Stability and comparison of systems in partially ordered space

Alexey G. Mazko

Institute of mathematics of NAS of Ukraine

3 Tereshchenkivska Str., 01601 Kiev-4, Ukraine

Various natural systems are positive and monotone. Positivity of a dynamic system is equivalent to positivity of some operator describing its motion with respect to a cone of phase space. Similarly, monotonicity of the system means that the motion operator is monotone. These properties of systems should be taken into consideration and be used in analysis and synthesis problems especially in stability and spectral characteristics investigation, in numerical procedures of construction of the solutions and appropriate controls etc. Stability investigation of linear autonomous positive systems is reduced to solving algebraic equations defined by operator coefficients of the systems. Well-known Lyapunov and Riccati differential equations are positive systems concerning the cone of symmetric positive semi-definite matrices.

In present paper, we study positivity, monotonicity and stability conditions for the classes of differential systems

, ,

where is a linear bounded operator in Banach space partially ordered by normal reproducing cone and is a nonlinear operator function. In robust stability problem, we assume that

, .

Positivity of the system can be utilized for estimation of its solutions. In the case of monotone evolutional operator, we have two-sided estimations for the solution in terms of solutions of auxiliary linear systems.

The main research results are the positivity conditions and algebraic criteria for asymptotic stability of linear systems formulated in terms of the monotone and monotonically invertible operators. Stability investigation of the linear positively reducible systems and time-varying systems described by functionally commutative operators is reduced to solving algebraic equations with monotonically invertible operators. The methods for robust stability analysis and analogs of comparison systems in partially ordered space are developed.

We have some examples of positive and monotone systems with respect to the cones of nonnegative vectors and positive semi-definite matrices. The matrix differential equation

,

where and , is positive with respect to the cone of positive semi-definite symmetric matrices. It can be used as a comparison system in stability problem for more complicated dynamic systems.

Well-known criteria for a mean quadratic asymptotic stability of the Ito stochastic systems are corollaries of obtained results.