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This paper studies the problem of event-triggered impulsive control for discrete-time systems. A novel periodic event-triggering scheme with two tunable parameters is presented to determine the moments of updating impulsive control signals which are called event times. Sufficient conditions are established to guarantee asymptotic stability of the resulting impulsive systems. It is worth mentioning that the event times are different from the impulse times, that is, the control signals are updated at each event time but the actuator performs the impulsive control tasks at a later time due to time delays. The effectiveness of our theoretical result with the proposed scheme is illustrated by three examples. [ABSTRACT FROM AUTHOR]

We consider a scalar linear mixed differential equation with several terms, both delayed and advanced arguments and a bounded right-hand side. Assuming that the deviations of the argument are bounded, we present sufficient conditions when there exists a unique bounded solution on the positive half-line. Explicit tests for a bounded solution of a homogeneous equation to decay exponentially are obtained. Existence of exponentially decaying solutions for this class of differential equations is studied for the first time, and we illustrate sharpness of the results with examples. We show that the standard approach when convergence of all solutions is stated does not work for mixed equations; in addition to an exponentially decaying, there may be a growing solution. All the coefficients and the mixed arguments are assumed to be Lebesgue measurable functions, not necessarily continuous. Though the equation is linear, some properties, as well as the methods applied, are more typical for nonlinear models, for example, fixed-point theorems are used in the proofs. [ABSTRACT FROM AUTHOR]

Pulse stabilization of cycles with Prediction-Based Control including noise and stochastic stabilization of maps with multiple equilibrium points is analyzed for continuous but, generally, non-smooth maps. Sufficient conditions of global stabilization are obtained. Introduction of noise can relax restrictions on the control intensity. We estimate how the control can be decreased with noise and verify it numerically. [ABSTRACT FROM AUTHOR]

We consider a system of s nonlinear differential equations with a distributed delay and obtain global asymptotic stability conditions, which are independent of delays. The ideas of the proofs are based on the notion of a strong attractor of a vector difference equation associated with a nonlinear vector differential equation. The results are applied to Hopfield neural networks and to compartment-type models of population dynamics with Nicholson's blowflies growth law. [ABSTRACT FROM AUTHOR]

A nonlinear cyclic system with delay and the overall negative feedback is considered. The characteristic equation of the linearized system is studied in detail. Sufficient conditions for the oscillation of all solutions and for the existence of monotone solutions are derived in terms of roots of the characteristic equation. [ABSTRACT FROM AUTHOR]

We present a review of known stability tests and new explicit exponential stability conditions for the linear scalar neutral equation with two delays x ˙ (t) − a (t) x ˙ (g (t)) + b (t) x (h (t)) = 0 , where | a (t) | < 1 , b (t) ⩾ 0 , h (t) ⩽ t , g (t) ⩽ t , and for its generalizations, including equations with more than two delays, integro-differential equations and equations with a distributed delay. [ABSTRACT FROM AUTHOR]

We obtain new explicit exponential stability conditions for linear scalar equations with positive and negative delayed terms ... and its modifications, and apply them to investigate local stability of Mackey-Glass type models ... and ... . [ABSTRACT FROM AUTHOR]

This paper considers a stabilizing stochastic control which can be applied to a variety of unstable and even chaotic maps. Compared to previous methods introducing control by noise, we relax assumptions on the class of maps, as well as consider a wider range of parameters for the same maps. This approach allows to stabilize unstable and chaotic maps by noise. The interplay between the map properties and the allowed neighbourhood where a solution can start to be stabilized is explored: as instability of the original map increases, the interval of allowed initial conditions narrows. A directed stochastic control aiming at getting to the target neighbourhood almost sure is combined with a controlling noise. Simulations illustrate that for a variety of problems, an appropriate bounded noise can stabilize an unstable positive equilibrium, without a limitation on the initial value. [ABSTRACT FROM AUTHOR]

In contrast with unstructured models, structured discrete population models have been able to fit and predict chaotic experimental data. However, most of the chaos control techniques in the literature have been designed and analyzed in a one-dimensional setting. Here, by introducing target-oriented control for discrete dynamical systems, we prove the possibility to stabilize a chosen state for a wide range of structured population models. The results are illustrated with introducing a control in the celebrated LPA model describing a flour beetle dynamics. Moreover, we show that the new control allows to stabilize periodic solutions for higher-order difference equations, such as the delayed Ricker model, for which previous target-oriented methods were not designed. [ABSTRACT FROM AUTHOR]

Given a deterministic difference equation xn+1=ƒ(xn) with a continuous ƒ increasing on [0,b], ƒ(0)≥0, we would like to stabilize any point x∗ ∈(ƒ(0),ƒ(b)), by introducing the proportional feedback (PF) control. We assume that PF control contains either a multiplicative xn+1=ƒ((ν+ℓχn+1)xn) or an additive noise xn+1=ƒ(λxn)+ℓχn+1. We study conditions under which the solution eventually enters some interval, treated as a stochastic (blurred) equilibrium. In addition, we prove that, for each ε>0, when the noise level ℓ is sufficiently small, all solutions eventually belong to the interval (x∗-ε,x∗+ε). [ABSTRACT FROM AUTHOR]

