Stability radii of positive higher order difference systems under fractional perturbations

In this paper we study stability radii of positive higher order difference systems under fractional perturbations and affine perturbations of the coefficient matrices. It is shown that real and complex stability radii coincide and can be computed by a simple formula. Finally, a simple example is given to illustrate the obtained results.     

Stability radii of positive linear systems under affine parameter perturbations in infinite dimensional spaces

In this paper we study the stability radii of positive linear discrete system under arbitrary affine parameter perturbations in infinite dimensional spaces. It is shown that complex, real, and positive stability radii of positive systems coincide. More importantly, estimates and computable formulas of these stability radii are also derived. The results are then illustrated by a simple example. The obtained results are extensions of the recent results in [3].     

Stability radius of linear parameter-varying systems and applications

In this paper, we present a unifying approach to the problems of computing of stability radii of positive linear systems. First, we study stability radii of linear time-invariant parameter-varying differential systems. A formula for the complex stability radius under multi perturbations is given. Then, under hypotheses of positivity of the system matrices, we prove that the complex, real and positive stability radii of the system under multi perturbations (or affine perturbations) coincide and they are computed via simple formulae. As applications, we consider problems of computing of (strong) stability radii of linear time-invariant time-delay differential systems and computing of stability radii of positive linear functional differential equations under multi perturbations and affine perturbations. We show that for a class of positive linear time-delay differential systems, the stability radii of the system under multi perturbations (or affine perturbations) are equal to the strong stability radii. Next, we prove that the stability radii of a positive linear functional differential equation under multi perturbations (or affine perturbations) are equal to those of the associated linear time-invariant parameter-varying differential system. In particular, we get back some explicit formulas for these stability radii which are given recently in [P.H.A. Ngoc, Strong stability radii of positive linear time-delay systems, Internat. J. Robust Nonlinear Control 15 (2005) 459-472; P.H.A. Ngoc, N.K. Son, Stability radii of positive linear functional differential equations under multi perturbations, SIAM J. Control Optim. 43 (2005) 2278-2295]. Finally, we give two examples to illustrate the obtained results.     

Robust stability of positive linear systems in Banach spaces

In this paper, we study the stability radii of positive linear systems with delays with respect to various classes of perturbations in infinite dimensional spaces. It is shown that the positive, real and complex stability radii coincide. Moreover, explicit formulas are derived for these stability radii and illustrated by a simple example.     

Robust stability of a class of positive quasi-polynomials in Banach spaces

In this paper, we study the stability radii of positive quasipolynomials associated with linear functional difference equations in infinite-dimensional spaces. It is shown that the positive, real and complex stability radii coincide. Moreover, explicit formulas are derived for these stability radii and illustrated by a simple example.     

Stability radii of positive linear Volterra-Stieltjes equations

We study stability radii of linear Volterra-Stieltjes equations under multi-perturbations and affine perturbations. A lower and upper bound for the complex stability radius with respect to multi-perturbations are given. Furthermore, in some special cases concerning the structure matrices, the complex stability radius can precisely be computed via the associated transfer functions. Then, the class of positive linear Volterra-Stieltjes equations is studied in detail. It is shown that for this class, complex, real and positive stability radius under multi-perturbations or multi-affine perturbations coincide and can be computed by simple formulae expressed in terms of the system matrices. As direct consequences of the obtained results, we get some results on robust stability of positive linear integro-differential equations and of positive linear functional differential equations. To the best of our knowledge, most of the results of this paper are new.     

On the strict monotonicity of spectral radii for classes of bounded positive linear operators

Strict monotonicity of the spectral radii of bounded, positive, ordered linear operators is investigated. It is well-known that under reasonable assumptions, the spectral radii of two ordered positive operators enjoy a non-strict inequality. It is also well-known that a “strict” inequality between operators does not imply strict monotonicity of the spectral radii in general—some additional structure is required. We present a number of sufficient conditions on both the cone and the operators for such a strict ordering to hold which generalise known results in the literature, and have utility in comparison arguments, ubiquitous in positive systems theory.     

Stability radii of higher order linear difference systems under multi-perturbations

We study stability radii of higher order linear difference systems under multi-perturbations. A formula for complex stability radius of higher order linear difference systems under multi-perturbations is given. Then, for the class of positive systems, we prove that the complex stability radius and real stability radius of the system under multi-perturbations coincide and they are computed via a simple formula. These are extensions of corresponding results of Hinrichsen and Son, Hinrichsen et al., Ngoc and Son, and Pappas and Hinrichsen. An example is given to illustrate the obtained results.     

Applications of Sturm sequences to bifurcation analysis of delay differential equation models

This paper formalizes a method used by several others in the analysis of biological models involving delay differential equations. In such a model, the characteristic equation about a steady state is transcendental. This paper shows that the analysis of the bifurcation due to the introduction of the delay term can be reduced to finding whether a related polynomial equation has simple positive real roots. After this result has been established, we utilize Sturm sequences to determine whether a polynomial equation has positive real roots. This work has extended the stability results found in previous papers and provides a novel theorem about stability switches for low degree characteristic equations.     

