Class Of Linear Inequalities Research Articles

Supervisory Control based on Behavioral Constraints using a Class of Linear Inequalities

This paper studies the design of supervisory controllers with a minimum number of monitor places for safe Petri Nets using a class of specification modeled as Behavioral Constraints. A logic equivalence is established between the set of linear inequalities induced by the Behavioral Constraint and a single inequality which, using the Invariant Based Control Design method, leads to a Petri Net supervisor with a single monitor place. The approach is illustrated with the synthesis of a supervisory controller for a section of a manufacturing system.

Andreev’s Theorem on hyperbolic polyhedra

In 1970, E.M.Andreev published a classification of all three-dimensional compact hyperbolic polyhedra (other than tetrahedra) having non-obtuse dihedral angles. Given a combinatorial description of a polyhedron, C, Andreev’s Theorem provides five classes of linear inequalities, depending on C, for the dihedral angles, which are necessary and sufficient conditions for the existence of a hyperbolic polyhedron realizing C with the assigned dihedral angles. Andreev’s Theorem also shows that the resulting polyhedron is unique, up to hyperbolic isometry.

A strongly polynomial algorithm for a new class of linear inequalities1

In this paper a method for solving perfect systems of linear inequalities is presented. It is based on selecting and removing inessential constraints. This method is a strongly polynomial one for the class of systems of inequalities with a constant difference between the number of constraints and the number of variables.

PMatrices and Solutions to a Class of Linear Inequalities

The main results of the paper are on pairs of real $n \times n$ matrices with all off diagonal elements nonpositive. Necessary and sufficient conditions are established for the following two important properties, proved to be equivalent, to hold for a pair $(A,B)$ : (i) there exists a solution $\lambda \in R^n $ to the set of linear inequalities $\lambda ^t > 0$, $\lambda ^t A > 0$, $\lambda ^t B > 0$; (ii) all elements of the inverse of $( {AD_1 + BD_2 } )$ are nonnegative for every pair of diagonal nonnegative matrices $D_1 $ and $D_2 $ such that the diagonal elements of $( {D_1 + D_2 } )$ are positive. The conditions are in terms of certain matrices being in P, the class of matrices with all principal minors positive. A complete, finite procedure for constructing a solution to the linear inequalities is also given.