In this note, we study a new class of ordinary differential equations with non-instantaneous impulses. Both existence and generalized Ulam-Hyers-Rassias stability results are established. Finally, an example is given to illustrate our theoretical results.

Microwave heating has been widely used in various fields during recent years. However, it also has a common problem of uneven heating. In this paper, optimal frequency control problem for microwave heating process is considered. The cost function is defined such that the temperature profile at the final stage has a relative uniform distribution in the field. The controlled system is a coupled by Maxwell equations with nonlinear heating equation. The existence of a weak solution for coupled system is proved. The weak continuity of the solution operator is also shown. Moreover, the existence of a global minimizer of the optimal frequency control problems is proved.

Summary This article investigates the finite‐time consensus control for stochastic multi‐agent systems (SMASs) by using adaptive techniques. First, we propose a finite‐time adaptive consensus protocol for SMASs with node‐based adaptive law design method. Second, a finite‐time adaptive consensus protocol is also proposed for SMASs by adding a dynamical scaling parameter to the weights on the edges of the communication graph. Finally, two simulation examples are provided to verify the effectiveness of the two consensus protocols.

Motivated by the definition of geometric-arithmetically s-convex functions in (Shuang et al. in Analysis 33:197-208, 2013) and second-order fractional integral identities in (Zhang and Wang in

<p style='text-indent:20px;'>In the framework of fixed topology and stochastic switching topologies, we study the mean-square bounded consensus(MSBC) of double-integrator stochastic multi-agent systems(SMASs) including additive system noises and communication noises. Combining algebra, graph theory and random analysis, we obtain several equivalent conditions for double-integrator SMASs to reach MSBC. In addition, the simulation examples also verify the correctness of the theoretical results.</p>

In this paper, we study the regularity of the weak solution of the coupled system derived from the microwave heating model with frequency variable. We first show that the weak solution E of the system is Hölder continuous near the boundary of S = ∂ Ω . The main idea of the proof is based on the estimation of linear degenerate system in Campanato space. Then we show that the solution u of the heat conduction equation is Hölder continuous with exponent α 2 . Finally, under the appropriate conditions we show that the coupled system with microwave heating has a weak solution. Moreover the regularity of the weak solution is studied.

No abstract

By using the definition of geometric-arithmetically s-convex functions in (Analysis 33:197-208, 2013)