This chapter studies the structure of approximate solutions for discrete-time control systems defined by a sequence of continuous functions $ v_i: X \times X \to \mathbb{R} $, where X is a complete metric space. It shows that for a generic sequence of functions $ \{v_i\} $, there exists a sequence $ \{y_i\} $ (the turnpike) such that any finite interval of $ \{y_i\} $ is optimal, and approximate solutions on large intervals are close to the turnpike. In Section 8.1, the focus is on convex infinite-dimensional control systems. X is a Banach space, and K is a closed convex bounded subset of X. The set A consists of bounded convex functions $ v: K \times K \to \mathbb{R} $ satisfying a Lipschitz condition. The space A is equipped with a metric $ \rho $, making it a complete metric space. The chapter studies optimization problems of the form $ \sum_{i=0}^{n-1} v(x_i, x_{i+1}) \to \min $, with $ \{x_i\} \subset K $ and $ x_0 = y, x_n = z $. These problems arise in various contexts, including continuous-time control systems, infinite-horizon control, and material analysis. In [99], it is shown that for a generic function $ v \in A $, there exists a point $ y_v \in K $ such that approximate solutions of the optimization problem are close to $ y_v $ for large n. This turnpike property is typically studied for single cost functions and compact convex state spaces. However, in [99], the turnpike property is investigated for the space of all such functions equipped with a natural metric, showing it holds for most functions without requiring compactness of the state space or strict convexity of the functions. This allows the turnpike property to be established without these assumptions.This chapter studies the structure of approximate solutions for discrete-time control systems defined by a sequence of continuous functions $ v_i: X \times X \to \mathbb{R} $, where X is a complete metric space. It shows that for a generic sequence of functions $ \{v_i\} $, there exists a sequence $ \{y_i\} $ (the turnpike) such that any finite interval of $ \{y_i\} $ is optimal, and approximate solutions on large intervals are close to the turnpike. In Section 8.1, the focus is on convex infinite-dimensional control systems. X is a Banach space, and K is a closed convex bounded subset of X. The set A consists of bounded convex functions $ v: K \times K \to \mathbb{R} $ satisfying a Lipschitz condition. The space A is equipped with a metric $ \rho $, making it a complete metric space. The chapter studies optimization problems of the form $ \sum_{i=0}^{n-1} v(x_i, x_{i+1}) \to \min $, with $ \{x_i\} \subset K $ and $ x_0 = y, x_n = z $. These problems arise in various contexts, including continuous-time control systems, infinite-horizon control, and material analysis. In [99], it is shown that for a generic function $ v \in A $, there exists a point $ y_v \in K $ such that approximate solutions of the optimization problem are close to $ y_v $ for large n. This turnpike property is typically studied for single cost functions and compact convex state spaces. However, in [99], the turnpike property is investigated for the space of all such functions equipped with a natural metric, showing it holds for most functions without requiring compactness of the state space or strict convexity of the functions. This allows the turnpike property to be established without these assumptions.