Related papers: Stabilization of a Chain of Three Hyperbolic PDEs …
We prove the well-posedness results, i.e. existence, uniqueness, and stability, of the solutions to a class of nonlocal fully nonlinear parabolic partial differential equations (PDEs), where there is an external time parameter $t$ on top of…
In this paper, we propose a data-driven model reduction method to solve parabolic inverse source problems efficiently. Our method consists of offline and online stages. In the off-line stage, we explore the low-dimensional structures in the…
In this paper, we prove the stabilizability of abstract Parabolic Integro-Differential Equations (PIDE) in a Hilbert space with decay rate $e^{-\gamma t} $ for certain $\gamma > 0,$ by means of a finite dimensional controller in the…
Motivated by engineering applications of subsea installation by deepwater construction vessels in oil drilling, and of aid delivery by unmanned aerial vehicles in disaster relief, we develop output-feedback boundary control of…
We propose novel less diffusive schemes for conservative one- and two-dimensional hyperbolic systems of nonlinear partial differential equations (PDEs). The main challenges in the development of accurate and robust numerical methods for the…
A system of partial differential equations (PDEs) is derived to compute the full-field stress from an observed kinematic field when the flow rule governing the plastic deformation is unknown. These equations generalize previously proposed…
The focal point of this paper is to theoretically investigate and numerically validate the effect of time delay on the exponential stabilization of a class of coupled hyperbolic systems with delayed and non-delayed dampings. The class in…
Systems modeled by partial differential equations (PDEs) are at least as ubiquitous as systems that are by nature finite-dimensional and modeled by ordinary differential equations (ODEs). And yet, systematic and readily usable…
In this work we study various continuous finite element discretization for two dimensional hyperbolic partial differential equations, varying the polynomial space (Lagrangian on equispaced, Lagrangian on quadrature points (Cubature) and…
This paper presents a control-oriented delay-based modeling approach for the exponential stabilization of a scalar neutral functional differential equation, which is then applied to the local exponential stabilization of a one-layer neural…
Stochastic differential equations (SDE) often exhibit large random transitions. This property, which we denote as pathwise stiffness, causes transient bursts of stiffness which limit the allowed step size for common fixed time step explicit…
Forecasting physical signals in long time range is among the most challenging tasks in Partial Differential Equations (PDEs) research. To circumvent limitations of traditional solvers, many different Deep Learning methods have been…
These notes are issued from a short course given by the author in a summer school in Chamb{\'e}ry in June 2015. We consider general semilinear PDE's and we address the following two questions: 1) How to design an efficient feedback control…
We present a nonlinear dynamical approximation method for time-dependent Partial Differential Equations (PDEs). The approach makes use of parametrized decoder functions, and provides a general, and principled way of understanding and…
In this work, we present the Domain of Dependence (DoD) stabilization for systems of hyperbolic conservation laws in one space dimension. The base scheme uses a method of lines approach consisting of a discontinuous Galerkin scheme in space…
This paper is devoted to a simple and new proof on the optimal finite control time for general linear coupled hyperbolic system by using boundary feedback on one side. The feedback control law is designed by first using a Volterra…
Physical processes evolving in both time and space are often modeled using Partial Differential Equations (PDEs). Recently, it has been shown how stability analysis and control of coupled PDEs in a single spatial variable can be more…
High-dimensional partial differential equations (PDEs) are ubiquitous in economics, science and engineering. However, their numerical treatment poses formidable challenges since traditional grid-based methods tend to be frustrated by the…
In this paper, we solve the problem of exponential stabilization for a class of cascade ODE-PDE system governed by a linear ordinary differential equation and a 1-d linearized Korteweg-de Vries equation (KdV) posed on a bounded interval.…
Metastability appears when a thermodynamic system, such as supercooled water (which is liquid below freezing temperature), lands on the "wrong" side of a phase transition, and remains for a very long time in a state different from its…