A Criterion for Stability of Matrices

Let A = (a ij ) be a real square matrix and 1 p ∞. We present two analogous developments. One for Schur stability and the discrete-time dynamical system x(t + 1) = Ax(t), and the other for Hurwitz stability and the continuous-time dynamical systemẋ(t) = Ax(t). Here is a description of the latter development.

2018

This paper examines the relation of the exponential dichotomy and the stability concepts for systems of linear differential equations. We are going to show some relationship between the studied concepts, more precisely we are presenting how the stability of a linear non-autonomous system is investigated with the help of the exponential dichotomy. Furthermore we are going to show how the stable and unstable subspace of an exponentially dichotomic system can be specified using the definition of the exponential dichotomy.

2014

This paper is dedicated to the study the problem of stability of some classes of gradient-like system of differential equations (both autonomous and non-autonomous cases). We present two main results. The first is a generalization of Absil & Kurdyka theorem about stability of gradient systems with analytic potential for non-gradient systems. Secondly we generalize for some classes of gradient-like non-autonomous systems the well-known Lagrange-Dirichet theorem.

1998

Chapter 2. ESTIMATES FOR MATRIX-VALUED FUNCTIONS 2.1. Norm estimates for matrix-valued functions 2.2. Estimates for absolute values 2.3. Impulse functions 2.4. Positivity conditions for impulse functions 2.5. The Lyapunov equation 2.6. Notes Chapter 3. LINEAR FINITE DIMENSIONAL SYSTEMS 3.1. General systems 3.2. The freezing method for linear systems 3.3. Systems with piecewise constant matrices 3.4. Triangular systems 3.5. Perturbations of triangular systems 3.6. Systems with the matrix Lipschitz property 3.7. Notes vi Chapter 4. LINEAR FINITE DIMENSIONAL SYSTEMS (CONTINUATION) 4.1. The multiplicative representation of solutions 4.2. The Lozinskii and Wazewski inequalities 4.3. Linear systems with majorants and minorants 4.4. Perturbations of autonomous systems 4.5. Second order systems 4.6. Scalar linear equations with real characteristic roots 71 4.7. Input-output stability of linear systems 4.8. Notes Chapter 5. NONLINEAR FINITE DIMENSIONAL SYSTEMS WITH AUTONOMOUS LINEAR PARTS 5.1. The Aizerman conjecture 5.2. The generalized Aizerman conjecture 5.3. Region of attraction and global stability R2 5.4. Stability and instability in the first approximation 5.5. Input-output stability 5.6. The input-output version of Aizerman's conjecture 5.7. Global feedback stabilization 5.8. Notes Chapter 6. NONLINEAR FINITE DIMENSIONAL SYSTEMS WITH TIME-VARIANT LINEAR PARTS 6.1. Stability of systems with general linear parts 6.2. Systems with the Lipschitz property 6.3. Boundedness of solutionsof systems with general linear parts 6.4. Boundedness of solutions of systems with the Lipschitz property 6.5. Input-output stability 6.6. Global feedback stabilization 6.7. Notes Chapter 7. ESSENTIALLY NONLINEAR. FINITE DIMENSIONAL SYSTEMS 7.1. The freezing method for nonlinear systems 7.2. Systems with differentiable right parts 7.3. The generalized Lozinskii and Wazewski inequalities 7.4. Nonlinear systems with linear majorants 7.5. Nonlinear triangular systems 7.6. Perturbations of nonlinear equations 7.7. Nonlinear systems which are "close" to triangular ones 7 .8. Nonlinear scalar equations with real characteristic roots vii 7.9. Input-output stability of essentially nonlinear systems 7.10. Notes Chapter 8. LINEAR AUTONOMOUS SYSTEMS WITH DELAY 8.1. Banach spaces and characteristic matrices 8.2. Representation of solutions 8.3. L 2-norm estimates for solutions of nonhomogeneous equations 8.4. Norm estimates for the Green function 8.5. The Lyapunov exponent of the Green function 8.6. Stability conditions 8.7. Estimates for the C-norm of solutions of nonhomogeneous equations 8.8. Perturbations of matrix-valued functions 8.9. Bounds for roots of characteristic functions of retarded systems 8.10. Systems with small delays 8.11. Stability with respect to arbitrary delay 8.12. Notes Chapter 9. LINEAR TIME-VARIANT SYSTEMS WITH DELAY 9.1. Definitions 9.2. Stability of retarded systems with bounded coefficients 9.3. The freezing method for systems with delay 9.4. Proof of Theorem 9.3.1 9.5. Integrally small perturbations 9.6. The generalized Wazewski and Lozinskii inequalities 9.7. Linear time-variant systems with small delays 9.8. Notes 183 Chapter 10. NONLINEAR SYSTEMS WITH DELAY 187 10.1. Definitions 187 10.2. L2_ estimates for solutions

