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Extended Lie algebraic stability analysis for switched systems with continuous-time and discrete–time subsystems

Zhai G., Xu X., Lin H., Liu D.

International Journal of Applied Mathematics and Computer Science

2007

Vol. 17, no 4

447-454

We analyze stability for switched systems which are composed of both continuous-time and discrete-time subsystems. By considering a Lie algebra generated by all subsystem matrices, we show that if all subsystems are Hurwitz/Schur stable and this Lie algebra is solvable, then there is a common quadratic Lyapunov function for all subsystems and thus the switched system is exponentially stable under arbitrary switching. When not all subsystems are stable and the same Lie algebra is solvable, we show that there is a common quadratic Lyapunov-like function for all subsystems and the switched system is exponentially stable under a dwell time scheme. Two numerical examples are provided to demonstrate the result.

On Construction of the First-Order Differential Equations in the Space M(1,3) x R(u) with Non-trivial Symmetry

Fedorchuk V., Fedorchuk V.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

2003

Vol. 9

35--39

We have constructed the first-order differential equations invariant under non-splitting subgroups of the group P(1,4) and defined in the space M(1,3) x R(u). The results obtained can be used in relativistic and non-relativistic physics.

Algebras associated with posets

Neggers J., Kim H. S.

Demonstratio Mathematica

2001

Vol. 34, nr 1

13-23

In this paper we introduce a class of algebras whose bases over a field K are pogroupoids. We discuss several properties of these algebras as they relate to the structure of their associated pogroupoids and through these to the associated posets also. In particular the Jacobi form is O precisely when the pogroupoid is a semigroup, precisely when the posets is (C2 + 1)-free. Thus, it also follows that a pg-algebra KS over a field K is a Lie algebra with respect to the commutator product iff its associated posets S(<) is (C2 +1)-free. The ideals generated by commutators have some easily identifiable properties m terms of the incomparability graph of the posets associated with the pogroupoid base of the algebra. We conjecture that a fundamental theorem on the relationship between isomorphic algebras and isomorphic pogroupoids holds as well.

A note on groups of symplectomorphisms

Rybicki T.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

1998

[Z] 38

115-126

It is shown that the groups of Hamiltonian diffeomorphisms of a symplectic manifold determine uniquely the smooth and symplectic structures themselves. An analogous result is true for the Lie algebras of Hamiltoniam vector fields.