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The p-th moment input-to-state stability and the p-th moment integral input-to-state stability of stochastic system with Markovian switching are investigated in this paper. To weaken the condition required by the classical Lyapunov stability theory, a novel comparison theorem is introduced, which makes infinitesimal operator of Lyapunov functions be able to take both positive and negative. A numerical example shows the effectiveness and advantages of the proposed results.

This paper mainly considers the stability of switched system composed of linear subsystems with time delay. The equation solutions under a pointed switching sequence are shown firstly. Through movement analysis of the switched system, the stability condition of switched linear scalar system are given. Then after analyzing the movements of this system, the stability condition for the switched linear scalar system with time-delay is obtained again. Meantime, by using the Comparison Theorem, the result can be extended to the multidimensional case. Finally, simulation results are illustrated to show the effectiveness of our results.

A one-dimensional optimal control problem with a state-dependent cost and a unimodular integrand is considered. It is shown that, under some standard assumptions, this problem can be solved without using the Pontryagin maximum principle, by simple methods of the classical analysis, basing on the Tchyaplygin comparison theorem. However, in some modifications of the problem, the usage of Pontryagin's maximum principle is preferable. The optimal synthesis for the problem and for its modifications is obtained.

This paper investigates synchronization of dynamical networks with both system delay and transmission delay by applying pinning impulsive control. Based on the comparison theorem and the theory of impulsive functional differential equations, some novel criteria are obtained, from which the proportion of pinned nodes is able to be determined. A numerical example is given to verify the effectiveness of the theoretical results.

This paper focuses on the problem of impulsive synchronization of a class of complex dynamical networks. Based on impulsive control theory and a comparison theorem, generic criteria for complete synchronization are derived. It is shown that these criteria provide a novel and effective control approach to synchronize a general dynamical network to a synchronization manifold by exposing the relationship between impulsive synchronization, the impulsive intervals, and the graph topology. A numerical example is given to illustrate the results.

A problem of controlling a group of independent identical agents is considered. Such problems arise in mathematical economics, in robotics (swarms of mini-robots). The model includes a stochastic component. The general method of the Bellman equation [6] cannot be applied to this problem. The solution is found by the comparison method for solutions of SDE using the technique of the Skorokhod reflection operators. The optimal policy admits a realization as a decentralized group control algorithm. The results of the preceding author's paper [1] are generalized: the state space changes from the line to a half-line with reflection at its base; thus the dynamics of the agent is extended to state dependent. Comparison with partial ordering (in ℝ d ) is a pioneering approach.

In this paper, several important inequalities for g-expectation and Choquet expectation are obtained by using back-ward stochastic differential equations (BSDEs) and comparison theorem. Markov inequality for g-probability is a special case of this inequality. If the generator g is independent on y and is linear in z, some classical inequalities for Choquet capacity are also presented.

In this paper, we investigate a n-patches nonautonomous predator-prey system with nonlinear diffusion. By the comparison theorem, the sufficient conditions are derived for uniform persistence of the system. We also prove the existence of positive periodic solution of the system by using Schauder's fixed point theorem. Furthermore, the sufficient conditions for global asymptotical stability of positive periodic solution are obtained by constructing an appropriate Lyapunov function. The conditions we obtained are more general. We also give the ecology interpretation to our results.

Considering the prey is a linear growth and is impulsively harvest, in this paper a predator-prey model with a stage structure for the predator which improves the assumption that each individual predator has the same ability to capture prey is proposed. It is assumed that immature individuals and mature individuals of the predator are divided by a fixed age and that immature predator do not have the ability to attach prey. By using the comparison theorem in impulsive differential equation,we obtain that when the death rate of the mature predator is less than the critical value and the impulsive period is larger than some critical value,the system is shown to be permanent, Our results indicate that the permanence of the system depends on impulsive perturbation.

The two-patch predator-prey system with diffusion and harvesting is concerned in this paper. By means of the coincidence degree theory and comparison theorem, we establish the existence of at least eight positive periodic solutions for this system. Further, an example is given to illustrate the effectiveness of our results.

A problem of controlling a group of independent identical agents is considered. Such problems arise in mathematical economics, in robotics (swarms of mini-robots). The model includes a stochastic component. The general method of the Bellman equation [1] cannot be applied to this problem. The solution is found by the comparison method for solutions of stochastic differential equations using the technique of the Skorokhod reflection operators. The optimal policy admits a realization as a decentralized group control algorithm.

