Comparison and ordering problems for dynamic systems set

A.G.Mazko

Institute of mathematics of NAS of Ukraine

3, Tereshchenkivs'ka Str., 01601, Kyiv, Ukraine

The well known comparison methods for dynamic systems represent essential development and generalization of Lyapunov functions method in stability theory. These methods allow to reduce the stability and state estimation problems for complicated differential and difference systems to studying similar problems for more simple systems in a partially ordered space so-called lower and upper comparison systems (see, for example, the works by V.M. Matrosov, L.Ju. Anapolsky, S.N. Vasiljev, V. Lakshmikantham, S. Leela, A.A. Martynyuk, N.S. Postnikov, E.F. Sabaev, Kozlov R.I., etc.). Here vector, matrix and operator analogs of Lyapunov functions and their derivatives along considered systems solutions are used. Construction of such functions for typical classes of dynamic systems is one of principal problems in practical application of comparison methods.

In previous works of the author, the solution of a robust stability problem, as one of corollaries of a comparison principle, is proposed for a set of differential systems in partially ordered Banach space corresponding to a certain cone interval of operators. Here both constants and time-varying normal reproducing cones are used.

In this work, the comparison problem for a finite set of differential systems is formulated in the form of a cone inequality for some operator (so-called comparison operator). We construct the conditions ensuring realization of the inequality on the basis of generalized derivation of a comparison operator and using elements of a dual cone. In the cases of sets consisting of two and three systems, known comparison theorems can be established as corollaries of the main result of the work. In the form of a general comparison problem, we formulate also the arrangement problems for a set of systems and, in particular, selection of dominating system problems in specified sense. Here we use specified structure of a comparison operator. We give an example of a set of nonlinear systems with an arrangement of their dynamics described in terms of the extreme eigenvalues of certain matrix pencils.

The comparison and arrangement problems for dynamic systems are connected with the invariant sets theory in a phase space. In the work, we propose a general technique for construction and research of invariant sets of the differential systems described in the form of cone inequalities. We generalize known algebraic positivity conditions for linear and nonlinear differential systems with respect to typical classes of cones (nonnegative vectors, Hermiteљ nonnegative definite matrices, ellipsoidal, etc.). The monotonicity conditions for considered classes of systems are similarly represented with help of the elements of corresponding dual cones.