Backstepping is a mature and powerful Lyapunov-based design approach for a specific set of systems. Throughout the development over three decades, innovative theories and practices have extended backstepping to stabilization and tracking problems for nonlinear systems with growing complexity. The attractions of the backstepping-like approach are the recursive design processes and modularized design. A nonlinear system can be transferred into a group of simple problems and solved it by a sequential superposition of the corresponding approaches for each problem. To handle the complexities, backstepping designs always come up with adaptive control and robust control. The survey aims to review the milestone theoretical achievements among thousands of publications making the state-feedback backstepping designs of complex ODE systems to be systematic and modularized. Several selected elegant methods are reviewed, starting from the general designs, and then the finite-time control enhancing the convergence rate, the fuzzy logic system and neural network estimating the system unknowns, the Nussbaum function handling unknown control coefficients, barrier Lyapunov function solving state constraints, and the hyperbolic tangent function applying in robust designs. The associated assumptions and Lyapunov function candidates, inequalities, and the deduction key points are reviewed. The nonlinearity and complexities lay in state constraints, disturbance, input nonlinearities, time-delay effects, pure feedback systems, event-triggered systems, and stochastic systems. Instead of networked systems, the survey focuses on stand-alone systems.

The \emph{turnpike property} in contemporary macroeconomics asserts that if an economic planner seeks to move an economy from one level of capital to another, then the most efficient path, as long as the planner has enough time, is to rapidly move stock to a level close to the optimal stationary or constant path, then allow for capital to develop along that path until the desired term is nearly reached, at which point the stock ought to be moved to the final target. Motivated in part by its nature as a resource allocation strategy, over the past decade, the turnpike property has also been shown to hold for several classes of partial differential equations arising in mechanics. When formalized mathematically, the turnpike theory corroborates the insights from economics: for an optimal control problem set in a finite-time horizon, optimal controls and corresponding states, are close (often exponentially), during most of the time, except near the initial and final time, to the optimal control and corresponding state for the associated stationary optimal control problem. In particular, the former are mostly constant over time. This fact provides a rigorous meaning to the asymptotic simplification that some optimal control problems appear to enjoy over long time intervals, allowing the consideration of the corresponding stationary problem for computing and applications. We review a slice of the theory developed over the past decade --the controllability of the underlying system is an important ingredient, and can even be used to devise simple turnpike-like strategies which are nearly optimal--, and present several novel applications, including, among many others, the characterization of Hamilton-Jacobi-Bellman asymptotics, and stability estimates in deep learning via residual neural networks.