The electronic journal of linear algebra ELA

For partitions of f1; : : : ; n g, the classes of P -matrices are deened, unifying the classes of the real P -matrices and of the real positive deenite matrices. Lyapunov scalar stability o f matrices is deened and characterized, and it is shown also that every real Lyapunov -scalar stable matrix is a P -matrix. Implication relations between Lyapunov scalar stability and H-stability are discussed.

Linear Algebra and its Applications, 1992

A complex matrix is said to be stable if all its eigenvalues have negative real part. Let J be a Jordan block with zeros on the diagonal and ones on the superdiagonal, and consider analytic matrix perturbations of the form A() = J + B + O(2), where is real and positive. A necessary condition on B for the stability of A() on an interval (0; 0), and a su cient condition on B for the existence of such a family A(), is (i) Re tr B 0; (ii) the sum of the elements on the rst subdiagonal of B has nonpositive real part and zero imaginary part; (iii) the sum of the elements on each of the other subdiagonals of B is zero. This result is extended to matrices with any number of nonderogatory eigenvalues on the imaginary axis, and to a stability de nition based on the spectral radius. A generalized necessary condition, though not a su cient condition, applies to arbitrary Jordan structures. The proof of our results uses two important techniques: the Puiseux-Newton diagram and the normal form of Arnold. In the nonderogatory case our main results were obtained by Levantovskii in 1980 using a di erent proof. Practical implications are discussed.

Optimization and Stability Problems in Continuum Mechanics

In this lecture~ it will be attempted to describe some recent results in the stability of general dynamical systems from the viewpoint of Liapunov theory, and to illustrate and motivate them through examples.

International Journal of Control, 1988

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Revista De Ciencias, 2015

In this paper we analyze asymptotic stability of the dynamical systemẋ = f (x) defined by a C 1 function f : D → R n where D ∈ R n + is an open set. We obtain a criterion of stability for the equilibrium solutionx ∈ D when the vector field f satisfies a) ∂ i f i (x) < 0 and b) (∂ i f i (x)) −1 ∂ i f j (x) + (∂ j f j (x)) −1 ∂ j f i (x) > 2 for i, j = 1,. .. , n.

1991 American Control Conference, 1991

In this paper we analyze asymptotic stability, instability and stabilization for the relative equilibria, i.e. equilibria modulo a group action, of natural mechanical systems. The practical applications of these results are to rotating mechanical systems where the group is the rotation group. We use a modification of the Energy-Casimir and Energy-Momentum methods for Hamiltonian systems to analyze systems with dissipation. Our work couples the modern theory of block diagonalization to the classical work of Chetaev. A central problem in control theory is that of stabilizing nonlinear systems. Very often the systems one is interested in are mechanical systems, which, in the absence of dissipation, are Hamiltonian in nature. Further, the equilibria we wish to stabilize are relative equilibria-equilibria modulo a group action, usually the rotation group. Recently two distinct but related methods have been developed to analyze the stability of the relative equilibria of Hamiltonian systems. The first, the "Energy-Casimir" method, originally goes back to Arnold [2] and was developed and formalized in Holm, Marsden, Ratiu and Weinstein [8] and Krishnaprasad