Viability Theorems for Partial Differential Inclusions

Abstract

We extend the viability theorems to the case of elliptic and parabolic differential equations and inclusions and consider the regulation of viable solutions to distributed control problems governed by a parabolic partial differential equation of the type:

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where ω ranges over an open subset Ω ⊂ ℝⁿ and where

EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiqaaeaafa % qabeWabaaabaGaamyAaiaacMcacaaMe8UaamOzaiaacQdacaWGsbGa % ey41aqRaamyDaiabgkziUkaadkfacaaMe8UaamyAaiaadohacaaMe8 % UaamyyaiaaysW7ciGGZbGaaiyAaiaac6gacaWGNbGaamiBaiaadwga % cqGHsislcaWG2bGaamyyaiaadYgacaWG1bGaamyzaiaadsgacaaMe8 % UaamyBaiaadggacaWGWbaaeaqabeaacaWGPbGaamyAaiaacMcacaaM % e8UaamyvaiaacQdacaaMe8UaeuyQdCLaey41aqRaamOuaiabgkziUk % aadwhacaaMe8UaamyAaiaadohacaaMe8UaamyyaiaaysW7caWGZbGa % amyzaiaadshacqGHsislcaWG2bGaamyyaiaadYgacaWG1bGaamyzai % aadsgacaaMe8UaamyBaiaadggacaWGWbaabaGaamyAaiaadMgacaWG % PbGaaiykaiaaysW7cqqHuoarcaGG6aGaeyypa0ZaaabmaeaadaWcaa % qaaiabgkGi2oaaCaaaleqabaGaaGOmaaaaaOqaaiabgkGi2kabeM8a % 3naaDaaaleaacaWGPbaabaGaaGOmaaaaaaGccaaMe8UaamyAaiaado % hacaaMc8UaamiDaiaadIgacaWGLbGaaGjbVlaadYeacaWGHbGaamiC % aiaadYgacaWGHbGaam4yaiaadMgacaWGHbGaamOBaaWcbaGaamyAai % abg2da9iaaigdaaeaacaWGUbaaniabggHiLdaaaOqaaaaaaiaawUha % aaaa!A551!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$\left\{ {\begin{array}{*{20}{c}} {i)\;f:R \times u \to R\;is\;a\;\sin gle - valued\;map} \\ \begin{gathered} ii)\;U:\;\Omega \times R \to u\;is\;a\;set - valued\;map \hfill \\ iii)\;\Delta : = \sum\nolimits_{i = 1}^n {\frac{{{\partial ^2}}}{{\partial \omega _i^2}}\;is\,the\;Laplacian} \hfill \\ \end{gathered} \\ {} \end{array}} \right.$$