Filippov’s theorem implies that, given an absolutely continuous function y: [t 0; T] → ℝd and a set-valued map F(t, x) measurable in t and l(t)-Lipschitz in x, for any initial condition x 0, there exists a solution x(·) to the differential inclusion x′(t) ∈ F(t, x(t)) starting from x 0 at the time t 0 and satisfying the estimation $$\left|x\left(t\right)-y\left(t\right)\right|\u2a7dr\left(t\right)=\left|{x}_{0}-y\left({t}_{0}\right)\right|{e}^{{\int}_{{t}_{0}}^{t}l\left(s\right)ds}+{\int}_{{t}_{0}}^{t}\gamma \left(s\right){e}^{{\int}_{s}^{t}l\left(\tau \right)d\tau}ds,$$
where the function γ(·) is the estimation of dist(y′(t), F(t, y(t))) ≤ γ(t). Setting P(t) = x ∈ ℝn: |x −y(t)| ≤ r(t), we may formulate the conclusion in Filippov’s theorem...