Computing the Shape of the Image of a Multi-Linear Mapping is Possible but Computationally Intractable: Theorems

Authors

Raul A. Trejo
Vladik Kreinovich, The University of Texas at El PasoFollow

Publication Date

3-2001

Comments

UTEP-CS-00-16a.

Publsihed in rnational Journal of General Systems2, Vol. 31, No. 1, pp. 17-27.

Abstract

In systems without inertia (or with negligible inertia), a change in the values of control variables x1,...,xn leads to the immediate change in the state z of the system. In more precise terms, for such systems, every component zi of the state vector z=(z1,...,zd) 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?

The 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).