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Stable-protocol (SP) output consensus analysis and design problems for high-order linear time-invariant swarm systems with time-varying delays are investigated. First, a dynamic output feedback consensus protocol is proposed on the basis of the observable decomposition, and a necessary and sufficient condition for SP output consensus is given, which transforms the SP output consensus problem into asymptotical stability problems of multiple lower-dimensional subsystems. Then, in terms of linear matrix inequalities (LMIs), a sufficient condition for SP output consensualisation is presented, which can guarantee the scalability of swarm systems since it includes only five LMI constrains independent of the number of agents. Furthermore, an explicit expression of the output consensus function, which is independent of the time-varying delay, is shown and the impacts of initial states of agents and consensus protocols on the output consensus function are determined, respectively. Finally, a numerical example is given to demonstrate theoretical results.

This study investigates a delayed predator–prey model with age-structure and a Holling III response function. The considered model is first presented as an abstract Cauchy problem, then the criteria of the presence for the positive equilibrium point are gained. The global asymptotic stability of the boundary equilibrium and the local asymptotic stability of the positive equilibrium are examined, respectively, through detailed investigation of the location of the solutions for the model’s characteristic equation. Then, using the Hopf bifurcation theorem, we show that Hopf bifurcation exists under certain criteria. At last, some numerical examples are performed to demonstrate the obtained theoretical results.

In this work, an SIRS model with age structure is proposed for recurrent infectious disease by incorporating temporary immunity and delay. We formulate the model as an abstract non-densely defined Cauchy problem and derive the conditions for the global stability of disease free equilibrium, the local stability of endemic equilibrium, and the existence of Hopf bifurcation. Both non-periodic and periodic behaviors are possible when the disease persists in population, where time delay plays an important role. Numerical examples are provided to illustrate our theoretical results.

We consider the initial-boundary value problem of reaction-diffusion-advection that has a solution of a front form. The statement comes from the theory of wave physics. We study the question of the solution stabilizing to the stationary one. Proof of the stabilization theorem is based on the concepts of upper and lower solutions and corollaries from comparison theorems. The upper and lower solutions with large gradients are constructed as modifications of the formal moving front asymptotic approximation in a small parameter. The main idea of the proof is to show that the upper and lower solutions of the initial-boundary value problem get into the attraction domain of the asymptotically stable stationary solution on a sufficiently large time interval. The study conducted in this work gives an answer about the non-local attraction domain of the stationary solution and can give some stationing criteria. The results are illustrated by computational examples.

A methodology of designing the block backstepping controller for a class of multi-input non-linear perturbed systems is proposed to solve the regulation problem in accordance with the Lyapunov stability theorem. The designed constant gains, which are embedded in the virtual input controllers, can effectively overcome more general mismatched perturbations than some works can. Furthermore, the asymptotical stability can be achieved by the proposed controller. Finally, an example of controlling the control-moment-gyro devices is used to demonstrate the feasibility of the proposed methodology.

In this paper, we consider how to design a kind of nonlinear neural networks for solving the inverse optimal value problem with convex constraints. Firstly, based on optimal theory, the inverse optimal value problem considered is changed to a class of nonlinear bilevel program problems. Secondly, under the given assumptions, the corresponding nonlinear bilevel programming problem can be reduced to the one-level programming problem. Thirdly, based on the gradient theory, a nonlinear neural network model is designed to solve this one-level programming problem. Moreover, by employing Lyapunov function approach, the proposed neural network is analyzed to be globally Lyapunov stable and capable of generating approximal optimal solution to the inverse optimal value problem. Finally, two illustrative examples are provided to verify the feasibility and the efficiency of the proposed method.