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

Sławomir PlaskaczMagdalena Wiśniewska — 2012

Open Mathematics

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 x ( t ) - y ( t ) r ( t ) = x 0 - y ( t 0 ) e t 0 t l ( s ) d s + t 0 t γ ( s ) e s t l ( τ ) d τ d s , 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...

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