Image inpainting methods recover true images from partial noisy observations. Natural images usually have two layers consisting of cartoons and textures. Methods using simultaneous cartoon and texture inpainting are popular in the literature by using two combined tight frames: one (often built from wavelets, curvelets or shearlets) provides sparse representations for cartoons and the other (often built from discrete cosine transforms) offers sparse approximation for textures. Inspired by the recent development on directional tensor product complex tight framelets ($\text{TP}\text{-}\mathbb{C}\text{TF}$s) and their impressive performance for the image denoising problem, we propose an iterative thresholding algorithm using tight frames derived from $\text{TP}\text{-}\mathbb{C}\text{TF}$s for the image inpainting problem. The tight frame $\text{TP}\text{-}\mathbb{C}\text{TF}_{6}$ contains two classes of framelets; one is good for cartoons and the other is good for textures. Therefore, it can handle both the cartoons and the textures well. For the image inpainting problem with additive zero-mean independent and identically distributed Gaussian noise, our proposed algorithm does not require us to tune parameters manually for reasonably good performance. Experimental results show that our proposed algorithm performs comparatively better than several well-known frame systems for the image inpainting problem. [ABSTRACT FROM AUTHOR]

Various types of stabilizing controls lead to a deterministic difference equation with the following property: once the initial value is positive, the solution tends to the unique positive equilibrium. Introducing additive perturbations can change this picture: we give examples of difference equations experiencing additive perturbations which have solutions staying around zero rather than tending to the unique positive equilibrium. When perturbations are stochastic with a bounded support, we give an upper estimate for the probability that the solution can stay around zero. Applying extra conditions on the behaviour of the map functionfat zero or on the amplitudes of stochastic perturbations, we prove that the solution tends to the unique positive equilibrium almost surely. In particular, this holds either for all amplitudes when the right derivative of the mapfat zero exceeds one or, independently of the behaviour offat zero, when the amplitudes are not square summable. [ABSTRACT FROM PUBLISHER]

Sufficient oscillation conditions involving lim sup and lim inf for first-order differential equations with several non-monotone deviating arguments and nonnegative coefficients are obtained. The results are based on the iterative application of the Grönwall inequality. Examples illustrating the significance of the results are also given. [ABSTRACT FROM AUTHOR]

In this paper, we propose a frame-based iterative algorithm to restore images which are corrupted by mixed Gaussian and impulse noise, under the assumption that the image region corrupted by impulse noise is unknown. The removal of mixed Gaussian and impulse noise by our proposed algorithm is split into two subproblems which are solved alternatively and iteratively. With an initial guessed region of location for impulse noise, the first subproblem is to inpaint a corrupted image by solving a frame-based convex minimization scheme using the balanced approach, where sparse and redundant directional representations play a key role. Motivated by our recent work on frame-based image denoising and image inpainting, we shall employ the tight frame generated from the directional tensor product complex tight framelets in our balanced approach to remove the mixed Gaussian and impulse noise. Such tensor product complex tight framelets provide sparse directional representations for natural images and can capture the cartoon and texture parts of images very well. The second subproblem is to estimate the image region of locations where the pixels are corrupted by impulse noise. We solve the second subproblem using an $$l_0$$ -minimization scheme. We consider both salt-and-pepper impulse noise and random-valued impulse noise. Numerical experiments show that our proposed algorithm compares favorably or often outperforms three well-known recent image-restoration methods employed for removing the mixed Gaussian and impulse noise. [ABSTRACT FROM AUTHOR]

For the nonlinear second order Lienard-type equations with time-varying delays global asymptotic stability conditions are obtained. The results are based on the new sufficient stability conditions for relevant linear equations and are applied to derive explicit stability conditions for the nonlinear Kaldor-Kalecki business cycle model. We also explore multistability of the sunflower non-autonomous equation and its modifications. [ABSTRACT FROM AUTHOR]

For a wide class of second-order nonlinear non-autonomous models, we illustrate that combining proportional state control with the feedback that is proportional to the derivative of the chaotic signal allows to stabilise unstable motions of the system. The delays are variable, which leads to more flexible controls permitting delay perturbations; only delay bounds are significant for stabilisation by a delayed control. The results are applied to the sunflower equation which has an infinite number of equilibrium points. [ABSTRACT FROM PUBLISHER]

