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This paper aims to address an interesting open problem, posed in the paper "Singular Optimal Control for a Transport-Diffusion Equation" of Sergio Guerrero and Gilles Lebeau in 2007. The problem involves studying the null controllability cost of a transport-diffusion equation with Neumann conditions, where the diffusivity coefficient is denoted by $\varepsilon>0$ and the velocity by $\mathfrak{B}(x,t)$. Our objective is twofold. First, we investigate the scenario where each velocity trajectory $\mathfrak{B}$ originating from $\overline{\Omega}$ enters the control region in a shorter time at a fixed entry time. By employing Agmon and dissipation inequalities, and Carleman estimate in the case $\mathfrak{B}(x,t)$ is the gradient of a time-dependent scalar field, we establish that the control cost remains bounded for sufficiently small $\varepsilon$ and large control time. Secondly, we explore the case where at least one trajectory fails to enter the control region and remains in $\Omega$. In this scenario, we prove that the control cost explodes exponentially when the diffusivity approaches zero and the control time is sufficiently small for general velocity.

This paper deals with the insensitizing controllability property of the quasilinear parabolic equation with dynamic boundary conditions. This problem can be reformulated as a null controllability problem for a cascade quasilinear system with dynamic boundary conditions. To this end, we approach the problem by first dealing with null controllability in the framework of an inhomogeneous linearized system. Next, we derive new estimates of control and state, allowing us to apply a local inversion theorem to obtain null controllability of the quasilinear system.

This paper is devoted to the theoretical and numerical analysis of the null controllability of a coupled ODE-heat system internally and at the boundary with Neumann boundary control. First, we establish the null controllability of the ODE-heat with distributed control using Carleman estimates. Then, we conclude by the strategy of space domain extension. Finally, we illustrate the analysis with some numerical experiments.

This paper aims to address an interesting open problem posed in the paper ''Singular Optimal Control for a Transport-Diffusion Equation'' of Sergio Guerrero and Gilles Lebeau in 2007. The problem involves studying the null-controllability cost of a transport-diffusion equation with Neumann conditions. Our objective is twofold. Firstly, we investigate the scenario where each trajectory of the tangential velocity enters the control region in a shorter time at a fixed entry time. By employing Agmon inequalities and Carleman estimates, we establish that the control cost remains bounded for sufficiently small diffusivity and large control time. Secondly, we explore the case where at least one trajectory fails to enter the control region. In this scenario, we prove that the control cost explodes exponentially when the diffusivity approaches zero and the control time is sufficiently small.