@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 -
