Abstract
Given any AC solution \({\overline{x} : [a,b] \rightarrow \mathbb{R}^{n}}\) to the convex ordinary differential inclusion
we aim at solving the associated nonconvex inclusion
under an extra pointwise constraint (e.g. on the first coordinate):
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 it existence of solution to the constrained inclusion (**) and (***). We also present many examples and counterexamples to the 2 × 2 case.
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Dedicated to Arrigo Cellina, on occasion of his 70th birthday. António has been maybe the second—out of a long list yet to end—earning a PhD under Arrigo’s supervision. He still recalls quite vividly the great pleasure aroused, back in 1985 and 1986, on his young and fresh mind, by Arrigo’s lectures at the great SISSA school, e.g. those on Smale’s continuous Newton method, those on Granas and Dugundji’s fixpoint theory, and those on Oxtoby’s Lebesgue measure versus Baire category. Arrigo still remains, to this day, António’s model mathematician.
The research leading to this paper was performed at: Cima-ue (Math Research Center of Universidade de Évora, Portugal) with financial support from the research project PEst-OE/MAT/UI0117/2011, FCT (Fundação para a Ciência e a Tecnologia, Portugal); and its resulting applications have been presented by A. Ornelas at the International Workshop “Nonlinear differential equations and control”, Milan September 2011.
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Carlota, C., Chá, S. & Ornelas, A. A pointwise constrained version of the Liapunov convexity theorem for single integrals. Nonlinear Differ. Equ. Appl. 20, 273–293 (2013). https://doi.org/10.1007/s00030-012-0199-5
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DOI: https://doi.org/10.1007/s00030-012-0199-5