Feedback control design for systems with x-discontinuous right-hand side

The problem of the admissible feedback synthesis for nonlinear systems with discontinuous right-hand side is considered. Sufficient conditions for solvability of this problem are proved. The neighborhood of the origin is broken in a finite number of domains G1,G2,...,Gk. In each Gj a control system x˙ = fj (x,u) is given. The problem of the admissible feedback synthesis is completely studied for control systems of the form x˙ = aj (x) + γj (x,u)bj (x), where u ∈ j ⊂ R for x ∈ Gj . The controllability function method is used to construct the feedback control.