Carja, Ovidiu; Postolache, Victor Necessary and sufficient conditions for local invariance for semilinear differential inclusions. (English) Zbl 1257.34045 Set-Valued Var. Anal. 19, No. 4, 537-554 (2011). The authors study local invariance of a set \(K\subseteq X\), where \(X\) is a separable Banach space, for the autonomous semilinear differential inclusion \[ {u^{\prime }}(t) \in Au(t)+F(u(t)), \quad {u(0) = \xi }.\tag{1} \] The assumptions generally include that \(A:D(A)\subseteq X\rightarrow X\) is the infinitesimal generator of a \(C_{0}\)-semigroup \(\left\{ S(t):X\rightarrow X\mid t\geq 0\right\} \), \(F:X\rightarrow 2^{X}\) is closed and bounded-valued, and satisfies a Lipschitz condition, and \(K\) is locally closed. The authors make use of a tangency concept in which a function \(f\in L_{\text{loc}}^{1}\left({R}_{+},X\right) \) is said to be \({A}\)-tangent to the set \({K}\) at the point \({\xi \in K}\) if \[ \lim \inf_{h\downarrow 0}\frac{1}{h}\operatorname{dist}\left( S(h)\xi +\int_{0}^{h}S\left( h-s\right) f(s)ds,K\right) =0. \] Both necessary and sufficient conditions are specified. The final section gives an application to an optimal control problem. Reviewer: Daniel C. Biles (Nashville) Cited in 4 Documents MSC: 34G25 Evolution inclusions 49K21 Optimality conditions for problems involving relations other than differential equations 49K27 Optimality conditions for problems in abstract spaces Keywords:invariance; differential inclusion; Lipschitz; tangency; optimal control; semilinear PDFBibTeX XMLCite \textit{O. Carja} and \textit{V. Postolache}, Set-Valued Var. Anal. 19, No. 4, 537--554 (2011; Zbl 1257.34045) Full Text: DOI References: [1] Aubin, J.-P.: Viability theory. Birkhäuser Boston, Inc., Boston, MA (1991) · Zbl 0755.93003 [2] Carja, O.: The minimum time function for semi-linear evolutions. (2011, to be published) · Zbl 1246.49025 [3] Carja, O., Necula, M., Vrabie, I.I.: Viability, Invariance and Applications. Elsevier Science B.V., Amsterdam (2007) [4] Carja, O., Necula, M., Vrabie, I.I.: Necessary and sufficient conditions for viability for semilinear differential inclusions. Trans. Am. Math. Soc. 361, 343–390 (2009) · Zbl 1172.34040 · doi:10.1090/S0002-9947-08-04668-0 [5] Castaing, C., Valadier, M.: Convex Analysis and Measurable Multifunctions. Springer-Verlag, Berlin-New York (1977) · Zbl 0346.46038 [6] Clarke, F.H., Ledyaev, Yu. S., Radulescu, M.L.: Approximate invariance and differential inclusions in Hilbert spaces. J. Dyn. Control Syst. 3(4), 493–518 (1997) · Zbl 0951.49007 [7] Donchev, T.: Functional-differential inclusion with monotone right-hand side. Nonlinear Anal. 16(6), 533–542 (1991) · Zbl 0722.34010 · doi:10.1016/0362-546X(91)90026-W [8] Donchev, T., Rios, V., Wolenski, P.: Strong invariance and one-sided Lipschitz multifunctions. Nonlinear Anal. 60(5), 849–862 (2005) · Zbl 1068.34010 · doi:10.1016/j.na.2004.09.050 [9] Frankowska, H.: A priori estimates for operational differential inclusions. J. Differ. Equ. 84(1), 100–128 (1990) · Zbl 0705.34016 · doi:10.1016/0022-0396(90)90129-D [10] Frankowska, H., Plaskacz, S., Rzeuchowski, T.: Measurable viability theorems and the Hamilton-Jacobi-Bellman equation. J. Differ. Equ. 116(2), 265–305 (1995) · Zbl 0836.34016 · doi:10.1006/jdeq.1995.1036 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.