A variable coefficients predator-prey system with Holling type IV functional response incorporating stage structure for both interacting populations and time delays has been presented in this paper. By analyzing the proposed system and using the standard comparison theorem, the sufficient conditions are derived for the permanence of the considered system and non-permanence of predators.

In this paper, we consider the following dynamic systems on time scales x(σ(t)) = f(t, x t ), xt 0 = x 0 y(σ(t)) = g(t, y t ), yt 0 = y 0 where σ(t) > t. By employing the concept of upper monotone increasing function and Lyapunov method, we obtain a new comparison theorem first, on this basis, some criteria on the boundedness in terms of two measures for these dynamic systems are obtained.

We present the convergent splitting and convergent multi splitting for linear system of algebraic equations Ax=b when the coefficient matrix is a non-hermitian positive definite matrix. Furthermore, we also establish the comparison theorems of different splitting or multisplittings by numerical radius. Finally, we give an application to solve the complex linear system.

In this paper, we consider the discrete hybrid system with different initial times. By employing variational cone-valued Lyapunov-like functions, we get some comparison theorems and several ⌽ 0 -stability criteria for discrete hybrid system with different initial values.

A set of population equations with feedback control is studied. By applying comparison theorem of differential equation and constructing a suitable Lyapunov function, sufficient conditions for the permanence, strictly positive (component wise), global asymptotic stability of the positive equilibrium point are obtained. Some new results are obtained.

This paper studies stability of solutions for a class of stochastic impulsive systems. The impulsive stochastic comparison theorem is established using stochastic differential inequalities. By using the comparison theorem, the comparison criterion of asymptotic stability of trivial solution of the systems is obtained based on vector Lyapunov-like function.

In this paper, a delayed stage-structured predator- prey model with the immature prey are taken care of by their parents and biological control strategies (that is releasing natural enemies at different fixed time) is considered. By using comparison theorem, the existence of the globally asymptotically stable prey-eradication periodic solution when the number of releasing natural enemies more than some critical value is proved. The conditions for the permanence of the system are given. Numerical simulation proved the academic results.

In this paper, we consider a three-population food chain model which concerning about impulsive releasing predators (natural enemies) at fixed moment. By using the Floquet theory and the comparison theorem of impulsive differential equation, conditions for global asymptotic stability of pest-eradication periodic solution and permanence of the system are obtained. Finally, the explanation of its biological significance is given.

A mathematical model for dynamics of a prey dependent consumption model concerning impulsive control strategy is proposed and analyzed. By using Floquet theorem and comparison theorem of impulsive differential equation, we show that there exists a globally stable pest-eradication periodic solutions when the impulsive periodic is less than some critical values. Further, the conditions for the permanence of the system are given. We show the existence of nontrivial periodic solution if the pest eradication periodic solution losses its stability. When the unique positive periodic solution lose its stability, numerical simulations shows there is a characteristic sequence of bifurcations, leading to a chaotic dynamics, which implies that the impulsive control model we considered has more complex dynamics including periodic doubling bifurcation, symmetry breaking bifurcation, period halving bifurcation, quasi-periodic oscillation and chaos.

A stage-structured predator-prey system with impulsive effect and Beddington-DeAngelis functional response is investigated. We shown that there exists a globally stable pest-eradication periodic solution by using the Floquet theorem and comparison theorem of impulsive differential equation. Further, conditions for the permanence of the system are established.

This paper generalizes the definition of classical g-expectations which is used as a nonlinear measurement in the risk management and mathematical finance. It considers properties of g-expectations, such as translation invariant property and converse comparison theorem.

In this paper we are concerned with the recursive zero-sum stochastic differential game problems. Using the backward stochastic differential equations techniques the existence result of a saddle point is obtained when the Isaacs' condition holds. An example and simulation result are also given to illustrate the application of theoretical result.

In this paper, we study deterministic algebratc Riccati equation, we obtain a comparison theorem under the condition of stablizability, which improves the previous result.

In this paper we refine the formulation of the combined frequency time domain analysis of the EM scattering from nonlinearly loaded scatterers or FSS. The response of the linear part is calculated in the frequency domain then the convolution theorem is used to obtain the nonlinear Volterra IE governing current through the nonlinear device terminals. Existence, uniqueness and asymptotic properties of the solution can be established using the comparison theorem which provides also a good initial approximation essential for high convergence rate of the numerical method of successive approximations.

