Filippov Solutions to Systems of Ordinary Differential Equations with Delta Function Terms as Summands

Abstract

This paper is devoted to the investigation of the Cauchy problem for the system of ordinary differential equations

$$ \dot{y}(t) = f \bigl(t,y(t)\bigr) + A{\delta}^{(s)}(t), \quad y(-1) = y_0 \in {\mathbb{R}}^n, $$

(1)

with a vector containing derivatives of the delta function and a possibly discontinuous function $f:[-1,T_{0}] \times{\mathbb{R}}^{n} \rightarrow{\mathbb{R}}^{n}$, T ₀>0, and a constant matrix A on the right-hand side. In our approach, the components of δ ^(s) are replaced by derivatives of different δ-sequences and the limiting behavior of approximating solutions is examined. Filippov’s notion of solution to a differential equation with discontinuous right-hand side is used.