New sufficient conditions of conditional stability of the trivial solution of a system of quadratic ordinary differential equations are obtained. For the homogeneous system the domain of unlocal stability is also found (it is a cone). This domain is then used for design of piecewise linear (discontinuous) control laws on output for a bilinear control system of any order. Thus, a system closed by such discontinuous feedback becomes globally stable in the Lyapunov sense on the whole state space. Examples are given.