Null Controllability of a Nonlinear Age Structured Model for a Two-Sex Population

This paper is devoted to study the null controllability properties of a nonlinear age and two-sex population dynamics structured model without spatial structure. Here, the nonlinearity and the couplage are at the birth level. In this work, we consider two cases of null controllability problem. The first problem is related to the extinction of male and female subpopulation density. The second case concerns the null controllability of male or female subpopulation individuals. In both cases, if A is the maximal age, a time interval of duration A after the extinction of males or females, one must get the total extinction of the population. Our method uses first an observability inequality related to the adjoint of an auxiliary system, a null controllability of the linear auxiliary system, and after Kakutani’s fixed-point theorem.

Optimization of non-cylindrical domains for the exact null controllability of the 1D wave equation

This work is concerned with the null controllability of the one-dimensional wave equation over non-cylindrical distributed domains. The controllability in that case has been obtained by Castro, C\^indea and M\"unch in SIAM J. Control Optim., 52 (2014) for domains satisfying the usual geometric optic condition. We analyze the problem of optimizing the non-cylindrical support $q$ of the control of minimal $L^2(q)$-norm. In this respect, we prove a uniform observability inequality for a class of domains $q$ satisfying the geometric optic condition. The proof based on the d'Alembert formula relies on arguments from graph theory. Numerical experiments are discussed and highlight the influence of the initial condition on the optimal domains.

Controllability of a Family of Nonlinear Population Dynamics Models

Considering a nonlinear dynamical system, we study the nonlinear infinite-dimensional system obtained by grafting an operator A and an age structure. This system is such that the nonlinearity is at the level of births. We show that there is a time T dependent on the constraints on the age and the observability minimal time T 0 of the pair A , B ( B is the control operator), from which the system is null controllable. We first establish an observability inequality useful for the proof of the null controllability of an auxiliary system. We also apply Schauder’s fixed point in the proof of the null controllability of the nonlinear system..

Null controllability for a class of stochastic singular parabolic equations with the convection term

<p style='text-indent:20px;'>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.</p>

Observability and unique continuation of the adjoint of a linearized simplified compressible fluid-structure model in a 2d channel.

We consider a compressible fluid structure interaction model in a 2D channel with a simplified expression of the net force acting on the structure appearing at the fluid boundary. Concerning the structure we will consider a damped Euler-Bernoulli beam located on a portion of the boundary. In the present article we establish an observability inequality for the adjoint of the linearized fluid structure interaction problem under consideration which in principle is equivalent with the null controllability of the linearized system. As a corollary of the derived observability inequality we also obtain a unique continuation property for the adjoint problem.

On the observability inequality of coupled wave equations: the case without boundary

In this paper, we study the observability and controllability of wave equations coupled by first or zero order terms on a compact manifold. We adopt the approach in Dehman-Lebeau’s paper [B. Dehman and G. Lebeau, SIAM J. Control Optim. 48 (2009) 521–550.] to prove that: the weak observability inequality holds for wave equations coupled by first order terms on compact manifold without boundary if and only if a class of ordinary differential equations related to the symbol of the first order terms along the Hamiltonian flow are exactly controllable. We also compute the higher order part of the observability constant and the observation time. By duality, we obtain the controllability of the dual control system in a finite co-dimensional space. This gives the full controllability under the assumption of unique continuation of eigenfunctions. Moreover, these results can be applied to the systems of wave equations coupled by zero order terms of cascade structure after an appropriate change of unknowns and spaces. Finally, we provide some concrete examples as applications where the unique continuation property indeed holds.