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

• Autorzy: Sławomir Plaskacz , Magdalena Wiśniewska
• Języki publikacji: EN

Abstrakty

EN

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 $$\left| {x(t) - y(t)} \right| \leqslant r(t) = \left| {x_0 - y(t_0 )} \right|e^{\int_{t_0 }^t {l(s)ds} } + \int_{t_0 }^t \gamma (s)e^{\int_s^t {l(\tau )d\tau } } ds,$$ 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.

Słowa kluczowe

EN

Lipschitz dependence on initial conditions; Viability for tubes; Contingent derivative

Kategorie tematyczne

• 49J53: Set-valued and variational analysis
• 93C15: Systems governed by ordinary differential equations
• 34A60: Differential inclusions
• 34A12: Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2012
• Tom: 10
• Numer: 6
• Strony: 1940-1943

Daty

wydano

2012-12-01

online

2012-10-12

Twórcy

autor

Sławomir Plaskacz

• Faculty of Mathematics and Computer Sciences, Nicholas Copernicus University, Chopina 12/18, 87-100, Toruń, Poland

autor

Magdalena Wiśniewska

• Faculty of Mathematics and Computer Sciences, Nicholas Copernicus University, Chopina 12/18, 87-100, Toruń, Poland

Bibliografia

• [1] Aubin J.-P., Cellina A., Differential Inclusions, Grundlehren Math. Wiss., 264, Springer, Berlin, 1984 http://dx.doi.org/10.1007/978-3-642-69512-4[Crossref]
• [2] Aubin J.-P., Frankowska H., Set-Valued Analysis, Systems Control Found. Appl., 2, Birkhäuser, Boston, 1990
• [3] Filippov A.F., Classical solutions of differential equations with multi-valued right-hand side, SIAM J. Control, 1967, 5(4), 609–621 http://dx.doi.org/10.1137/0305040[Crossref]
• [4] Frankowska H., Plaskacz S., Rzezuchowski T., Measurable viability theorems and the Hamilton-Jacobi-Bellman equation, J. Differential Equations, 1995, 116(2), 265–305 http://dx.doi.org/10.1006/jdeq.1995.1036[Crossref]

Identyfikatory

DOI

10.2478/s11533-012-0100-0