A new method of proof of Filippov’s theorem based on the viability theorem

Abstract

Filippov’s theorem implies that, given an absolutely continuous function y: [t0; T] → ℝd and a set-valued map F(t, x) measurable in t and l(t)-Lipschitz in x, for any initial condition x0, there exists a solution x(·) to the differential inclusion x′(t) ∈ F(t, x(t)) starting from x0 at the time t0 and satisfying the estimation $$\left| {x(t) - y(t)} \right| \leqslant r(t) = \left| {x_0 - y(t_0 )} \right|e^{\int_{t_0 }^t {l(s)ds} } + \int_{t_0 }^t \gamma (s)e^{\int_s^t {l(\tau )d\tau } } ds,$$ 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 as x(t) ∈ P(t). We calculate the contingent derivative DP(t, x)(1) and verify the tangential condition F(t, x) ∩ DP(t, x)(1) ≠ ø. It allows to obtain Filippov’s theorem from a viability result for tubes.