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

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>.

@article{SławomirPlaskacz2012,
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 - Sławomir Plaskacz
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 -