The maximum principle in the theory of optimal processes of control

Study of an Optimum Non-linear Control System†

“…A corollary of the Pontryagin theory in some cases when the performance criterion is an improper integral is that along an optimum trajectory H is identically zero (Boltyanski et al 1960). Let us confirm this result for the example in hand.…”

Tent Method in Optimal Control Theory

“…As Dubovitski and Miljutin have written in their papers, the main necessary criteriom in their theory (called the Euler equation in their papers and in this paper as well) was discovered by them after examining the proof of the Farkas lemma (that is well known in mathematical programming) and Boltyanski's first proof of the maximum principle [5]. So, the tent method and the Dubovitski-Miljutin method one are twins.…”

“…Finally, while preparing the complete account [10] of the article [8], this author understood that the maximum principle is not a sufficient condition, but only a necessary condition of optimality. This author gave the first proof of the maximum principle [5] in this correct statement as a necessary condition of optimality in the general, nonlinear case.…”

“…The tent method is more general and does not require the cones to be bodies. Remark 6.3 In the author's first proof of the maximum principle [5], the construction of a tent of the controllability domain n1 for the system :i: = g(x, u), u E U (cf. Example 2.2) was given.…”

“…So, this author had to use the title "Pontryagin's maximum principle" in his paper [5]. This is why today mathematics and engineering researchers know the main optimization criteriom as the Pontryagin maximum principle.…”