Phase Portrait of CDS

Abstract

At the present time, it is apparently difficult to give a general and exhaustive definition of the phase portait of CDSs considered here. At the present stage of development of the theory, it can be adequately done for a second-order CDS defined on the plane or on a two- dimensional manifold (Sec. 34). In the general case, by the phase portrait of a CDS we mean an aggregate of geomatric and analytic aids for representing the CDS that give as far as possible a complete, clear and vivid description of admissible trajectories, domains of reachability and controllability as well as other characteristic features of the given CDS. The aids and notions of a geometrical nature will be referred to as elements of the phase portrait of the CDS. It will be useful to include among these aids a number of notions introduced in the present book such as the Hamiltonian and the Lagrangian of the CDS, types of cones of CDS, trajectory and integral funnels and their hatched surfaces, separating surfaces, admissible surfaces, singular surfaces of reverse hatching, invariant surfaces etc. These notions serve as examples of elements of the phase portrait of a CDS.