The theory of duality in systems with aftereffect

Some problems of control and observation /1-3/ of linear dynamic systems with aftereffect, defined by differential and integral equations with deviating arguments are considered. The theory of duality for the problem of minimizing the Boltz convex functional on the trajectories of a functionally differentiable system of the neutral type with a lag in the control, state, and velocity variables is developed. New concepts of controllability are introduced into the system with aftereffects and phase constraints, as well as dual concepts of ideal observability of their conjugate system of integral equations with a lead in conditions of incomplete information. The observability concepts introduced here are connected with the restitution of the generalized final state of the system containing minimum information to enable the future motion to be calculated uniquely. The schemes and results obtained enable them to be used in differential-game problems of dynamic systems with aftereffects /4-6/.