Synthesis of impulse and fast controls under uncertainty

The synthesis of impulse controls in problems without uncertainty was addressed. Since impulses are ideal elements, they can be implemented in practice by applying approximations based on bounded functions. Such approximations of ideal impulse controls are known as fast controls. Impulse and fast controls have been designed for a linear system with an unknown bounded disturbance. The dynamic programming method has been used for the purpose, which is extended below to impulse controls. It is proved that the corresponding value function, which is used to find the desired control, is the solution of a variational inequality of the Hamilton-Jacobi-Bellman (HJB) type. As a result, an impulse control strategy is obtained under uncertainty and proposes a method for designing fast controls as based on impulse control functions.