Local Exact Controllability to the Trajectories of Burgers–Fisher Equation

This paper is addressed to study local exact controllability to the trajectories of the Burgers–Fisher (BF) equation. By using the global Carleman estimate for the second-order parabolic operator, we establish the observable inequality and obtain the exact controllability to the trajectories of the linear system. Then, by local inverse theory, we consider the controllability result for the Burgers–Fisher equation.

Protection Zones in Periodic-Parabolic Problems

This paper characterizes whether or not\Sigma_{\infty}\equiv\lim_{\lambda\uparrow\infty}\sigma[\mathcal{P}+\lambda m(% x,t),\mathfrak{B},Q_{T}]is finite, where {m\gneq 0} is T-periodic and {\sigma[\mathcal{P}+\lambda m(x,t),\mathfrak{B},Q_{T}]} stands for the principal eigenvalue of the parabolic operator {\mathcal{P}+\lambda m(x,t)} in {Q_{T}\equiv\Omega\times[0,T]} subject to a general boundary operator of mixed type, {\mathfrak{B}}, on {\partial\Omega\times[0,T]}. Then this result is applied to discuss the nature of the territorial refuges in periodic competitive environments.

Carleman estimates for time-discrete parabolic equations and applications to controllability

In this paper, we prove a Carleman estimate for a time-discrete parabolic operator under some condition relating the large Carleman parameter to the time step of the discretization scheme. This estimate is then used to obtain relaxed observability estimates that yield, by duality, some controllability results for linear and semi-linear time-discrete parabolic equations. We also discuss the application of this Carleman estimate to the controllability of time-discrete coupled parabolic systems.

Exact Controllability of the Heat Equation by Temporarily Strategic Actuators Borders

The use of the strategic zone actuators, where possible, to control a system is simpler because the unknown functions depend only on the time variable. However, the restrictions and difficulties for establishing the coercivity inequality of the parabolic operator, requires looking for other methods of internal controls. So, we will generalize the concept of actuators strategic areas and deduce results of exact controllability.