Discrete-continuous systems with impulse control

Consider the evolution of a dynamical system, which state is described by the variable X(t) ∈ Rn defined in some interval [0, T]. Suppose that X(t) satisfies the differential equation 2.1$$ \mathop{X}\limits^{.} \left( t \right) = F(X(t),t), $$with a given initial condition $$ X(0) = {{x}_{0}} \in {{R}^{n}} $$ and the following intermediate conditions 2.2$$ X({{\tau }_{i}}) = X({{\tau }_{i}} - ) + \Psi (X({{\tau }_{i}} - ),{{\tau }_{i}},{{\omega }_{i}}), $$ which are given for some sequence of instants $$ \{ {{\tau }_{i}},i = 1,...,N\underline < \infty \} , $$, satisfying the inequalities 2.3$$ 0\underline < {{\tau }_{1}} < {{\tau }_{2}} < ... < {{\tau }_{i}} < ... < {{\tau }_{N}}\underline < T. $$