactors (with C.D. Siebenthal). Chem. Eng. ScL 19,747-761 (1964). 

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.). 

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 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. 

Moreover, quite often are the cases when the derivatives of functions and functionals in specific points and regions do not really exist (e.g. in diseontinuous or broken-line funetions). To get over these difficulties, non-classic methods of ealeulus of variations are applied. Among these the most effective and naturally the more popular ones are Bellman s method of dynamic programming and Pontryagin s principle of maximum. 

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]. 

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

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