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By Greg Knowles (Eds.)

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Wiley, New York, 1968. H. Hermes and J. P. " Academic Press, New York, 1969. M. R. Hestenes, Multiplier and gradient methods, J. Optim. Theory Appl. 4, 303-320 (1969). E. B. Lee and L. " Wiley, New York, 1967. L. Pontryagin, V. Boltyanskii, R. Gramkrelidze, and E. " Wiley (lnterscience), New York, 1962. G. " Prentice-Hall, Englewood Cliffs, New Jersey, 1980. Chapter III 1. The Pontryagin Maximum Principle THE MAXIMUM PRINCIPLE Consider the autonomous control problem: (1) Xi = h(Xl>' .. ,Xn , Ub' differentiable in R" x n.

At a(t)b(t) 2h +2 - + Cd = 0 a(T) = 0, , which, when solved, gives (14) substitution of (14) and (13) into (5) gives the feedback control law U*(t'X)=~[Cd-X]tanh(~(T-t))+d(t), O::;;t::;;T. (15) As a consequence the optimum control rate is equal to the demand rate plus an inventory correction factor, which tends to restore the inven­ tory to the desired level Cd. Further computation gives the optimal inventory level as x*(t) = Cd + (xo -fi:T:. Cd) cosh [~ - (T - t)] , cosh('\I hlc T) C 0::;; t s: T.

If we denote by J1. = Jb 2 + 1 in the lower right­ the solution of(3) passing through (0,0) with u = hand quadrant and by the solution of (3) passing through (0,0) with u = - 1 in the upper left­ hand quadrant, then the switching locus X2 = W(x 1 ) is as pictured in Fig. 11. The optimal control synthesizer is then for for X2 X2 > W(xd and on (I' _) < W(x 1) and on (r +). The verification of these details is exactly the same as in Section 2 Example 1. u ~-1 -------------t-------------~Xl u ~ ----_ ...

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