Linear third order inclusions: the adjacent vector

Stability of linear inclusions arising in absolute stability problem for control systems with one sector nonlinearity is studied. It is shown that asymptotic stability of this inclusion is equivalent to asymptotic stability of special three dimensional autonomous system with switches at points with zero output and at points orthogonal to a special vector, which is called the adjacent vector. The Lyapunov exponent of each nonzero solution of corresponding autonomous system is proved to be equal to the Lyapunov exponent of the original linear inclusion. Thus, the result known for two dimensional inclusions is generalized to inclusions of dimension three.