A note on minimum time–energy control

Elementary observations on a free-end-time problem of minimizing a functional of the form u↦∫ ₀ ^{t _u} L(u(t))dt, over controls t↦u(t)∈U, U a convex, compact neighbourhood of 0∈R ^m, are provided in the context of systems ẋ=f(x,u) with prescribed initial and terminal states: x(0)=ξ and x(t _u)=0. Conditions on f and L are identified which are sufficient for existence and continuity of minimizers. For m=1, U=[−1,1] and [Formula presented] (reflecting a twofold performance objective with equal weight placed on transition time and energy expenditure), an exposition of aspects of the problem is given in the case of integrator chains.