On control problems of minimum time for Lagrangian systems similar to a swing. II Application of convexity criteria to certain minimum time problems

Bressan, Aldo ; Motta, Monica

Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 5 (1994) no. 3, pp. 255-264

Résumé

This Note is the Part II of a previous Note with the same title. One refers to holonomic systems \( \Sigma = \mathcal{A} \bigcup \mathcal{U} \) with two degrees of freedom, where the part \( \mathcal{A} \) can schemetize a swing or a pair of skis and \( \mathcal{U} \) schemetizes whom uses \( \mathcal{A} \). The behaviour of \( \mathcal{U} \) is characterized by a coordinate used as a control. Frictions and air resistance are neglected. One considers on \( \Sigma \) minimum time problems and one is interested in the existence of solutions. To this aim one determines a certain structural condition \( \Gamma \) which implies a well known convexity condition (briefly WCC) just ensuring the afore-mentioned existence. These proofs are based on the results of Part I. The condition \( \Gamma \) becomes equivalent to the WCC in both the cases of the swing or of the ski having constant curvature trajectory. An other equivalent structural condition is established in a simple case regarding the ski. The WCC fails to be verified, e.g., for the simple pendulum of variable length. One observes that, also in the absence of the WCC, for certain initial and terminal data, the solution still exists.