# 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.

## How to cite

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Sławomir Plaskacz, and Magdalena Wiśniewska. "A new method of proof of Filippov’s theorem based on the viability theorem." Open Mathematics 10.6 (2012): 1940-1943. <http://eudml.org/doc/269546>.

abstract = {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.},
author = {Sławomir Plaskacz, Magdalena Wiśniewska},
journal = {Open Mathematics},
keywords = {Lipschitz dependence on initial conditions; Viability for tubes; Contingent derivative; viability for tubes; contingent derivative},
language = {eng},
number = {6},
pages = {1940-1943},
title = {A new method of proof of Filippov’s theorem based on the viability theorem},
url = {http://eudml.org/doc/269546},
volume = {10},
year = {2012},
}

TY - JOUR
AU - Magdalena Wiśniewska
TI - A new method of proof of Filippov’s theorem based on the viability theorem
JO - Open Mathematics
PY - 2012
VL - 10
IS - 6
SP - 1940
EP - 1943
AB - 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.
LA - eng
KW - Lipschitz dependence on initial conditions; Viability for tubes; Contingent derivative; viability for tubes; contingent derivative
UR - http://eudml.org/doc/269546
ER -

## References

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1. [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. [2] Aubin J.-P., Frankowska H., Set-Valued Analysis, Systems Control Found. Appl., 2, Birkhäuser, Boston, 1990 Zbl0713.49021
3. [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. [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] Zbl0836.34016

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