Sufficient Conditions For Positivity Research Articles

Stability analysis of delayed neural networks based on a relaxed delay-product-type Lyapunov functional

This paper deals with the stability analysis of delayed neural networks. A necessary and sufficient condition for positivity or negativity of the high-order polynomial over a finite interval is derived. An appropriate Lyapunov–Krasovskii functional (LKF), including a relax delay-product-type Lyapunov functional, is constructed. The necessary and sufficient condition of the polynomial inequality and a relaxed delay-product-type Lyapunov functional are employed to derive less conservative stability criteria. Finally, three commonly used numerical examples are presented to demonstrate the effectiveness and less conservativeness of the proposed method.

Finite-time stability and stabilisation of singular discrete-time linear positive systems with time-varying delay

ABSTRACTIn this paper, the problem of finite-time stability and stabilisation for positive singular discrete-time linear systems with time-varying delay is investigated. We first present novel delay-dependent sufficient conditions for positivity and finite-time stability of the unforced systems. We then apply the obtained results to solve finite-time stabilisation problem of the considered systems. The sufficient conditions for the positivity and finite-time stabilisable of such systems are formulated in terms of a standard linear programming (LP) problem. Numerical examples are provided to illustrate the effectiveness and advantages of our results.

Fundamental Limits of Communication Over State-Dependent Channels With Feedback

The fundamental limits of communication over state-dependent discrete memoryless channels with noiseless feedback are studied, under the assumption that the communicating parties are allowed to use variable-length coding schemes. Various cases are analyzed, with the employed coding schemes having either bounded or unbounded codeword lengths, and with state information revealed to the encoder and/or decoder in a strictly causal, causal, or non-causal manner. In each of these settings, necessary and sufficient conditions for positivity of the zero-error capacity are obtained and it is shown that, whenever the zero-error capacity is positive, it equals the conventional vanishing-error capacity. Moreover, it is shown that the vanishing-error capacity of state-dependent channels is not increased by the use of feedback and variable-length coding. Both these kinds of capacities of state-dependent channels with feedback are thus fully characterized.

On Finite-Time Stability of Linear Positive Differential-Algebraic Delay Equations

This brief addresses finite-time stability of linear positive differential-algebraic equations with delay. Being different from the Lyapunov function method, delay-dependent sufficient conditions for positivity and finite-time stability of the system are established in terms of standard linear programming problems. A numerical example is given to illustrate the obtained results.

A new class of rational cubic spline fractal interpolation function and its constrained aspects

This paper pertains to the area of shape preservation and sets a theoretical foundation for the applications of preserving constrained nature of a given constraining data in fractal interpolation functions (FIFs) techniques. We construct a new class of rational cubic spline FIFs (RCSFIFs) with a preassigned quadratic denominator with two shape parameters, which includes classical rational cubic interpolant [Appl. Math. Comp., 216 (2010), pp. 2036–2049] as special case and improves the sufficient conditions for positivity. Convergence analysis of RCSFIF to the original function in C1 is studied. In order to meet the needs of practical design or overcome the drawback of the tension effect in the proposed RCSFIFs, we improve our method by introducing a new tension parameter wi and construct a new class of rational cubic spline FIFs with three shape parameters. The scaling factors and shape parameters have a predictable adjusting role on the shape of curves. The elements of the rational iterated function system in each subinterval are identified befittingly so that the graph of the resulting C1-rational cubic spline FIF constrained (i) within a prescribed rectangle (ii) above a prescribed straight line (iii) between two piecewise straight lines. These parameters include, in particular, conditions on the positivity of the C1-rational cubic spline FIF. Several numerical examples are presented to ascertain the correctness and usability of developed scheme and to suggest how these schemes outperform their classical counterparts.

