APPROXIMATION AND RELAXATION OF MECHANICAL
SYSTEMS WITH DISCONTINUOUS VELOCITIES
Elena Goncharova, Maxim Staritsyn
We study mechanical systems controlled by “shock impacts”, i.e., external signals of, possibly, negligible duration and very high intensity. From a physical point of view, the signals are due to fast vibrations of segments of a rigid body, and impactive blocking of a part of its degrees of freedom. A mathematical idealization of such phenomena leads to systems with discontinuous velocities described by distributional (measure differential) equations with square and affine impulses, at that blocking of degrees of freedom formally results in a “complementarity” constraint relating states with the affine impulsive control. We raise two closely connected issues: first, we provide a correct approximation of the prototypical impulsive system by ordinary control processes; second, seeing that the trajectory tube of the system occurs to lose the property of compactness, we design its constructive relaxation (a compactification). The final goal is to discover the limit behavior of the system driven by the two types of impulsive controls.