The project is devoted to the development of the theory of optimal impulse control and the qualitative theory of differential equations with the generalized right-hand side. Problems of optimization of degenerate quadratic functional on trajectories of quasilinear systems, optimal stabilization problems in systems with impulse control and delay will be considered. Pulsed position controls, leading to pulse-sliding modes, will also be investigated. Extensions will be constructed for new types of differential equations with a generalized action on the right-hand side in particular, involving the apparatus of finitely additive measures. Problems of attainability under conditions of asymptotic compliance with constraints will also be investigated. Planned development of applications for modeling the motion of objects in a viscous medium with variable parameters. Such problems are singular and for their solutions having an impulse structure the development of a special mathematical apparatus is required.