Minimal solutions for discrete-time control systems in metric spaces

In this work we study the structure of minimal solutions for an autonomous discrete-time control system in a metric space X determined by a continuous function v : X x X --> R-1. A sequence {x(i)} (i =) (-infinity) (infinity) subset of X is called (v)-minimal if for each pair of integers m(2) > m(1) and each sequence {y(i)) (m2)(i = m1) satisfying y(j) = x(j), Sigma(i=m1)(m2-1) v(x(i) , x(i), x(i+1)) less than or equal to (2) the inequality Sigma(i=m1)(m2-1) v(x(i), x(i+1)) less than or equal to Sigma(i=m1)(m2-1) v(y(i), y(i+1)) is valid. We con sider a space of functions v : X x X --> R-1 equipped with a natural complete metric and show that for a generic function v there exists a (v)-minimal sequence.