The membrane diffraction is a new imaging method for space telescopes. It makes a hot research topic in space telescope technology with lots of advantages, such as light weight, easy foldability and high optical imaging accuracy. The active vibration control of the truss structure of a kind of membrane diffraction space telescope was investigated, and an active vibration control strategy was proposed based on cable actuators. Firstly, the dynamic model for the telescope truss structure was established. Then the particle swarm optimization algorithm was used to study the arrangement optimization of cable actuators. The active control law for the structure vibration was designed with the classical linear quadratic regulator method. Finally, the numerical simulation results verify the effectiveness of the proposed method. In the numerical simulations, the relationship between the number of cable actuators and the required time for the structure to regain stability was studied in detail.

An adaptive and robust backstepping method based on wavelet network approximation was proposed to solve the problems of load variation, unmodeled uncertainties, physical parameter perturbation and external disturbance in the sprayer boom profiling system. Firstly, a complete mathematical model for the boom system with uncertainties, unknowns and nonlinear terms was established and transformed into a state space form with strict feedback. Secondly, the designed wavelet primitive was used to construct the neural network, to approximate the virtual equivalent control part of the backstepping method under the condition that the optimal error is bounded. The adaptive update law was selected to estimate the unknown parameters. The robust compensation term was introduced to reduce the adverse effect of the composite interference on the system. The input command signal order requirement was reduced. Finally, suitable functions were constructed by means of the Lyapunov stability theory, to prove that the position tracking error of the closed-loop system asymptotically converges to the origin. The simulation results show that, the proposed control method can realize the rapid maneuver adjustment of the sprayer boom position and posture, and effectively enhance the robust stability and control accuracy of the boom system.

A set of nonlinear feedback control strategies were designed to realize the bifurcation solutions of codimensional bifurcations in discrete dynamical systems with 1∶2 resonance from the perspective of bifurcation anti-controlling. Firstly, aimed at the limitation of traditional bifurcation criteria for determination of high codimensional bifurcation points, a new explicit criterion for codimension-2 bifurcation in 1∶2 resonance was proposed. Based on this explicit criterion, the linear control gain was designed to ensure the existence of such codimension-2 bifurcation. Then, the central manifold of 1∶2 resonance was derived. Based on the normal form method, the types and stability of codimension-2 bifurcation solutions in 1∶2 resonance were analyzed through design of nonlinear control gain. Finally, an Arneodo-Coullet-Tresser mapping was taken as an example, and various bifurcation solutions with 1∶2 resonance bifurcation properties were realized by control at the specified parameter points, to further validate the theoretical analysis.

The transverse deformation of the beam will lead to the longitudinal shortening deformation, and this transverse-longitudinal deformation coupling will bring the dynamic stiffening effect term on the generalized rigidity of the beam model. For the rotating beam structure, the centrifugal force will cause axial tension, with coupling axial and transverse deformation of the beam and bring additional geometric stiffness, which is more obvious for the thick short beam. The central rigid body-Timoshenko beam model with a large-range-motion center was investigated. Firstly, the dynamic model with centrifugal forces was established by means of the Timoshenko beam theory and the Hamilton principle. Secondly, the unconstrained mode concept was introduced, and the unconstrained mode shape functions and natural frequencies were solved with the Frobenius method. Finally, numerical simulations were carried out to explore the difference of generalized stiffness between the unconstrained mode and the constrained mode at different constant speeds, and the effects of centrifugal forces on the model under unconstrained mode condition were discussed.

The dynamic model was built for rotating pipes conveying fluid based on the Lagrange principle and the assumed mode method. The eigenvalue problem of the system was solved via the method of “reducing the order and increasing the dimension”. The free vibration characteristics of the rotating pipe conveying fluid were analyzed. The variations of the eigenvalue trajectories with the fluid velocity were illustrated under different tip masses and rotating speeds. The effects of system parameters on the critical fluid velocity were revealed. It is found that, the flowing fluid has significant effects on the dynamic characteristics of the rotating pipe. Different internal resonances between the 1st several modes of the system could exist under certain parameter conditions. The work reveals rich dynamic phenomena of the rotating pipe conveying fluid.

A rotated flux mixed scheme was proposed for solving 2D shallow water equations. Spatially, the algorithm uses the rotation invariance of the shallow water equations. In the normal direction and tangent direction of the element interface, both the HLL, which can eliminate the carbuncle, and the entropy stable weighted hybrid numerical flux function satisfying the 2nd law of thermodynamics, were applied to give fine numerical results. Temporally, the 3rd-order strongly stable Runge-Kutta method was used. The numerical results show that, the new scheme has high resolution for solving 2D shallow water equations.

Based on the 4th-order compact difference scheme for spatial discretization, the Taylor series expansion and the error remainder correction method for temporal discretization, a high-order compact finite difference scheme for solving the 3D unsteady convection diffusion reaction equations was proposed. The unconditional stability was proved with the Fourier analysis method. The proposed scheme has 2nd-order accuracy in time and 4th-order accuracy in space. At last, numerical examples validate the theoretical results.

The research of gene regulatory networks (GRNs) and their dynamic models is important in the post-genome era. Qualitative analysis of GRNs and their dynamics is of great significance to the understanding of organisms from a systematic perspective. A stochastic GRN model with time-varying delay and Markovian switching was proposed to study the properties of mean-square synchronization and stochastically passive synchronization. Through the design of an appropriate Lyapunov-Krasovskii functional (LKF), the sufficient conditions for mean-square synchronization and stochastically passive synchronization were obtained by means of the Lyapunov stability theory, the linear matrix inequality method and the random analysis techniques. In addition, the comparison between the results of this paper and some other literatures shows that, the present results have markable theoretical meaning. The numerical simulation illustrates the validity of the obtained sufficient conditions.