Characterisation of the factors of quasi-differential expressions

1992 ◽  
Vol 120 (3-4) ◽  
pp. 297-312
Author(s):  
D. Race ◽  
A. Zettl
Keyword(s):  
Differential Expression ◽  
Null Space ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
General Scalar ◽  
Linear Differential ◽  
Necessary And Sufficient

SynopsisA necessary and sufficient condition for a general, scalar, quasi-differential expression of order n to be factorisable into a product of expressions of order n − k and k, for any 0 < k < n, is given. The factors are characterised completely in terms of elements of the null space of the expression and its adjoint. The results obtained extend existing results due to both Polya and Zettl from the case of classical linear differential expressions to quasi-differential expressions.

20.—Deficiency Indices of Polynomials in Symmetric Differential Expressions, II

1975 ◽  
Vol 73 ◽  
pp. 301-306 ◽  
Author(s):  
Anton Zettl
Keyword(s):  
Hilbert Space ◽  
Differential Expression ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Real Polynomial ◽  
Deficiency Indices ◽  
Linear Differential ◽  
Image Position ◽  
Necessary And Sufficient ◽  
The Relationship

SynopsisGiven a symmetric (formally self-adjoint) ordinary linear differential expression L which is regular on the interval [0, ∞) and has C∞ coefficients, we investigate the relationship between the deficiency indices of L and those of p(L), where p(x) is any real polynomial of degree k > 1. Previously we established the following inequalities: (a) For k even, say k = 2r, N+(p(L)), N−(p(L)) ≧ r[N+(L)+N−(L)] and (b) for k odd, say k = 2r+1where N+(M), N−(M) denote the deficiency indices of the symmetric expression M (or of the minimal operator associated with M in the Hilbert space L2(0, ∞)) corresponding to the upper and lower half-planes, respectively. Here we give a necessary and sufficient condition for equality to hold in the above inequalities.

The uniform exponential stability of a class of linear differential–difference equations in a Hilbert space

1981 ◽  
Vol 89 (3-4) ◽  
pp. 201-215 ◽  
Author(s):  
Richard Datko
Keyword(s):  
Hilbert Space ◽  
Phase Space ◽  
Exponential Stability ◽  
Difference Equations ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Uniform Exponential Stability ◽  
Linear Differential ◽  
Necessary And Sufficient

SynopsisA necessary and sufficient condition is given for the uniform exponential stability of certain autonomous differential–difference equations whose phase space is a Hilbert space. It is shown that this property is preserved when the delays depend homogeneously on a nonnegative parameter.

scholarly journals Positive Stabilization of Linear Differential Algebraic Equation System

2016 ◽  
Vol 2016 ◽  
pp. 1-3 ◽  
Author(s):  
Muhafzan
Keyword(s):  
Algebraic Equation ◽  
State Feedback ◽  
Closed Loop ◽  
Equation System ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Closed Loop System ◽  
Differential Algebraic Equation ◽  
Linear Differential ◽  
Necessary And Sufficient

We study in this paper the existence of a feedback for linear differential algebraic equation system such that the closed-loop system is positive and stable. A necessary and sufficient condition for such existence has been established. This result can be used to detect the existence of a state feedback law that makes the linear differential algebraic equation system in closed loop positive and stable.

5.—The Uniform Asymptotic Stability of Certain Linear Differential-difference Equations

1976 ◽  
Vol 74 ◽  
pp. 71-79
Author(s):  
R. Datko
Keyword(s):  
Asymptotic Stability ◽  
Differential Equations ◽  
Difference Equations ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Uniform Asymptotic Stability ◽  
Linear Ordinary Differential Equations ◽  
Linear Differential ◽  
Necessary And Sufficient

SynopsisA necessary and sufficient condition is developed for determination of the uniform stability of a class of non-autonomous linear differential-difference equations. This condition is the analogue of the Liapunov criterion for linear ordinary differential equations.

scholarly journals STABILITY OF LYAPUNOV EXPONENTS, WEAK INTEGRAL SEPARATION AND NONUNIFORM DICHOTOMY SPECTRUM

2021 ◽  
Author(s):  
Hailong Zhu ◽  
Zhaoxiang LI
Keyword(s):  
Lyapunov Exponents ◽  
Exponential Dichotomy ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Dichotomy Spectrum ◽  
Integral Separation ◽  
Linear Differential ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Nonuniform Exponential Dichotomy

In this paper, a necessary and sufficient condition for the stability of Lyapunov exponents of linear differential system is proved in the sense that the equations satisfy the weaker form of integral separation instead of its classical one. Furthermore, the existence of full nonuniform exponential dichotomy spectrum under the condition of weak integral separateness is also presented.

scholarly journals On the Necessary and Sufficient Condition for a Set of Matrices to Commute and Some Further Linked Results

2009 ◽  
Vol 2009 ◽  
pp. 1-24 ◽  
Author(s):  
M. De La Sen
Keyword(s):  
Null Space ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Complex Matrices ◽  
Kronecker Sum ◽  
Structured Matrix ◽  
Commuting Matrices ◽  
The Matrix ◽  
Necessary And Sufficient

This paper investigates the necessary and sufficient condition for a set of (real or complex) matrices to commute. It is proved that the commutator[A,B]=0for two matricesAandBif and only if a vectorv(B)defined uniquely from the matrixBis in the null space of a well-structured matrix defined as the Kronecker sumA⊕(−A∗), which is always rank defective. This result is extendable directly to any countable set of commuting matrices. Complementary results are derived concerning the commutators of certain matrices with functions of matricesf(A)which extend the well-known sufficiency-type commuting result[A,f(A)]=0.

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