Constrained controllability of semilinear dynamical systems with delay in control

In the paper finite-dimensional dynamical control systems described by first order semilinear both stationary and nonstationary ordinary differential state equations with single variable point delay in control are considered. Using a generalized open mapping theorem, sufficient conditions for constrained local controllability in a given time interval are formulated and proved. These conditions require verification of constrained global controllability of the associated linear first-order stationary or nonstationary dynamical control systems. It is generally assumed, that the values of admissible controls are in a convex and closed cone with vertex at zero. Moreover, several remarks and comments on the existing results for controllability of semilinear dynamical control systems are also presented. Finally, simple numerical example which illustrates theoretical considerations is also given. It should be pointed out, that the results given in the paper extend for the case of semilinear nonstationary first-order dynamical systems constrained controllability conditions, which were previously known only for linear stationary first-order systems.