Summary
We provide two examples concerning the relaxation properties of a model problem in multidimensional control: EquationSource % % 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.
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Wagner, M. (2006). On Nonconvex Relaxation Properties of Multidimensional Control Problems. In: Seeger, A. (eds) Recent Advances in Optimization. Lecture Notes in Economics and Mathematical Systems, vol 563. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-28258-0_15
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