In this paper we prove the convexity and the compactness of the cores of targets for neutral control systems. We make use of a weak compactness argument; but in the crucial part where we establish the boundedness of the cores of the target we make use of the notion of asymptotic direction from Convex Set Theory. Let E n be n-dimensional Euclidean space. We prove that the core of the target is convex, and is compact if, and only if, the system is Euclidean controllable. Downloaded from

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... w.cambridge.org/core. IP address: 207.241.231.83, on 28 Jul 2018 at 02:37:12, subject to the Cambridge Core terms of use, Definition 1.2. The system (1.1) is said to be Euclidean controllable if for each £ Wj' and each Si € E n there exist a U ^ 0 and an admissible control u such that the solution x(t"u) = x(t), say, of (1.1) satisfies z o (O"u) = and x{t\"u) -x t . Definition 1.3. The system (1.1) is said to be proper on [0,<i] if and only if q T X(t 1 -a)D = 0 a.e. where s € [0, t,J, and q € E n implies q -0. The system (1.1) is controllable on [0, ti) if and only if it is proper on [0, «j]. Remark. The above was shown to be true in Chukwu and Silliinan [1]. Hence, we have the following lemma LEMMA 1.1. The system (1-1) is Euclidean controllable on [0,ti] if and only if q T X(ti -s)D = 0, qe E n , s G [0, tj] implies q = 0. available at https://www.cambridge.org/core/terms. https://doi.