DETERMINATION OF STABILITY BY LINEAR APPROXIMATION OF A PERIODIC SOLUTION OF A SYSTEM OF DIFFERENTIAL EQUATIONS WITH DISCONTINUOUS RIGHT-HAND SIDES†

摘要

In the works of Liapounoff (1) the problem of stability of a periodic solution zi=z˜i(t) (with periodτ) of a system of differential equationsdzidt=fi(z1,…,zn,t) fi(z1,…znt+τ)≡fi(z1,…zn,t)(0.1)was reduced in the principal (non-critical) cases to the problem of stability of the zero-solution xi=0 of the system of linear approximation‡dxidt=∑k(∂fi∂zk)z=z˜(t)xkThis reduction is applicable when the right-hand sides of equations (0.1) are analytic functions, or functions continuous in the neighbourhood of the integral curve and differentiable with respect to z1…, zn, uniformly in regard totat the points of this curve.It is the purpose of this paper to analyse the stability of a periodic solutionzi=z˜i(t)z˜i(t+τ)≡z˜i(t)of system (0.1) when the right-hand sides fiare discontinuous at some points of the integral curve zi=z˜i(t). An explanation of the meaning of the linear approximation for this‘discontinuous’case is offered, and theorems, analogous to those of Liapounoff on stability by linear approximation in