This paper concerns the null controllability for a strongly coupled stochastic degenerate reaction-diffusion system in cardiac electrocardiology, which describes the electrical activity in the cardiac tissue with random effects. To deal with degeneracy of this system, we first consider an approximate problem of this coupled stochastic degenerate system. By a weighted identity method, we then prove a uniform Carleman estimate for the adjoint system of this approximate problem, which is a strongly coupled backward stochastic parabolic system with homogeneous Neumann boundary conditions. Based on this Carleman estimate and a limit process, we finally obtain the null controllability result. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

This paper concerns the null controllability of stochastic degenerate parabolic equation with convection term in weakly degenerate case. Due to the degeneracy, we first transfer to study an approximate nondegenerate system. Next, we establish the Carleman estimate for backward stochastic degenerate parabolic equation with convection term. In order to deal with the convection term, the Carleman estimate is established by introducing an auxiliary function to transfer the diffusion and the convection term into a whole complex divergence-like form. By this Carleman estimate, we then obtain an observability inequality for the adjoint system of the approximate system. Based on the observability inequality and an approximate argument, we proved our null controllability result.

In this paper, we consider a null controllability and an inverse source problem for stochastic Grushin equation with boundary degeneracy and singularity. We construct two special weight functions to establish two Carleman estimates for the whole stochastic Grushin operator with singular potential by a weighted identity method. One is for the backward stochastic Grushin equation with singular weight function. We then apply it to prove the null controllability for stochastic Grushin equation for any T and any degeneracy γ > 0, when our control domain touches the degeneracy line { x = 0}. In order to study the inverse source problem of determining two kinds of sources simultaneously, we prove the other Carleman estimate, which is for the forward stochastic Grushin equation with regular weight function. Based on this Carleman estimate, we obtain the uniqueness of the inverse source problem.

This paper concerns the null controllability for a class of stochastic singular parabolic equations with the convection term in one dimensional space. Due to the singularity, we first transfer to study an approximate nonsingular system. Next we establish a new Carleman estimate for the backward stochastic singular parabolic equation with convection term and then an observability inequality for the adjoint system of the approximate system. Based on this observability inequality and an approximate argument, we obtain the null controllability result.

This paper concerns an inverse problem for a stochastic degenerate wave equation. This inverse problem aims to determine a source term and an initial displacement and an initial velocity from the boundary observation and the final time observation. More precisely, we will prove a global uniqueness for our inverse problem by a suitable Carleman estimate for the stochastic degenerate wave equation.

The paper is concerned with the inverse source problem for an ADMB-KdV equation, which describes the nonlinear waves generated by a long-wave instability in a viscous film flowing down an inclined rigid surface. The inverse problem aims to determine a spatially varying source function from internal observation data on a suitable subdomain and the whole spatial observation data at a time. We first prove a Carleman inequality for ADMB-KdV equation, and then apply this Carleman inequality to derive Lipschitz stability for this inverse source problem.

该文研究了一类带有非奇异系数矩阵的2×2强耦合偏微分方程组的卡勒曼估计.文献[7]和[15]利用对角化的技巧将方程组解耦,证明了一个2×2强耦合双曲方程组的卡勒曼估计.不同于此,该文考虑将微分方程组的两个方程作为整体来建立逐点的卡勒曼,然后进一步得到了这类强耦合方程组的全局卡勒曼估计.最后,作为卡勒曼估计的应用,该文建立了一个反源问题的H(o)lder稳定性.

We consider an inverse problem for an integro-differential parabolic equation with free boundary. This inverse problem aims to identify the memory kernel function from a global measurement data. Based on a fixed point argument, we derive the local in time existence and uniqueness of our inverse problem. Additionally, we present a numerical experiment.

This paper concerns Carleman estimate and its applications for a linearized bidomain model in electrocardiology, which describes the electrical activity in the cardiac tissue. We first establish a new Carleman estimate for this reaction–diffusion system. By means of this Carleman estimate, we study two problems for the linearized bidomian model, a Cauchy problem and an inverse conductivities problem. We prove a conditional stability result for the Cauchy problem and a Hölder stability result for the inverse conductivities problem.