On pointwise nontriviality of the maximum principle in problems with phase constraints (Q1114918)

For a very general control problem for systems described by ordinary differential equations with constraints imposed on state, control, initial and terminal conditions some criteria under which a conjugate function \(\psi\) (t) in the maximum principle or in the local maximum principle is not vanishing at any point of an interval \([t_ 0,t_ 1]\) are given. These criteria are formulated in terms of: accordance of phase and terminal functions on an optimal trajectory at \(t_ 0\) and \(t_ 1\), controllability of an optimal trajectory with respect to phase functions [see the authors, Usp. Mat. Nauk 40, No.2(242), 175-176 (1985; Zbl 0572.49006); English translation in Russ. Math. Surv. 40, No.2, 209-210 (1985)] and the so-called \(\gamma\)-regularity of an optimal trajectory on \([t_ 0,t_ 1]\). Some examples which illustrate the importance of the given conditions are also reported. The main result of the paper is contained in the following alternative: If an optimal trajectory is controllable (\(\Gamma\)-regular) on \(\Delta \subset [t_ 0,t_ 1]\), then either \(\inf_{t\in \Delta}| \psi (T)| >0\) or \(\psi =0\) on \(\Delta\) in the maximum principle (local maximum principle).