On Nonconvex Relaxation Properties of Multidimensional Control Problems

Marcus Wagner ()
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Marcus Wagner: Cottbus University of Technology

A chapter in Recent Advances in Optimization, 2006, pp 233-250 from Springer

Abstract: Summary We provide two examples concerning the relaxation properties of a model problem in multidimensional control: % % MathType!MTEF!2!1!+- % feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D % aebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY-Hhbbf9v8qqaq % Fr0xc9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qq % Q8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacq % GHRiI8daWgaaWcbaGaeuyQdCfabeaaruWrHjxyT9MBKbacfaGccaWF % sgGaaiikaiaadQeacaWG4bGaaiikaiaadshacaGGPaGaaiykaiaads % gacaWG0bGaeyOKH4QaciyAaiaac6gacaGGMbGaaiyiaaaa!4D08! \[ \smallint _\Omega f(Jx(t))dt \to \inf !$$ , Ω ⊂ ℝm, x ∈ W{sk0/1,∞} (Ω, ℝn), Jx(t) ∈ K ⊂ ℝnm a. e. where n ≤ 2, m ≤ 2, Jx(t) is the Jacobian of x, and K is a convex body. The first example justifies the use of quasiconvex functions with infinite values in the relaxation process. In the second one, we examinate the relaxation properties of a restricted quasiconvex envelope function ƒ* introduced by Dacorogna/Marcellini.

Keywords: Convex Body; Relaxation Property; Convex Envelope; Bound Lipschitz Domain; Quasiconvex Function (search for similar items in EconPapers)
Date: 2006
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Persistent link: https://EconPapers.repec.org/RePEc:spr:lnechp:978-3-540-28258-7_15

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DOI: 10.1007/3-540-28258-0_15

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