Stability of discrete-time positive evolution operators on ordered Banach spaces and applications

In this paper we discuss stability problems for a class of discrete-time evolution operators generated by linear positive operators acting on certain ordered Banach spaces. Our approach is based upon a new representation result that links a positive operator with the adjoint operator of its restriction to a Hilbert subspace formed by sequences of Hilbert–Schmidt operators. This class includes the evolution operators involved in stability and optimal control problems for linear discrete-time stochastic systems. The inclusion is strict because, following the results of Choi, we have proved that there are positive operators on spaces of linear, bounded and self-adjoint operators which have not the representation that characterize the completely positive operators. As applications, we introduce a new concept of weak-detectability for pairs of positive operators, which we use to derive sufficient conditions for the existence of global and stabilizing solutions for a class of generalized discrete-time Riccati equations. Finally, assuming weak-detectability conditions and using the method of Lyapunov equations we derive a new stability criterion for positive evolution operators.     

On α-convex operators

In this paper, the existence and uniqueness of positive fixed points for a class of convex operators is obtained by means of the properties of cone, concave operators and the monotonicity of set-valued maps. In the end, we give a simple application to certain integral equations.     

On Positive Linear Volterra-Stieltjes Differential Systems

We first introduce the notion of positive linear Volterra-Stieltjes differential systems. Then, we give some characterizations of positive systems. An explicit criterion and a Perron-Frobenius type theorem for positive linear Volterra-Stieltjes differential systems are given. Next, we offer a new criterion for uniformly asymptotic stability of positive systems. Finally, we study stability radii of positive linear Volterra-Stieltjes differential systems. It is proved that complex, real and positive stability radius of positive linear Volterra-Stieltjes differential systems under structured perturbations coincide and can be computed by an explicit formula. The obtained results in this paper include ones established recently for positive linear Volterra integro-differential systems [36] and for positive linear functional differential systems [32]-[35] as particular cases. Moreover, to the best of our knowledge, most of them are new. The first author is supported by the Alexander von Humboldt Foundation.     

ON THE WELL-POSEDNESS AND STABILITY FOR CARBON NANOTUBES AS COUPLED TWO TIMOSHENKO BEAMS WITH FRICTIONAL DAMPINGS

The objective of this paper is to study the well-posedness and stability questions for double wall carbon nanotubes modeled as linear one-dimensional coupled two Timoshenko beams in a bounded domain under frictional dampings. First, we prove the well-posedness of our system by applying the semigroups theory of linear operators. Second, we show several strong, non-exponential, exponential, polynomial and non-polynomial stability results depending on the number of frictional dampings, their position and some connections between the coefficients. In some cases, the optimality of the polynomial decay rate is also proved. The proofs of these stability results are based on a combination of the energy method and the frequency domain approach.     

Robustness of strongly and polynomially stable semigroups

In this paper we study the robustness properties of strong and polynomial stability of semigroups of operators. We show that polynomial stability of a semigroup is robust with respect to a large and easily identifiable class of perturbations to its infinitesimal generator. The presented results apply to general polynomially stable semigroups and bounded perturbations. The conditions on the perturbations generalize well-known criteria for the preservation of exponential stability of semigroups. We also show that the general results can be improved if the perturbation is of finite rank or if the semigroup is generated by a Riesz-spectral operator. The theory is applied to deriving concrete conditions for the preservation of stability of a strongly stabilized one-dimensional wave equation.     

Approximation properties of Szász‐Mirakyan‐Kantorovich type operators

In this paper, we introduce and study new type Szász‐Mirakyan‐Kantorovich operators using a technique different from classical one. This allow to analyze the mentioned operators in terms of exponential test functions instead of the usual polynomial type functions. As a first result, we prove Korovkin type approximation theorems through exponential weighted convergence. The rate of convergence of the operators is obtained for exponential weights.     

Liénard-Chipart稳定性准则的一个初等证明

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Zur Konstruktion gewisser Integraloperatoren für partielle Differentialgleichungen Teil II

Using results of Part I of this paper, we shall now develop two methods of constructing linear partial differential equations which admit Bergman operators with polynomial kernels; these equations will be obtained explicitly. Those methods will also yield general representations of solutions of such an equation which are holomorphic in some domain of complex two-space. For generating all those solutions, one needs a pair of Bergman operators. Whereas in Part I of this paper we required at least one of the two operators to have a polynomial kernel, we now impose the condition that both operators be of that kind. This entails further basic results about the existence, construction, and uniqueness of solutions.     

Linear operators and positive semidefiniteness of symmetric tensor spaces

We study symmetric tensor spaces and cones arising from polynomial optimization and physical sciences.We prove a decomposition invariance theorem for linear operators over the symmetric tensor space,which leads to several other interesting properties in symmetric tensor spaces.We then consider the positive semidefiniteness of linear operators which deduces the convexity of the Frobenius norm function of a symmetric tensor.Furthermore,we characterize the symmetric positive semidefinite tensor(SDT)cone by employing the properties of linear operators,design some face structures of its dual cone,and analyze its relationship to many other tensor cones.In particular,we show that the cone is self-dual if and only if the polynomial is quadratic,give specific characterizations of tensors that are in the primal cone but not in the dual for higher order cases,and develop a complete relationship map among the tensor cones appeared in the literature.     

On the stability of some classical operators from approximation theory

We obtain some results on Hyers–Ulam stability for some classical operators from approximation theory. For Bernstein operators we determine the Hyers–Ulam constant using a result concerning coefficient bounds of Lorentz representation for a polynomial.     

On stable operator polynomials

We attempt to generalize the classical theorem of Wall on the stability of an ordinary numerical polynomial to the operator-valued case. It is shown that such a generalization is admissible if the coefficients of the operator polynomial are uniformly positive commuting operators in Hilbert space. Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 39, No. 2, 1996, pp. 144–147.