Maximum principle, Bellman

Consider now two steps, the last two of the multibed adiabatic reactor. From Bellman s maximum principle it follows that the optimal policy of bed 1 is preserved. This time X2 and Ti have to be chosen in an optimal way to arrive at... 

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 ... 

To find this maximum it suffices to choose the conditions in the first of the two beds, since the optimal policy of the second has been calculated already. Equation (11.5.d-2) is Bellman s optimum principle. Consider now the last three beds. These can be decomposed into a first bed (in the direction of process flow) and a pseudo stage consisting of the last two beds, for which the optimal policy has already been calculated for a series of inlet conditions. The procedure is continued in the same way towards the inlet of the multibed reactor. 

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. 

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