By Morton M Denn
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We define a new variable yn by yn - yn-1 - 61n(xn_1)un) = 0 It follows then that 8= YN yo = 0 (3) (4) OPTIMIZATION BY VARIATIONAL METHODS 10 and substitution of Eqs. (1) and (3) into the generalized Euler equation [Eq. (7), Sec. 1 a6i,. -1- 0f/au,1 n= 1,2, . . ,N (5) with the boundary condition acRrv atN - 0 (6) The difference equations (1) and (5) are then solved by varying ul until Eq. (6) is satisfied. Fan and Wangt have collected a number of examples of processes which can be modeled by Eqs.
M. Denn and R. Aris: Z. Angew. Math. , 16:290 (1965) Applications to several elementary one-dimensional design problems are contained in L. T. Fan and C. S. 15: The interpretation of Lagrange multipliers as sqp'sitwity coefficients follows the books by Bellman and Dreyfus and Hadley. The chain-rule development for one-dimensional staged processes is due'to F. Horn and R. Jackson: Ind. Eng. Chem. 16: The use of penalty functions appears to be due to Courant: R. Courant: Bull. Am. Math. , 49:1 (1943) The theoretical basis is contained in supplements by H.
Ously for X1, xi, then X2, x2, etc. We have already noted, however, that the particular problem we are considering has a closed-form solution, and our past experience, as well as the structure of Eq. (6), suggests that we seek a solution of the form x. = C2Mnxn (7a)4 U. X. (7b) or We then obtain xn+l - axn - Mn+lxn+l = 0 (8a) x. -C2=0 (9) with the boundary condition MN+1 = 0 (10) for a finite control period. Note that as N - w, we cannot satisfy Eq. (10), since the solution of the Riccati equation becomes a constant, but in that case XN - 0 and the boundary condition AN+1 = 0 is still satisfied.