In this paper, we consider the time decay of the solutions to some problems arising in strain gradient thermoelasticity. We restrict to the two-dimensional case, and we assume that two dissipative mechanisms are introduced, the temperature and the mass dissipation. First, we show that this problem is well-posed proving that the operator defining it generates a contractive semigroup of linear operators. Then, assuming that the function involving the coupling terms is elliptic, the exponential decay of the solutions is concluded as well as the analyticity of the solutions. Finally, we describe how to obtain the exponential stability in the case of hyperbolic dissipation.

This paper formulates and solves a new problem of global practical inverse optimal exponential path-tracking control of mobile robots driven by Lévy processes with unknown characteristics. The control design is based on a new inverse optimal control design for nonlinear systems driven by Lévy processes and ensures global practical exponential stability almost surely and in the pth moment for the path-tracking errors. Moreover, it minimizes cost function that penalizes tracking errors and control torques without having to solve a Hamilton–Jacobi–Bellman or Hamilton–Jaccobi–Isaacs equation.

We analyse the decay properties of the solution semigroup S(t) generated by the linear integrodifferential equation

where the operator A is strictly positive self-adjoint with A –1 not necessarily compact. The asymptotic stability of S(t) is investigated in terms of the dependence of the parameter γ ∈ ℝ. In particular, we show that S(t) is not exponentially stable when γ ≠ 1.

In the present article, we show that there exists a critical number that stabilizes the Reissner–Mindlin–Timoshenko system with frictional dissipation acting on rotation angles. We identify two speed characteristics v12:=K/ρ1 and v22:=D/ρ2, and we show that the system is exponentially stable if and only if

\begin{equation*} v_{1}^{2}=v_{2}^{2}. \end{equation*}

This paper analyses the pseudo almost periodicity of the impulsive neoclassical growth model. We investigate the existence, uniqueness and exponential stability of the pseudo almost periodic solution. Moreover, an example is given to illustrate the significance of the main findings.

We consider a new kind of simple repairable system consisting of a repairman with multiple delayed-vacation strategy. A common technique in reliability studies is to substitute the steady-state reliability indexes for instantaneous ones because the dynamic solution of the system is difficult or even impossible to obtain. However, this substitution is not always valid. Therefore, it is important to study the existence, uniqueness and expression for the system’s dynamic solution, and to discuss the system’s stability. The purpose of this paper is threefold: to study the uniqueness and existence of the dynamic solution, and its expression, using C0-semigroup theory; to discuss the exponential stability of the system by analysing the spectral distribution and quasi-compactness of the system operator; to derive some reliability indexes of the system from an eigenfunction point of view, which is different from the traditional Laplace transform technique, and present a profit analysis to determine the optimal vacation time in order to achieve the maximum system profit.


In this paper, the stability of a Timoshenko beam with time delaysin the boundary input is studied. The system is fixed at the leftend, and at the other end there are feedback controllers, in which time delays exist. We prove that this closed loop system iswell-posed. By the complete spectral analysis, we show that there isa sequence of eigenvectors and generalized eigenvectors of thesystem operator that forms a Riesz basis for the state Hilbert space. Hence the system satisfies the spectrum determined growth condition. Then we conclude the exponential stability of the system under certain conditions. Finally, we give some simulations to support our results.


This paper deals with the problem of discrete and distributed time-delay exponential stability for deterministic and uncertain stochastic high-order neural networks. The concept of a parameter weak coupling linear matrix inequality set (PWCLMIS) is developed. New results are derived in terms of PWCLMIS. Large mixed time delays can be obtained by using this approach. Furthermore, these results are more general than some previous existence results. Two numerical examples are given to show the merit of the approach.

In this paper we study asymptotic behaviour of distributed parameter systems governedby partial differential equations (abbreviated to PDE). We first review some recently developed resultson the stability analysis of PDE systems by Lyapunov's second method. On constructing Lyapunov functionalswe prove next an asymptotic exponential stability result for a class of symmetric hyperbolic PDEsystems. Then we apply the result to establish exponential stability of various chemical engineeringprocesses and, in particular, exponential stability of heat exchangers. Through concrete examples weshow how Lyapunov's second method may be extended to stability analysis of nonlinear hyperbolic PDE.Meanwhile we explain how the method is adapted to the framework of Banach spaces Lp , $1<p\leq \infty$ . 


We study the dynamic behavior and stability of two connectedRayleigh beams that are subject to, in addition to two sensors andtwo actuators applied at the joint point, one of the actuators alsospecially distributed along the beams. We show that with thedistributed control employed, there is a set of generalizedeigenfunctions of the closed-loop system, which forms a Riesz basiswith parenthesis for the state space. Then both thespectrum-determined growth condition and exponential stability areconcluded for the system. Moreover, we show that the exponentialstability is independent of the location of the joint. The range ofthe feedback gains that guarantee the system to be exponentiallystable is identified.

In this paper, we consider the boundary stabilization of asandwich beam which consists of two outer stiff layers and acompliant middle layer. Using Riesz basis approach, we show thatthere is a sequence of generalized eigenfunctions, which forms aRiesz basis in the state space. As a consequence, thespectrum-determined growth condition as well as the exponentialstability of the closed-loop system are concluded. Finally, thewell-posedness and regularity in the sense of Salamon-Weiss classas well as the exact controllability are also addressed.

In this paper we study the frequency and time domain behaviour of a heat exchanger network system. The system is governed by hyperbolic partial differential equations. Both the control operator and the observation operator are unbounded but admissible. Using the theory of symmetric hyperbolic systems, we prove exponential stability of the underlying semigroup for the heat exchanger network. Applying the recent theory of well-posed infinite-dimensional linear systems, we prove that the system is regular and derive various properties of its transfer functions, which are potentially useful for controller design. Our results remain valid for a wide class of processes governed by symmetric hyperbolic systems.

The problem of partial exact boundary controllability and exponential stability for the higher-dimensional linear system of thermoelasticity isconsidered. By introducing a velocity feedback on part of the boundary of the thermoelastic body, which is clamped along the rest of its boundary, toincrease the loss of energy, we prove that the energy in the system of thermoelasticity decays to zero exponentially. We also give a positive answer toa related open question raised by Alabau and Komornik for the Lamé system. Via Russell's “Controllability via Stabilizability" principle, we thenprove that the thermoelastic system is partially controllable with boundary controls without smallness restrictions on the coupling parameters.