In this paper, we consider the existence of multiple solutions for second-order equation with Stieltjes integral boundary condition using the three-critical-point theorem and variational method. Firstly, a novel space is established and proved to be Hilbert one. Secondly, based on the above work, we obtain the existence of multiple solutions for our problem. Finally, in order to illustrate the effectiveness of our problem better, the example is listed.
This paper presents the output feedback stabilization of a first-order hyperbolic equation with multi-point nonlocal term by boundary control. First of all, we utilize the backstepping method to design a boundary state feedback controller.On the basis of equivalence of original and target system, we guarantee the exponential stability of closed-loop system. Hereafter, the output feedback control in the light of an infinite-dimensional observer is constructed. We demonstrate that the resulting closed-loop system is exponentially stable via operator semigroup theory. Finally, we display the validity of the proposed controller by some numerical simulations.
The ricochet behaviour of the air–water trans-media vehicle (AWTMV) during water-entry crossing was experimentally investigated. Three types of small-scale AWTMV including cone, ogive, and flat nose were used in the test. The underwater trajectory, velocity, and inclination angle of projectiles during the ricochet process were obtained using a high-speed camera. The angle change of the AWTMV and the ratio of the residual velocity are introduced. Based on this result, the relationship between the ricochet responses and initial conditions was derived. The results of this study show that (1) a small incident angle and great velocity make the occurrence of ricochet behaviour easier, (2) the stability of the trajectory of projectiles with cone, ogive, and flat nose weakened in turn at the same initial conditions, (3) the angle change and the ratio of the residual velocity are linear functions of the incident angle and velocity.
In this paper, we consider boundary stabilization problem of heat equation with multi-point heat source. Firstly, a state feedback controller is designed mainly by backstepping approach. Under the designed state controller, the exponential stability of closed-loop system is guaranteed. Then, an observer-based output feedback controller is proposed. We prove the exponential stability of resulting closed-loop system using operator semigroup theory. Finally, the designed state and output feedback controllers are effective via some numerical simulations.
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