This chapter focuses on the structure of "approximate" solutions for discrete-time control systems, where the control functions are sequences of continuous functions \( v_i : X \times X \to \mathbb{R}^1 \). For a generic sequence of functions \( \{v_i\}_{i=-\infty}^{\infty} \), the chapter demonstrates the existence of a "turnpike" sequence \( \{y_i\}_{i=k_1}^{\infty} \subset X \) that satisfies two key properties: (i) \( \{y_i\}_{i=k_1}^{k_2} \) is an optimal solution for any finite interval \( [k_1, k_2] \); (ii) for any \( \epsilon > 0 \), each "approximate" solution on an interval \( [k_1, k_2] \) with sufficiently large \( k_2 - k_1 \) is within \( \epsilon \) of the turnpike for all \( i \in \{L + k_1, \ldots, k_2 - L\} \), where \( L \) is a constant depending only on \( \epsilon \). The chapter also delves into convex infinite-dimensional control systems, where \( X \) is a Banach space and \( K \subset X \) is a closed convex bounded set. The set \( \mathcal{A} \) consists of all bounded convex functions \( v : K \times K \to \mathbb{R}^1 \) that satisfy a specific Lipschitz condition. The metric space \( \mathcal{A} \) is equipped with a metric \( \rho \) defined by the supremum of the absolute difference in function values over \( K \). The study of these systems is motivated by various optimization problems, including continuous-time control systems with discounting factors, infinite-horizon control problems, and the analysis of long slender bars under tension. The chapter highlights the turnpike property, which states that for a generic function \( v \in \mathcal{A} \), there exists a unique solution \( y_v \in K \) such that "approximate" solutions are contained in a small neighborhood of \( y_v \) for large \( n \). This property is established without compactness assumptions on \( K \) or additional assumptions on the functions \( v \).This chapter focuses on the structure of "approximate" solutions for discrete-time control systems, where the control functions are sequences of continuous functions \( v_i : X \times X \to \mathbb{R}^1 \). For a generic sequence of functions \( \{v_i\}_{i=-\infty}^{\infty} \), the chapter demonstrates the existence of a "turnpike" sequence \( \{y_i\}_{i=k_1}^{\infty} \subset X \) that satisfies two key properties: (i) \( \{y_i\}_{i=k_1}^{k_2} \) is an optimal solution for any finite interval \( [k_1, k_2] \); (ii) for any \( \epsilon > 0 \), each "approximate" solution on an interval \( [k_1, k_2] \) with sufficiently large \( k_2 - k_1 \) is within \( \epsilon \) of the turnpike for all \( i \in \{L + k_1, \ldots, k_2 - L\} \), where \( L \) is a constant depending only on \( \epsilon \). The chapter also delves into convex infinite-dimensional control systems, where \( X \) is a Banach space and \( K \subset X \) is a closed convex bounded set. The set \( \mathcal{A} \) consists of all bounded convex functions \( v : K \times K \to \mathbb{R}^1 \) that satisfy a specific Lipschitz condition. The metric space \( \mathcal{A} \) is equipped with a metric \( \rho \) defined by the supremum of the absolute difference in function values over \( K \). The study of these systems is motivated by various optimization problems, including continuous-time control systems with discounting factors, infinite-horizon control problems, and the analysis of long slender bars under tension. The chapter highlights the turnpike property, which states that for a generic function \( v \in \mathcal{A} \), there exists a unique solution \( y_v \in K \) such that "approximate" solutions are contained in a small neighborhood of \( y_v \) for large \( n \). This property is established without compactness assumptions on \( K \) or additional assumptions on the functions \( v \).