On Abstract Control Problems with Non-Smooth Data

The aim of this paper is to extend and generalize the known necessary optimality conditions to non-smooth as well as higher-order setting concerning the optimization problem (which is called an abstract control problem)

% MathType!MTEF!2!1!+- % feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D % aebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY-Hhbbf9v8qqaq % Fr0xc9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qq % Q8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaada % GabaqaamaaDaaaleaaruWrHjxyT9MBKbacfaGaa8NKbmaaBaaameaa % caWFXaaabeaaliaa-HcacaWF4bGaa8hlaiaa-vhacaWFPaGaeyizIm % Qaa8hmaiaa-XcacqWIMaYscaWFSaGaa8NKbmaaBaaameaacaWFRbaa % beaaliaacIcacaWG4bGaaiilaiaadwhacaGGPaGaeyizImQaaGimai % aacYcacaWGgbGaaiikaiaadIhacaGGNaGaamyDaiaacMcacqGH9aqp % caaIWaGaaiilaiaacIcacaWG4bGaaiilaiaadwhacaGGPaGaeyicI4 % SaamiraiaadIhacaWGvbGaaiilaaqaaiaaywW7caaMf8UaaGzbVlaa % ywW7caWGnbGaamyAaiaad6gacaWGPbGaamyBaiaadMgacaWG6bGaam % yzaiaaykW7caWFsgWaaSbaaWqaaiaaicdaaeqaaSGaaiikaiaadIha % caGGSaGaamyDaiaacMcacaaMc8Uaam4DaiaadMgacaWG0bGaamiAai % aaykW7caWGYbGaamyzaiaadohacaWGWbGaamyzaiaadogacaWG0bGa % aGPaVlaadshacaWGVbaaaaGccaGL7baaaaa!875A! \[ \left\{ {_{f_1 (x,u) \leqslant 0, \ldots ,f_k (x,u) \leqslant 0,F(x'u) = 0,(x,u) \in DxU,}^{\quad \quad \quad \quad Minimize\,f_0 (x,u)\,with\,respect\,to} } \right. $$

where

D

is an open set of a Banach space

X, U

is a nonempty set, and the data fulfill a certain convexity condition which can often be verified in the context of ordinary optimal control problems.