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**Extra info for Approximation of Hilbert Space Operators: v. 2**

**Sample text**

In this paper we present two techniques for analysis of discrete approximations in optimal control. In Section 2 we study convergence properties of the optimal value and optimal solutions. In Section 3 we obtain an estimate for the optimal control error in the case when the Euler discretization scheme is used for solving the first-order optimality conditions. Section 4 contains a survey on related results. 1. Introduction. When solving an optimal control problem we deal with functions which, except in very special cases, are to be replaced by numerically tractable approximations.

Further developments in this direction can be found in Capuzzo Dolcetta-Lions [CDL] and in the review paper by Crandall-Ishii-Lions [CIL]. e. 1) and the constraint n, is a necessary requirement to set the optimal control problem. We limit ourselves to a rather simple situation assuming that n is convex and has a smooth boundary, however a collection of results for more general classes of constraints (including the non convex case) can be found in Aubin [A]. We should also mention that some ideas coming from viability theory are at the origin of the scheme studied in Falcone-Digrisolo [FD] where some tools of nonsmooth analysis have also been used to establish its convergence.

We have lim S(h, y, ¢(y), ¢) h ;;=0 > - /3¢(x + hb(x, a)) - ¢(x) · -1 -- '/3I"'() I1m > ' y -;;=0 h h - I( y a ) -_ ' A¢(X) - b(x, a)V¢(x) - I(x, a). 39). 40). 41) S(h, y, t(y), ¢) < (1 ~ /3) ¢(y) _ /3¢(y + hb(Yha*) - ¢(y)) + I(y, a*) + e. Since ¢ E COO(fi), for all y E fi we have Ib(y, a)V¢(y) _ ¢(y + hb(y~ a)) - ¢(y) I ~ Ch. 42) imply that lim S(h,y,¢(y),¢) < h_O h - II-Z ~ l~ 1 ~ /3 ¢(y) - /3¢(y + hb(Yha*)) - ¢(y) - I(y, a*) + e ~ II-Z -1-/3 ~ l~ -h-¢(Y) - /3b(y, a*)· V¢(y) - I(y, a*) + e + Ch ~ lim 1 ~ /3 ¢(y) + sup{ -b(y, a) .