exponentially stable
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AbstractA lot of attention has been paid recently to the study of mixtures and also to the Moore–Gibson–Thompson (MGT) type equations or systems. In fact, the MGT proposition can be used to describe viscoelastic materials. In this paper, we analyze a problem involving a mixture composed by a MGT viscoelastic type material and an elastic solid. To this end, we first derive the system of equations governing the deformations of such material. We give the suitable assumptions to obtain an existence and uniqueness result. The semigroups theory of linear operators is used. The paper concludes by proving the exponential decay of solutions with the help of a characterization of the exponentially stable semigroups of contractions and introducing an extra assumption. The impossibility of location is also shown.

Abstract We investigate the interaction between a human and a miniature autonomous blimp using a wand as pointing device. The wand movement generated by the human is followed by the blimp through a tracking controller.The Vector Integration to Endpoint (VITE) model, previously applied to human-computer interface (HCI), has been applied to model the human generated wand movement when interacting with the blimp. We show that the closed-loop human-blimp dynamics are exponentially stable. Similar to HCI using computer mouse, overshoot motion of the blimp has been observed. The VITE model can be viewed as a special reset controller used by the human to generate wand movements that effectively reduce the overshoot of blimp motion. Moreover, we have observed undershoot motion of the blimp due to its inertia, which does not appear in HCI using computer mouse. The asymptotic stability of the human-blimp dynamics is beneficial towards tolerating the undershoot motion of the blimp.

This article presents a qualitative mathematical model to simulate the relationship between supplied water and plant growth. A novel aspect of the construction of this phenomenological model is the consideration of a structure of three phases: (1) The soil water availability, (2) the available water inside the plant for its growth, and (3) the plant size or amount of dry matter. From these phases and their interactions, a model based on a three-dimensional nonlinear dynamic system was proposed. The results obtained showed the existence of a single equilibrium point, global and exponentially stable. Additionally, considering the framework of the perturbation theory, this model was perturbed by incorporating irrigation to the available soil water, obtaining some stability results under different assumptions. Later through the control theory, it was demonstrated that the proposed system was controllable. Finally, a numerical simulation of the proposed model was carried out, to depict the soil water content and plant growth dynamic and its agreement with the results of the mathematical analysis. In addition, a specific calibration for field data from an experiment with wheat was considered, and these parameters were then used to test the proposed model, obtaining an error of about 6% in the soil water content estimation.

<abstract><p>Few results seem to be known about the stability with general decay rate of nonlinear neutral stochastic function differential equations driven by $ G $-Brownain motion ($ G $-NSFDEs in short). This paper focuses on the $ G $-NSFDEs, and the coefficients of these considered $ G $-NSFDEs can be allowed to be nonlinear. It is first proved the existence and uniqueness of the global solution of a $ G $-NSFDE. It is then obtained the trivial solution of the $ G $-NSFDE is mean square stable with general decay rate (including the trivial solution of the $ G $-NSFDE is mean square exponentially stable and the trivial solution of the $ G $-NSFDE is mean square polynomially stable) by $ G $-Lyapunov functions technique. In this paper, auxiliary functions are used to dominate the Lyapunov function and the diffusion operator. Finally, an example is presented to illustrate the obtained theory.</p></abstract>

<p>In this paper, we study the L^p-asymptotic stability of the one dimensional linear damped<br />wave equation with Dirichlet boundary conditions in <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math>, with <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>&#8712;</mo><mo>(</mo><mn>1</mn><mo>,</mo><mo>&#8734;</mo><mo>)</mo></math>. The damping<br />term is assumed to be linear and localized&nbsp; to an arbitrary open sub-interval of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math>. We prove that the&nbsp;<br />semi-group <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mi>p</mi></msub><mo>(</mo><mi>t</mi><msub><mo>)</mo><mrow><mi>t</mi><mo>&#8805;</mo><mn>0</mn></mrow></msub></math> associated with the previous equation is well-posed and exponentially stable.<br />The proof relies on the multiplier method and depends on whether <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>&#8805;</mo><mn>2</mn></math>&nbsp;or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>&#60;</mo><mi>p</mi><mo>&#60;</mo><mn>2</mn></math>.</p>

Robustness refers to the ability of a system to maintain its original state under a continuous disturbance conditions. The deviation argument (DA) and stochastic disturbances (SDs) are enough to disrupt a system and keep it off course. Therefore, it is of great significance to explore the interval length of the deviation function and the intensity of noise to make a system remain exponentially stable. In this paper, the robust stability of Hopfield neural network (VPHNN) models based on differential algebraic systems (DAS) is studied for the first time. By using integral inequalities, expectation inequalities, and the basic control theory method, the upper bound of the interval of the deviation function and the noise intensity are found, and the system is guaranteed to remain exponentially stable under these disturbances. It is shown that as long as the deviation and disturbance of a system are within a certain range, there will be no unstable consequences. Finally, several simulation examples are used to verify the effectiveness of the approach and are described below.

In this paper, a class of non-autonomous stochastic Nicholson’s blowflies systems with patch structure and time delays is formulated and studied. By constructing suitable Lyapunov functions and using the stochastic technical, the pth moment boundedness and almost sure growth bounds are discussed, which reveal that solutions of the system do not exceed the time value [Formula: see text] and the sample Lyapunov exponent is no more than zero. Then, the system is proved to be exponentially stable (or extinct) if the production rate is less than the mortality rate, which provides an effective reference for the population control. Moreover, taking into account the specific form of time-varying coefficients, related results for several classical stochastic Nicholson’s blowflies systems are studied, and they show the significant improvement of this paper. Finally, numerical simulations for several specific examples are carried out to illustrate our theoretical conclusions.

This paper is concerned with spreading phenomena of the classical two-species Lotka-Volterra reaction-diffusion system in the weak competition case. More precisely, some new sufficient conditions on the linear or nonlinear speed selection of the minimal wave speed of travelling wave fronts, which connect one half-positive equilibrium and one positive equilibrium, have been given via constructing types of super-sub solutions. Moreover, these conditions for the linear or nonlinear determinacy are quite different from that of the minimal wave speeds of travelling wave fronts connecting other equilibria of Lotka-Volterra competition model. In addition, based on the weighted energy method, we give the global exponential stability of such solutions with large speed $c$ . Specially, when the competition rate exerted on one species converges to zero, then for any $c>c_0$ , where $c_0$ is the critical speed, the travelling wave front with the speed $c$ is globally exponentially stable.

In solving the problem of exponential tracking and disturbance rejection, it has been long always assumed that the reference to be tracked and the disturbance to be rejected are generated by an exosystem such as a finite dimensional system with pure imaginary eigenvalues. The aim of this note is to show that this assumption can be removed. For any nonlinear control system subject to a general disturbance, it can be split into a linear exponentially-stable system and a dynamical regulator system. If the dynamical regulator system has a solution, then there exists a feedback and feedforward controller such that an output of the control system exponentially tracks a desired general reference. The result is applied to the blood glucose regulation system.