Null controllability of a population dynamics with interior degeneracy

In this paper, we deal with the null controllability of a population dynamics model with an interior degenerate diffusion. To this end, we proved first a new Carleman estimate for the full adjoint system and afterwards we deduce a suitable observability inequality which will be needed to establish the existence of a control acting on a subset of the space which lead the population to extinction in a finite time

Nonlinear stability and ergodicity of ensemble based Kalman filters

The ensemble Kalman filter (EnKF) and ensemble square root filter (ESRF) are data assimilation methods used to combine high dimensional, nonlinear dynamical models with observed data. Despite their widespread usage in climate science and oil reservoir simulation, very little is known about the long-time behavior of these methods and why they are effective when applied with modest ensemble sizes in large dimensional turbulent dynamical systems. By following the basic principles of energy dissipation and controllability of filters, this paper establishes a simple, systematic and rigorous framework for the nonlinear analysis of EnKF and ESRF with arbitrary ensemble size, focusing on the dynamical properties of boundedness and geometric ergodicity. The time uniform boundedness guarantees that the filter estimate will not diverge to machine infinity in finite time, which is a potential threat for EnKF and ESQF known as the catastrophic filter divergence. Geometric ergodicity ensures in addition that the filter has a unique invariant measure and that initialization errors will dissipate exponentially in time. We establish these results by introducing a natural notion of observable energy dissipation. The time uniform bound is achieved through a simple Lyapunov function argument, this result applies to systems with complete observations and strong kinetic energy dissipation, but also to concrete examples with incomplete observations. With the Lyapunov function argument established, the geometric ergodicity is obtained by verifying the controllability of the filter processes; in particular, such analysis for ESQF relies on a careful multivariate perturbation analysis of the covariance eigen-structure.Comment: 38 page

Controllability of the heat and wave equations and their finite difference approximations by the shape of the domain

In this article we study a controllability problem for a parabolic and a hyperbolic partial differential equations in which the control is the shape of the domain where the equation holds. The quantity to be controlled is the trace of the solution into an open subdomain and at a given time, when the right hand side source term is known. The mapping that associates this trace to the shape of the domain is nonlinear. We show (i) an approximate controllability property for the linearized parabolic problem and (ii) an exact local controllability property for the linearized and the nonlinear equations in the hyperbolic case. We then address the same questions in the context of a finite difference spatial semi-discretization in both the parabolic and hyperbolic problems. In this discretized case again we prove a local controllability result for the parabolic problem, and an exact controllability for the hyperbolic case, applying a local surjectivity theorem together with a unique continuation property of the underlying adjoint discrete system.Comment: 27 page

Interior degenerate/singular parabolic equations in nondivergence form: well-posedness and Carleman estimates

We consider non smooth general degenerate/singular parabolic equations in non divergence form with degeneracy and singularity occurring in the interior of the spatial domain, in presence of Dirichlet or Neumann boundary conditions. In particular, we consider well posedness of the problem and then we prove Carleman estimates for the associated adjoint problem.Comment: Accepted in Journal of Differential Equations. arXiv admin note: text overlap with arXiv:1507.0778

Carleman estimates, observability inequalities and null controllability for interior degenerate non smooth parabolic equations

We show Carleman estimates, observability inequalities and null controllability results for parabolic equations with non smooth coefficients degenerating at an interior point.Comment: Accepted in Memoirs of the American Mathematical Societ

Hierarchical control for the semilinear parabolic equations with interior degeneracy

This paper concerns with the hierarchical control of the semilinear parabolic equations with interior degeneracy. By a Stackelberg-Nash strategy, we consider the linear and semilinear system with one leader and two followers. First, for any given leader, we analyze a Nash equilibrium corresponding to a bi-objective optimal control problem. The existence and uniqueness of the Nash equilibrium is proved, and its characterization is given. Then, we find a leader satisfying the null controllability problem. The key is to establish a new Carleman estimate for a coupled degenerate parabolic system with interior degeneracy