Theory of cone methods for null controllability of nonlinear control systems

Cone-valued Lyapunov functions have been used by several authors to examine the stability of nonlinear systems. However, the general problem of constructing a suitable cone given the nature of nonlinearity seems to be as difficult as that of constructing a suitable Lyapunov function. The construction of a suitable cone, at least in special cases, deserves further investigation. Lakshmikantham [1] has used the theory of cones for stability of nonlinear systems. A more unified theory of cones can be found in the book by Krasnosel'skii [2]. The objective of this paper is to use methods of the theory of cones to establish a sufficient condition of null controllability of
\dot{x}(t)= A(t)+ g(t,x,u)
.