Capture in linear functional differential games of pursuit

AbstractThe problem of capture in a pursuit game which is described by a linear retarded functional differential equation is considered. The initial function belongs to the Sobolev space W2(1). The target is either a subset of W2(1) a point in W2(1), a subset of the Euclidean space En or a point of En. There is capture if the initial function can be forced to the target by the pursuer no matter what the quarry does. The concept of capture therefore formalizes the concepts of controllability under unpredictable disturbances. This is proved to be equivalent to the controllability of an associated linear retarded functional differential equation. There is nothing in (2) (6) or (7) below which restricts the control sets to be of the same dimension as the phase space. Our results can be applied in (2) for example, if the constraint sets Q′, P′ are subsets of Em and Ei respectively with q(t) = C(t) q′(t), − p(t) = B(t) p′(t), q′(t) ϵ Emp′(t) ϵ Er and B(t) is an n × r′-matrices and C(t) an n × m-matrix