  The turnpike property refers to the phenomenon that in many optimal control
problems, the solutions for different initial conditions and varying horizons
approach a neighborhood of a specific steady state, then stay in this
neighborhood for the major part of the time horizon, until they may finally
depart. While early observations of the phenomenon can be traced back to works
of Ramsey and von Neumann on problems in economics in 1928 and 1938, the
turnpike property received continuous interest in economics since the 1960s and
recent interest in systems and control. The present chapter provides an
introductory overview of discrete-time and continuous-time results in finite
and infinite-dimensions. We comment on dissipativity-based approaches and
infinite-horizon results, which enable the exploitation of turnpike properties
for the numerical solution of problems with long and infinite horizons. After
drawing upon numerical examples, the chapter concludes with an outlook on
time-varying, discounted, and open problems.
