A new method of proof of Filippov’s theorem based on the viability theorem
Sławomir Plaskacz ; Magdalena Wiśniewska
Open Mathematics, Tome 10 (2012), p. 1940-1943 / Harvested from The Polish Digital Mathematics Library
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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 x(t)-y(t)⩽r(t)=x0-y(t0)e∫t0tl(s)ds+∫t0tγ(s)e∫stl(τ)dτds, 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.

Publié le : 2012-01-01
EUDML-ID : urn:eudml:doc:269546 
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     author = {S\l awomir Plaskacz and Magdalena Wi\'sniewska},
     title = {A new method of proof of Filippov's theorem based on the viability theorem},
     journal = {Open Mathematics},
     volume = {10},
     year = {2012},
     pages = {1940-1943},
     zbl = {1268.34046},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0100-0}
}

Sławomir Plaskacz; Magdalena Wiśniewska. A new method of proof of Filippov’s theorem based on the viability theorem. Open Mathematics, Tome 10 (2012) pp. 1940-1943. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0100-0/

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