### Compactness and convexity of cores of targets for neutral systems

Anthony N. Eke
1989 Bulletin of the Australian Mathematical Society
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
more » ... 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 . 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.