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Numerical checking method for positive invariance of polyhedral sets for linear dynamical systems

Yang H., Hu Y.

Bulletin of the Polish Academy of Sciences. Technical Sciences

2020

Vol. 68, nr 3

593--599

Positively invariant sets play an important role in the theory and applications of dynamical systems. The stability in Lyapunov sense of equilibrium x = 0 is equivalent to the existence of the ellipsoidal positively invariant sets. The constraints on the state and control vectors of dynamical systems can be formulated as polyhedral positively invariant sets in practical engineering problems. Numerical checking method of positive invariance of polyhedral sets is addressed in this paper. The validation of the positively invariant sets can be done by solving LPs which can be easily implemented numerically. The effectiveness of the proposed checking method is illustrated by examples. Compared with the now existing algebraic methods, numerical checking method is attractive and, importantly, easy to be implemented.

Higher-order conditions for strict local Pareto minima for problems with partial order introduced by a polyhedral cone

Stasiak A.

Journal of Applied Analysis

2018

Vol. 24, nr 1

45--54

Using the definitions of μ-th order lower and upper directional derivatives of vector-valued functions, introduced in Rahmo and Studniarski (J. Math. Anal. Appl. 393 (2012), 212-221), we provide some necessary and sufficient conditions for strict local Pareto minimizers of order μ for optimization problems where the partial order is introduced by a pointed polyhedral cone with non-empty interior.