Positivity for non-Metzler systems and its applications to stability of time-varying delay systems

In 1950, Wazewski obtained the following necessary and sufficient condition for positivity of linear systems of ordinary differential equations: the matrix of coefficients is Metzler (i.e. off-diagonal elements in the matrix of coefficients are nonnegative). No results on the positivity of solutions to delay systems in the case where the matrix is non-Metzler were expected to be obtained. It was demonstrated that for delay systems the Wazewski condition is not a necessary one. We prove results on positivity of solutions for non-Metzler systems of delay differential equations. New explicit tests for exponential stability are obtained as applications of results on positivity for non-Metzler systems. Examples demonstrate possible applications of our theorems to stabilization. For instance, in view of our results, the implicit requirement on dominance of the main diagonal can be skipped. Our approach is based on nonoscillation of solutions and positivity of the Cauchy functions of scalar diagonal delay differential equations.

Operator means and positivity of block operators

The aim of this paper is to find some sufficient conditions for positivity of block matrices of positive operators. It is shown that for positive operators A, B, C and for every non-negative operator monotone function f on $$ (0,\infty )$$ , the block matrix $$\begin{aligned} \left( \begin{array}{cc} f(A) &{} f(C) \\ f(C) &{} f(B) \\ \end{array} \right) \end{aligned}$$ is positive if and only if $$C \le A!B$$ . In particular, if $$C \le A!B$$ then $$\begin{aligned} \left( \begin{array}{cc} A &{} C \\ C &{} B \\ \end{array} \right) , \end{aligned}$$ is positive.

Stabilization of positive descriptor fractional discrete-time linear system with two different fractional orders by decentralized controller

Abstract Positive descriptor fractional discrete-time linear systems with fractional different orders are addressed in the paper. The decomposition of the regular pencil is used to extend necessary and sufficient conditions for positivity of the descriptor fractional discrete-time linear system with different fractional orders. A method for finding the decentralized controller for the class of positive systems is proposed and its effectiveness is demonstrated on a numerical example.

On positivity of spectral shift functions

We show that every spectral shift function of an even order η2k is nonnegative outside the convex hull of the spectrum of an initial operator cvhσ(H); every spectral shift function of an odd order η2k−1 is nonnegative (respectively, nonpositive) outside cvhσ(H) whenever a perturbation is nonnegative (respectively, nonpositive). We also derive several sufficient conditions for positivity of η2k and η2k−1 on the whole real line.

Positive tensor products of maps and n-tensor-stable positive qubit maps

We analyze positivity of a tensor product of two linear qubit maps, . Positivity of maps and is a necessary but not a sufficient condition for positivity of . We find a non-trivial sufficient condition for positivity of the tensor product map beyond the cases when both and are completely positive or completely co-positive. We find necessary and (separately) sufficient conditions for n-tensor-stable positive qubit maps, i.e. such qubit maps that is positive. Particular cases of 2- and 3-tensor-stable positive qubit maps are fully characterized, and the decomposability of 2-tensor-stable positive qubit maps is discussed. The case of non-unital maps is reduced to the case of appropriate unital maps. Finally, n-tensor-stable positive maps are used in characterization of multipartite entanglement, namely, in the entanglement depth detection.

Positivity and stability analysis for linear implicit difference delay equations

This paper deals with positivity and stability of linear implicit difference delay equations. Being different from the Lyapunov function approach commonly used in stability analysis, the method employed in this paper gives a way to solve the exponential stability of linear implicit difference equations with time-varying delay. By decomposition state-space and mathematical induction method, new necessary and sufficient conditions for positivity and stability of such systems are derived. Numerical examples are given to illustrate the proposed results.

An Improved Positivity Preserving Odd Degree-N Said-Ball Boundary Curves on Rectangular Grid Using Partial Differential Equation

This paper discusses the sufficient conditions for positivity preserving odd degree-n Said-Ball boundary curves defined on a rectangular grid. We derive a sufficient condition on boundary curves of rectangular Said-Ball patches where the lower bound ordinates are adjusted independently. To construct the boundary curves for each rectangular patch, the Said-Ball polynomial solution of fourth order PDE will be considered where its coefficients can be calculated using edge Said-Ball ordinates which fulfill the positivity preserving conditions. Graphical examples are presented using well-known test functions.

