Discontinuous feedback in nonlinear control: Stabilization under persistent disturbances

We consider a nonlinear control system which, under persistently acting disturbances, can be asymptotically driven to the origin by some non-anticipating strategy with infinite memory (such a strategy determines a value of control $u(t)$ at moment $t$ using complete information on the prehistory of disturbances until moment $t$). We demonstrate that this property is equivalent to the existence of a robust stabilizing (possibly discontinuous) feedback $k(x)$.