In this paper, the stability, stabilization, and $L_{2}$ -gain problems are investigated for periodic piecewise systems with time-varying subsystems. Continuous Lyapunov function with time-varying Lyapunov matrix is adopted. A condition guaranteeing the negative definiteness of a matrix polynomial, deriving from the Lyapunov derivative, is first obtained. Based on such a condition, an exponential stability condition is provided. Moreover, a state-feedback controller with time-varying gain is developed to stabilize the unstable periodic piecewise time-varying system. The $L_{2}$ -gain criterion for periodic piecewise time-varying system is also studied. Numerical examples are given to show the validity of the proposed techniques.

Stability and $L_2$ Synthesis of a Class of Periodic Piecewise Time-Varying Systems

In this paper, a predator-prey model with Holling type-I functional response and multi state impulsive feedback control is established, where the intensity of pesticide spraying and the release amount of natural enemies are linearly dependent on the given threshold in the second impulse. Firstly, the existence of order-1 periodic solution of the system is investigated by successor functions and Bendixson theorem of impulsive differential equations, then the stability of periodic solutions is proved by the analogue of the Poincare criterion. Furthermore, in order to reduce the actual total cost and obtain the best economic benefit, the optimal economic threshold is obtained, which provides the optimal strategy for the practical application. Finally, numerical simulations for specific examples are carried out to illustrate the feasibility of the above conclusions.

/pdf/geometrical-analysis-and-control-optimization-of-a-predator-1qxekj3223.pdf

Geometrical analysis and control optimization of a predator-prey model with multi state-dependent impulse

In this paper, we establish a unilateral diffusion Gompertz model of a single population in two patches in a theoretical way. Firstly, we prove the existence and uniqueness of an order-one periodic solution by the geometry theory of differential equations and the method of successor function. Secondly, we prove the stability of the order-one periodic solution by imitating the theory of the limit cycle of an ordinary differential equation. Finally, we verify the theoretical results by numerical simulations.

/pdf/dynamic-analysis-of-unilateral-diffusion-gompertz-model-with-3qxwxhiyz9.pdf

Dynamic analysis of unilateral diffusion Gompertz model with impulsive control strategy

Global dynamic behavior of a predator-prey model under ratio-dependent state impulsive control

Dynamic behaviors of the FitzHugh–Nagumo neuron model with state-dependent impulsive effects

The order-1 periodic solution of the system with impulsive state feedback control is investigated. We get the sufficient condition for the existence of the order-1 periodic solution by differential equation geometry theory and successor function. Further, we obtain a new judgement method for the stability of the order-1 periodic solution of the semi-continuous systems by referencing the stability analysis for limit cycles of continuous systems, which is different from the previous method of analog of Poincare criterion. Finally, we analyze numerically the theoretical results obtained.

Periodic solution of the system with impulsive state feedback control

In this paper, a Leslie predator–prey system with ratio-dependent and state impulsive feedback control is investigated by applying the geometry theory of differential equation. When the economic threshold level is under the positive equilibrium, the existence, uniqueness and orbital asymptotical stability of order-1 periodic solution for the system can be obtained. When the economic threshold level is above the positive equilibrium, and the positive equilibrium is a focus point, sufficient conditions of the existence, uniqueness and orbital asymptotical stability of order-1 periodic solution for the system are also acquired. Furthermore, when the positive equilibrium is an unstable focus point, the existence of order-1 periodic solution of the impulsive system can be obtained within limit cycle of the continuous system. The mathematical results can be verified by numerical simulations.

Periodic solution of a Leslie predator–prey system with ratio-dependent and state impulsive feedback control

In this paper, a predator–prey system with mutual interference and impulsive state feedback control is constructed. Firstly, by using the stability theorem, we obtain the global asymptotically stability of the positive equilibrium for the system without impulse effects. Secondly, by using the geometric theory of differential equations, the method of successor functions and mathematical analysis, we obtain some sufficient conditions for the existence, uniqueness and orbital asymptotically stability of the order-1 periodic solution for the system with impulse effects. Finally, the main conclusions are verified by numerical simulation, and the pest prevention and control strategies on combining light trapping and water-gun spraying are proposed.

Qualitative analysis of a predator–prey system with mutual interference and impulsive state feedback control

In this paper, the system with impulsive state feedback control corresponding to the sulphitation reaction in process of manufacture of sugar is considered. By means of square approximation and a s...

Impulsive state feedback control during the sulphitation reaction in process of manufacture of sugar

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Papers

Periodic solution of the system with impulsive state feedback control

Periodic solution of a Leslie predator–prey system with ratio-dependent and state impulsive feedback control

Qualitative analysis of a predator–prey system with mutual interference and impulsive state feedback control

Impulsive state feedback control during the sulphitation reaction in process of manufacture of sugar