We consider population dynamics models dN/dt = f(N(tτ)) − d(N(t)) with an increasing fecundity function f and any mortality function d which can be quadratic, as in the logistic equation, or have a different form provided that the equation has at most one positive equilibrium. Here the delay in the production term can be distributed and unbounded. It is demonstrated that the positive equilibrium is globally attractive if it exists, otherwise all positive solutions tend to zero. Moreover, we demonstrate that solutions of the equation are intrinsically non-oscillatory: once the initial function is less/greater than the equilibrium K > 0, so is the solution for any positive time value. The assumptions on f, d and the delay are rather nonrestrictive, and several examples demonstrate that none of them can be omitted. [ABSTRACT FROM AUTHOR]

A map which experiences a period doubling route to chaos, under a stochastic perturbation with a positive mean, can have a stable blurred two-cycle for large enough values of the parameter. The limit dynamics of this cycle is described, and it is demonstrated that most well-known population dynamics models (e.g. Ricker, truncated logistic, Hassel and May, Bellows maps) have this stable blurred two-cycle. For a general type of maps, in addition, there may be a blurred stable area near the equilibrium. [ABSTRACT FROM AUTHOR]

The relation between the following two properties of linear difference equations with infinite delay is investigated: (i) exponential stability and (ii) -input -state stability (Perron's property) which means that solutions of the non-homogeneous equation with zero initial data belong to when non-homogeneous terms are in . It is assumed that at each moment the prehistory (the sequence of preceding states) belongs to some weighted -space with an exponentially fading weight (the phase space). Our main result states that whenever and a certain boundedness condition on coefficients is fulfilled. This condition is sharp and ensures that, to some extent, exponential and -input -state stabilities do not depend on the choice of a phase space and parameters p and q. -input -state stability corresponds to uniform stability. We provide some evidence that similar criteria should not be expected for non-fading memory spaces. [ABSTRACT FROM AUTHOR]

We investigate the stabilization of periodic orbits of one-dimensional discrete maps by using a proportional feedback method applied in the form of pulses. We determine a range of the parameter μ values representing the strength of the feedback for which all positive solutions of the controlled equation converge to a periodic orbit. An important feature of our approach is that the required assumptions for which our results hold are met by a general class of maps, improving in this way some previous results. We discuss the applicability of our scheme to some models of population dynamics, and give numerical simulations to illustrate our analytical results. [ABSTRACT FROM AUTHOR]

We discuss the famous constants of $ 1/e$ in necessary and/or sufficient oscillation and stability conditions for delay differential equations with one or more delays: $\displaystyle x^{\prime }(t)= - \sum_{k=1}^ma_k(t) x(t-h_k(t)), $ including equations with oscillatory coefficients. Some counterexamples (which refer to necessary oscillation and stability conditions) are presented and open problems are stated. [ABSTRACT FROM AUTHOR]

The Bohl-Perron result on exponential dichotomy for a linear difference equation[image omitted] states (under some natural conditions) that if all solutions of the non-homogeneous equation with a bounded right hand side are bounded, then the relevant homogeneous equation is exponentially stable. According to its corollary, if a given equation is close to an exponentially stable comparison equation (the norm of some operator is less than one), then the considered equation is exponentially stable. For a difference equation with several variable delays and coefficients we obtain new exponential stability tests using the above results, representation of solutions and comparison equations with a positive fundamental function. [ABSTRACT FROM AUTHOR]

Existence of nonoscillatory solutions for the second-order dynamic equation (A0xΔ)Δ(t) + ∑i∈[1,n]ℕ Ai(t)x(αi(t)) = 0 for t ∈ [t0,∞)핋 is investigated in this paper. The results involve nonoscillation criteria in terms of relevant dynamic and generalized characteristic inequalities, comparison theorems, and explicit nonoscillation and oscillation conditions. This allows to obtain most known nonoscillation results for second-order delay differential equations in the case A0(t) ≡ 1 for t ∈ [t0,∞)ℝ and for second-order nondelay difference equations (αi(t) = t + 1 for t ∈ [t0,∞)ℕ). Moreover, the general results imply new nonoscillation tests for delay differential equations with arbitrary A0 and for second-order delay difference equations. Known nonoscillation results for quantum scales can also be deduced. [ABSTRACT FROM AUTHOR]

New explicit conditions of asymptotic and exponential stability are obtained for the scalar nonautonomous linear delay differential equation ẋ(t)+ ∑ Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed.ak(t)x(hk(t) = 0 with measurable delays and coefficients. These results are compared to known stability tests. [ABSTRACT FROM AUTHOR]

We obtain new explicit exponential stability results for difference equations with several variable delays and variable coefficients. Several known results, such as Clark's asymptotic stability criterion, are generalized and extended to a new class of equations. [ABSTRACT FROM AUTHOR]

For the delay differential equations and explicit exponential stability conditions are obtained. [ABSTRACT FROM AUTHOR]

We study existence of nonoscillatory solutions and positiveness of the fundamental function for a scalar linear difference equation with several delays:[image omitted] where al can be either positive or negative. Coefficients of arbitrary signs are also considered. [ABSTRACT FROM AUTHOR]