About stability of dynamic systems with unilateral constraints

A.P.Ivanov

Moscow Physical Technical Institute (SRU)

Russia

 

Dynamical system on a part of plane, restricted by a piecewise smooth curve (not necessary closed), is considered. From a mathematical point of view, the phase variables are connected with certain inequalities, which are called unilateral constraints in mechanics. The one-side character of such restrictions is motivated by specific character of the variables: in multibody systems distances between the bodies can not be negative, in biological and social systems the population is non-negative, in economics oil supply is restricted by quotas etc. If a phase trajectory comes to the boundary, it will pass along the boundary or rebound from it. To ensure such behavior, we should add some additional terms to the right-hand sides of the equations of motion.љ In multibody systems, the additional term corresponds to reactions of the unilateral constraints. These reactions are continuous functions of time during joint motion of the bodies and have an impulsive character at collisions. In biological systems, vanishing of population in an area is similar to inelastic impact. Delivery of new animals to the area is an impulsive action similar to the rebound of a body from barrier. In given paper, the vector field on the boundary is defined by analogy with mechanical systems subject to an ideal unilateral constraint (impact is inelastic). Thus, the subject of analysis is the dynamical system with discontinuous right-hand side and continuous trajectories.љ Stability conditions are derived for equilibrium points at the boundary. These conditions have different form in the cases where the constraint is tense or relaxed (i.e. the compensating terms vanish). The rest points at corner points have more complicated classification due to the direction of vector field with respect to one-side tangents. The orbital stability of such periodic trajectories, which have common part with the boundary, is studied. In particular, it is proved that any trajectory, including intervals with non-zero reaction, is asymptotically stable. The results areљ applied to the analysis ofљ rest points and periodic trajectories of the linear dynamical system with the unilateral constraint and arbitrary coefficients. An exhaustive list of stable rest points and closed trajectories is presented for this example.