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ACP(S,E admissible initial data Banach space characterize the solutions completely determined continuous function continuously differentiable Corollary cription defined in 9 description of HDS differen differentiable on a,b differential operator differential state space domain eventual modification everywhere exists a locally fa,b ferentiable function g function spaces function with segments functional differential equation g(t+s Hence hereditary differential systems iff there exists implies initial condition int(T+S intervals t+S lemma linear locally absolutely continuous locally integrable function Mp spaces nonlinear operators nonlinear semi-group theory null set open loop control open set operator differential equation optimal control optimal open loop original functional differential phase space problem DSSD problem of differential prop satisfies segment function semi-group of nonlinear sets of initial solu solution of 29 solution of z(t space description space equation space trajectory specified Suppose tion tQ,b Universitšt Bremen x(a+s x(t+s XQ(t z(tQ