Computing the shape of the image of a multi-linear mapping is possible but computationally intractable: Theorems

Abstract

In systems without inertia (or with negligible inertia), a change in the values of control variables x (1) ,…, x ( n ) leads to the immediate change in the state z of the system. In more precise terms, for such systems, every component z i of the state vector z =( z 1 ,…, z d ) is a function of the control variables. When we know what state z we want to achieve, the natural question is: can we achieve this state, i.e. are there values of the control variables which lead to this very state? This simplest possible functional dependence is described by linear functions . For such functions, the question of whether we can achieve a given state z reduces to the solvability of the corresponding system of linear equations; this solvability can be checked by using known (and feasible) algorithms from linear algebra. Next in complexity is the case when instead of a linear dependence, we have a multi-linear dependence. In this paper, we show that for multi-linear functions, the controllability problem is, in principle, algorithmically solvable, but it is computationally hard (NP-hard).