A class of mechanical systems with some coordinates as controls. A reduction of certain optimization problems for them. Solution methods

The paper under review is connected with a previous one by Bressan [Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Mem. (9) Mat. Appl. 1 (1991), no. 6, 147–196; MR1119158 (92h:49017)] and has two parts. In the first part, a general class of controlled mechanical (Lagrangian) systems and a certain optimization problem (P) for them are considered in order to study more deeply (than in the above-cited paper) the relation between problems connected with strictly increasing solutions of the ordinary differential equations involved and the reduced versions of those problems. Under suitable regularity assumptions (weaker than those in the above paper), the problem (P) is reduced to a certain (Mayer) problem (RP). It is shown that any nondecreasing admissible process for (RP) has a unique corresponding such process for (P) which is strongly increasing. Relying on that, one can know whether or not from a given solution of (RP) one can construct a certain possibly asymptotic solution of (P). The second part of the paper is dedicated to a special version of Pontryagin’s maximum principle and its usefulness for constructing solutions for problems of type (P).