On the existence in a problem of the calculus of variations without convexity assumptions (Q2368604)

The author considers Lie classical problem in the calculus of variations of finite stationary values of \[ \int^b_a L (x(t)),\overset{\circ}{x}(t)) \,dt. \] He shows that the problem is equivalent to the time-control problem connected by \textit{R.~V. Gamkrelidze} [Principles of optimal control theory, Plenum Press (1978; Zbl 0401.49001)] and points out that the bang-bang property in optimal control problems is not of use here. There are long detailed analyses.