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Optimal programs on infinite horizon. I. (English) Zbl 0847.49021
The author investigates the infinite-horizon problem \[ \sum^{N- 1}_{i= 0} v(x_i, x_{i+ 1})= \min!, \] where \(\{x_i\}^\infty_{i= 0}\subset K\) is called a program, \(K\) is a compact metric space, \(v\) is continuous on \(K\times K\). Such kind of problems allows the study of a large class of optimization problems.
Three concepts of optimality are considered: 1. (v)-overtaking optimal program, 2. (v)-weakly optimal program, 3. (v)-good program. The structure of (v)-good programs is studied. For an arbitrary initial data \(x_0\in K\) the existence of a (v)-optimal program is shown.
[See also the review of Part II below].

49J99 Existence theories in calculus of variations and optimal control
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