Asymptotic cooperative stability is studied in the paper, and explicit conditions are found for heterogeneous nonlinear systems to reach a consensus. Specifically, a new comparison theorem is proposed for concluding both cooperative stability and Lyapunov stability, and it is in terms of vector nonlinear differential inequalities (on Lyapunov function components). It is unique that the proposed result admits both heterogeneous dynamics of nonlinear systems and intermittent unpredictable changes in their associated sensing/communication network. Its proof is done using a combination of Lyapunov argument (in terms of the Lyapunov function components) and topology- dependent argument (in terms of structural properties of reducible matrices). Consequently, the proposed result does not impose any of the following assumptions required in the existing results: the knowledge of a successful Lyapunov function, system dynamics being convex, nonsmooth analysis, fixed or certain types of communication patterns, quasi- monotone property on differential inequalities. If the systems under consideration are all linear, the theorem reduces to the necessary and sufficient condition of cooperative controllability obtained using the matrix-theoretical approach, and the inequalities become equalities. For nonlinear systems, the proposed cooperative stability conditions are straightforward to verify. Several types of nonlinear systems are used as examples to illustrate application potentials of the comparison theorem in both cooperative stability analysis and cooperative control design.

In this paper, memoryless linear state or state estimation feedback controller design is introduced for delayed systems with linear or nonlinear time-varying uncertainties. Comparison theorem is employed to investigate the robust stability conditions which are based on the upper norm bounds of the uncertainties. The feedback gain and observer gain are selected to satisfy the requirement of robust stability under plant uncertainties.

The stability of a bilinear system with stated delay and saturating actuator is discussed. Delay-dependent sufficient conditions for the uniform asymptotic stability of the closed-loop systems via a memoryless static state feedback controller is developed. The matrix measure and comparison theorem are employed to investigate the problem. The results obtained omit solving the Lyapunov function or Riccati equation and is less conservative than the delay-dependent criterion for the same example when delay is small. The authors' result is demonstrated by an illustrate example.

The stabilization problem of uncertain time-delay systems with a saturating actuator is considered. These uncertainties may be linear or nonlinear, but only the upper bounds are known. The properties of norm and comparison theorem are employed to investigate the robust stability condition which assure asymptotic stability rather than ultimate boundedness of trajectories. Moreover, an algorithm is proposed to synthesize a dynamic feedback controller for guaranteeing the asymptotic stability of the uncertain time delay saturating systems. The sufficient condition for robust stability of input delay constrained systems subjected to parametric perturbations is also derived. The result obtained eliminates the need to solve the Riccati equation. The stability criterion is delay-dependent and less conservation than the delay-independent criteria and delay-dependent criterion when delays are small. Finally, simulation examples are given to demonstrate the application of the authors' results.

The delay-dependent stability of singularly perturbed multiple time-delay systems with saturating actuators in slow states is discussed. A delay-dependent sufficient condition for the uniform asymptotic stability of the closed-loop systems via a memoryless state feedback controller are developed. The matrix measure and comparison theorem are employed to investigate the problem. The results obtained omit solving the Lyapunov function or Riccati equation. The result obtained can be used to estimate the delay time /spl tau/. The significance of the result obtained is illustrated by an example.

When filtering a diffusion process with small polynomial nonlinearities, the conditional density may be expanded in an additive asymptotic expansion about a Kalman filter. Because such an expansion may fail to be a probability density, finite dimensional multiplicative asymptotic expansions are developed here as an alternative approach. Using the robust version of the filtering equation, an existence theorem and upper and lower estimates for the tail behavior of the unnormalized conditional density are obtained, and error bounds are derived using an integral representation of the error and a comparison theorem for partial differential inequalitites.

Since systems are generally often subject to some forms of perturbation, such as unmodelled dynamics, environmental variations, and parameter uncertainties, the robust stability is one of the most important things to be considered. Owing to its complexity and largeness, the large scale system is more easy to suffer uncertainties than the small one. Thus in this paper, we consider the robust stability of large scale system with parametric perturbation. The sufficient condition of the robust stability is derived via comparison theorem and norm theory from the viewpoint of time domain. Also, a design procedure is proposed to find a controller to satisfy the robust stability condition.