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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.