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

Open Mathematics (2012)

• Volume: 10, Issue: 6, page 1940-1943
• ISSN: 2391-5455

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Abstract

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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 $\left|x\left(t\right)-y\left(t\right)\right|⩽r\left(t\right)=\left|{x}_{0}-y\left({t}_{0}\right)\right|{e}^{{\int }_{{t}_{0}}^{t}l\left(s\right)ds}+{\int }_{{t}_{0}}^{t}\gamma \left(s\right){e}^{{\int }_{s}^{t}l\left(\tau \right)d\tau }ds,$ 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 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.

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