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Please use this identifier to cite or link to this item:
http://hdl.handle.net/10174/7254
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Title: | A pointwise constrained version of the Liapunov convexity theorem for single integrals |
Authors: | Ornelas, António Carlota, Clara Chá, Sílvia |
Keywords: | Liapunov convexity theorem for single integrals Nonconvex differential inclusions Pointwise constraints Convexity of the range of vector measures |
Issue Date: | 16-Oct-2012 |
Citation: | Journal article - NoDEA (Nonlinear Differ. Equ. Appl.), Special issue dedicated to Arrigo Cellina on the occasion of his
70th birthday, DOI 10.1007/s00030-012-0199-5, ISSN: 1021-9722 (Print) 1420-9004 (Online), Published online: 16 October 2012, 23 pgs |
Abstract: | Given any AC solution x : [a, b] → Rn to the convex ordinary
differential inclusion
x
(t) ∈ co{v1(t), . . . , vm(t)} a.e. on [a, b], (*)
we aim at solving the associated nonconvex inclusion
x
(t) ∈ {v1(t), . . . , vm(t)} a.e., x(a) = x(a), x(b) = x(b), (**)
under an extra pointwise constraint (e.g. on the first coordinate):
x1(t) ≤ x1(t) ∀t ∈ [a, b]. (***)
While the unconstrained inclusion (**) had been solved already in 1940
by Liapunov, its constrained version, with (***), was solved in 1994 by
Amar and Cellina in the scalar n = 1 case. In this paper we add an
extra geometrical hypothesis which is necessary and sufficient, in the vector
n > 1 case, for existence of solution to the constrained inclusion (**)
and (***). We also present many examples and counterexamples to the
2 × 2 case. |
URI: | http://hdl.handle.net/10174/7254 |
ISSN: | 1021-9722 (print) 1420-9004 (online) |
Type: | article |
Appears in Collections: | CIMA - Publicações - Artigos em Revistas Internacionais Com Arbitragem Científica
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