# Extremal selections of multifunctions generating a continuous flow

Alberto Bressan; Graziano Crasta

Annales Polonici Mathematici (1994)

- Volume: 60, Issue: 2, page 101-117
- ISSN: 0066-2216

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topAlberto Bressan, and Graziano Crasta. "Extremal selections of multifunctions generating a continuous flow." Annales Polonici Mathematici 60.2 (1994): 101-117. <http://eudml.org/doc/262467>.

@article{AlbertoBressan1994,

abstract = {Let $F:[0,T] × ℝ^n → 2^\{ℝ^n\}$ be a continuous multifunction with compact, not necessarily convex values. In this paper, we prove that, if F satisfies the following Lipschitz Selection Property:
(LSP) For every t,x, every y ∈ c̅o̅F(t,x) and ε > 0, there exists a Lipschitz selection ϕ of c̅o̅F, defined on a neighborhood of (t,x), with |ϕ(t,x)-y| < ε,
then there exists a measurable selection f of ext F such that, for every x₀, the Cauchy problem
ẋ(t) = f(t,x(t)), x(0) = x₀,
has a unique Carathéodory solution, depending continuously on x₀.
We remark that every Lipschitz multifunction with compact values satisfies (LSP). Another interesting class for which (LSP) holds consists of those continuous multifunctions F whose values are compact and have convex closure with nonempty interior.},

author = {Alberto Bressan, Graziano Crasta},

journal = {Annales Polonici Mathematici},

keywords = {differential inclusion; extremal selection; Cauchy problem; unique Carathéodory solution},

language = {eng},

number = {2},

pages = {101-117},

title = {Extremal selections of multifunctions generating a continuous flow},

url = {http://eudml.org/doc/262467},

volume = {60},

year = {1994},

}

TY - JOUR

AU - Alberto Bressan

AU - Graziano Crasta

TI - Extremal selections of multifunctions generating a continuous flow

JO - Annales Polonici Mathematici

PY - 1994

VL - 60

IS - 2

SP - 101

EP - 117

AB - Let $F:[0,T] × ℝ^n → 2^{ℝ^n}$ be a continuous multifunction with compact, not necessarily convex values. In this paper, we prove that, if F satisfies the following Lipschitz Selection Property:
(LSP) For every t,x, every y ∈ c̅o̅F(t,x) and ε > 0, there exists a Lipschitz selection ϕ of c̅o̅F, defined on a neighborhood of (t,x), with |ϕ(t,x)-y| < ε,
then there exists a measurable selection f of ext F such that, for every x₀, the Cauchy problem
ẋ(t) = f(t,x(t)), x(0) = x₀,
has a unique Carathéodory solution, depending continuously on x₀.
We remark that every Lipschitz multifunction with compact values satisfies (LSP). Another interesting class for which (LSP) holds consists of those continuous multifunctions F whose values are compact and have convex closure with nonempty interior.

LA - eng

KW - differential inclusion; extremal selection; Cauchy problem; unique Carathéodory solution

UR - http://eudml.org/doc/262467

ER -

## References

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- [10] A. LeDonne and M. V. Marchi, Representation of Lipschitz compact convex valued mappings, Atti Accad. Naz. Lincei Rend. 68 (1980), 278-280. Zbl0481.54011
- [11] A. F. Filippov, On certain questions in the theory of optimal control, SIAM J. Control Optim. 1 (1962), 76-84. Zbl0139.05102
- [12] A. Ornelas, Parametrization of Carathéodory multifunctions, Rend. Sem. Mat. Univ. Padova 83 (1990), 33-44. Zbl0708.28005
- [13] A. A. Tolstonogov, Extreme continuous selectors of multivalued maps and their applications, preprint, 1992.

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