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In this paper, we focus on a particular class of nonlinear affine control systems of the form _ = ( ) + , where the drift is a multi-affine vector field (i.e., affine in each state component), the control distribution is constant, and the control is constrained to a convex set. For such a system, we first derive necessary and sufficient conditions for the existence of a multiaffine feedback control law keeping the system in a rectangular invariant. We then derive sufficient conditions for driving all initial states in a rectangle through a desired facet in finite time. If the control constraints are polyhedral, we show that all these conditions translate to checking the feasibility of systems of linear inequalities to be satisfied by the control at the vertices of the state rectangle. This work is motivated by the need to construct discrete abstractions for continuous and hybrid systems, in which analysis and control tasks specified in terms of reachability of sets of states can be reduced to searches on finite graphs. We show the application of our results to the problem of controlling the angular velocity of an aircraft with gas jet actuators.

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In this paper the notion of autoregressive systems over an integral domain R is introduced, as a generalization of AR-systems over the rings R s] a n d R s s ;1 ]. Unlike the behavioral approach, the signal space is considered as a module M over the ring R. In this setup the problem of system equivalence is studied: when do two di erent AR-representations characterize the same behavior? This problem is solved using a ring extension of R, that explicitly depends on the choice of the signal space M. In this way the usual divisibility conditions on the system-de ning matrices can be recovered. The results apply to the class of delay-di erential systems with (in)commensurable delays. In this particular application, the ring extension of R is characterized explicitly.