A necessary and sufficient condition for positivity of linear maps on M_4 constructed from permutation pairs

A necessary and sufficient condition for positivity of linear maps on M_4 constructed from permutation pairs

On the existence of positive solutions for systems of differential equations with a distributed delay

For the system with a distributed delay Ẋ(t)+∑k=1m∫hk(t)t[dsRk(t,s)]X(s)=0 sufficient conditions for positivity of the fundamental matrix are obtained. In the case when the fundamental matrix is positive, comparison results and sufficient positivity conditions for solutions are established.

Equilibration and asymptotic stationarity for non-Markovian master equations

We show that certain positivity-preserving non-Markovian generalizations of the Kossakowski–Lindblad master equation can exhibit equilibration to an asymptotic state which is stationary with respect to the shifted system Hamiltonian for general system–bath coupling. This is in sharp contrast to results for Markovian forms which require strong relations between these operators (e.g. commutation of isolated system Hamiltonian and coupling operator). We also expand the list of sufficient conditions for positivity of non-Markovian master equations.

Positive 2D Discrete-Time Linear Lyapunov Systems

Positive 2D Discrete-Time Linear Lyapunov SystemsTwo models of positive 2D discrete-time linear Lyapunov systems are introduced. For both the models necessary and sufficient conditions for positivity, asymptotic stability, reachability and observability are established. The discussion is illustrated with numerical examples.

Sufficient conditions for positivity of non-Markovian master equations with Hermitian generators

We use basic physical motivations to develop sufficient conditions for positive semidefiniteness of the reduced density matrix for generalized non-Markovian integrodifferential Lindblad–Kossakowski master equations with Hermitian generators. We show that it is sufficient for the memory function to be the Fourier transform of a real positive symmetric frequency density function with certain properties. These requirements are physically motivated, and are more general and more easily checked than previously stated sufficient conditions. We also explore the decoherence dynamics numerically for some simple models using the Hadamard representation of the propagator. We show that the sufficient conditions are not necessary conditions. We also show that models exist in which the long time limit is in part determined by non-Markovian effects.

Positivity constraints on the conditional variances in the family of conditional correlation GARCH models

In this article, we derive a set of necessary and sufficient conditions for positivity of the vector conditional variance equation in multivariate GARCH models with explicit modelling of conditional correlation. These models include the constant conditional correlation GARCH model of Bollerslev [1990. Review of Economics and Statistics 72, 498–505] and its extensions. Under the new conditions, it is possible to introduce negative volatility spillovers in the model. An empirical example illustrates usefulness of having such conditions in practice.

On stability of a class of positive linear functional difference equations

We first give a sufficient condition for positivity of the solution semigroup of linear functional difference equations. Then, we obtain a Perron–Frobenius theorem for positive linear functional difference equations. Next, we offer a new explicit criterion for exponential stability of a wide class of positive equations. Finally, we study stability radii of positive linear functional difference equations. It is proved that complex, real and positive stability radius of positive equations under structured perturbations (or affine perturbations) coincide and can be computed by explicit formulae.

Nonoscillation and positivity of solutions to first order state-dependent differential equations with impulses in variable moments

The state-dependent delay differential equation x ˙ ( t ) + ∑ i = 1 m p i ( t ) x ( t − ( H i x ) ( t ) ) = f ( t ) , t ∈ [ 0 , ∞ ) , x ( ξ ) = φ ( ξ ) , ξ < 0 , with state-dependent impulses is under consideration. Sufficient conditions for positivity of solutions to the Cauchy and periodic problems as well as conditions for positivity of solutions to the problem with a condition on the right end of the interval [ 0 , ω ] are given. Sufficient conditions for nonoscillation of solutions to the homogeneous equation ( f = 0 , φ = 0 ) on the half-line are formulated.