Mazko A. G.

UDC 517.925.51; 681.5.03
We establish necessary and sufficient conditions for the existence of dynamic regulators guaranteeing the prescribed estimation of the weighted damping level of bounded disturbances and the asymptotic stability of linear descriptor systems. An algorithm of construction of these regulators in the problems of robust stabilization and generalized $H_\infty$-optimization is proposed for the descriptor systems with controlled and observed outputs. The main computational procedures of the algorithm are reduced to the solution of linear matrix inequalities with additional rank restrictions. The efficiency of the algorithm is demonstrated with the help of an illustrative example of descriptor stabilization system with bounded disturbances.

We establish necessary and sufficient conditions for the validity of the upper bounds for the performance criteria of linear descriptor systems characterizing the weighted damping level of external and initial disturbances. The verification of these conditions is reduced to solving matrix equations and inequalities. The main statements are formulated with an aim of their subsequent application in the problems of robust stabilization and in the $H_{\infty}$ -optimization problems for descriptor control systems.

New necessary and sufficient conditions for the asymptotic stability and localization of the spectra of linear autonomous systems are proposed by using the matrix trace functions. The application of these conditions is reduced to the solution of two scalar inequalities for a symmetric positive-definite matrix. As a corollary, for linear control systems, we present a procedure aimed at the construction of the set of stabilizing measurable output feedbacks.

We consider the problem of localization of eigenvalues of polynomial matrices. We propose sufficient conditions for the spectrum of a regular matrix polynomial to belong to a broad class of domains bounded by algebraic curves. These conditions generalize the known method for the localization of the spectrum of polynomial matrices based on the solution of linear matrix inequalities. We also develop a method for the localization of eigenvalues of a parametric family of matrix polynomials in the form of a system of linear matrix inequalities.

We investigate generalizations of classes of monotone dynamical systems in a partially ordered Banach space. We establish algebraic conditions for the stability of equilibrium states of differential systems on the basis of linearization and application of derivatives of nonlinear operators with respect to a cone. Conditions for the positivity and absolute stability of a certain class of differential systems with delay are proposed. Several illustrative examples are given.

We present a method for the investigation of the stability and positivity of systems of linear differential equations of arbitrary order. Conditions for the invariance of classes of cones of circular and ellipsoidal types are established. We propose algebraic conditions for the exponential stability of linear positive systems based on the notion of maximal eigenpairs of a matrix polynomial.

We investigate classes of dynamical systems in a partially ordered space with properties of monotonicity type with respect to specified cones. We propose new methods for the stability analysis and comparison of solutions of differential systems using time-varying cones. To illustrate the results obtained, we present examples using typical cones in vector and matrix spaces.

We investigate properties of positive and monotone dynamical systems with respect to given cones in the phase space. Stability conditions for linear and nonlinear differential systems in a partially ordered space are formulated. Conditions for the positivity of dynamical systems with respect to the Minkowski cone are established. By using the comparison method, we solve the problem of the robust stability of a family of systems.

We investigate properties of positive and monotone differential systems with respect to a given cone in the phase space. We formulate criteria for the stability of linear positive systems in terms of monotonically invertible operators and develop methods for the comparison of systems in a partially ordered space.

We establish criteria of asymptotic stability for positive differential systems in the form of conditions of monotone invertibility of linear operators. The structure of monotone and monotonically invertible operators in the space of matrices is investigated.

We develop a general method for the localization of eigenvalues of matrix polynomials and functions based on the solution of matrix equations. For a broad class of equations, we formulate theorems that generalize the known properties of the Lyapunov equation. A new method for the representation of solutions of linear differential and difference systems is proposed.

We propose algebraic methods for the investigation of the spectrum and structure of solutions of degenerate dynamical systems. These methods are based on the construction and solution of new classes of matrix equations. We prove theorems on the inertia of solutions of the matrix equations, which generalize the well-known properties of the Lyapunov equation.

We develop a method for the localization of spectra of multiparameter matrix pencils and matrix functions, which reduces the problem to the solution of linear matrix equations and inequalities. We formulate algebraic conditions for the stability of linear systems of differential, difference, and difference-differential equations.

We develop a method for localization of the eigenvalues of a matrix polynomial. This method is related to a generalization and solution of the Lyapunov equation.

Linear equations and operators in a space of matrices are investigated. The transformations of matrix equations which allow one to find the conditions of solvability and the inertial properties of Hermite solutions are determined. New families of matrices (collectives) are used in the theory of inertia and positive invertibility of linear operators and, in particular, in the problems of localization of matrix spectra and matrix beams.