Eduardo Cerpa

Boundary Stabilization of a 1-D Wave Equation with In-Domain Antidamping

We consider the problem of boundary stabilization of a 1-D (one-dimensional) wave equation with an internal spatially varying antidamping term. This term puts all the eigenvalues of the open-loop system in the right half of the complex plane. We design a feedback law based on the backstepping method and prove exponential stability of the closed-loop system with a desired decay rate. For plants with constant parameters the control gains are found in closed form. Our design also produces a new Lyapunov function for the classical wave equation with passive boundary damping.

Exact Controllability of a Nonlinear Korteweg-de Vries Equation on a Critical Spatial Domain

We consider the boundary controllability problem for a nonlinear Korteweg-de Vries equation with the Dirichlet boundary condition. We study this problem for a spatial domain with a critical length for which the linearized control system is not controllable. In order to deal with the nonlinearity, we use a power series expansion of second order. We prove that the nonlinear term gives the local exact controllability around the origin provided that the time of control is large enough.

Rapid Stabilization for a Korteweg-de Vries Equation From the Left Dirichlet Boundary Condition

This paper deals with the stabilization problem for the Korteweg-de Vries equation posed on a bounded interval. The control acts on the left Dirichlet boundary condition. At the right end-point, Dirichlet and Neumann homogeneous boundary conditions are considered. The proposed feedback law forces the exponential decay of the system under a smallness condition on the initial data. Moreover, the decay rate can be tuned to be as large as desired. The feedback control law is designed by using the backstepping method.

Lipschitz stability in an inverse problem for the Kuramoto-Sivashinsky equation

In this article, we present an inverse problem for the nonlinear 1D Kuramoto–Sivashinsky (KS) equation. More precisely, we study the nonlinear inverse problem of retrieving the anti-diffusion coefficient from the measurements of the solution on a part of the boundary and also at some positive time in the whole space domain. The Lipschitz stability for this inverse problem is our main result and it relies on the Bukhgeĭm–Klibanov method. The proof is indeed based on a global Carleman estimate for the linearized KS equation.

Output feedback control of the linear Korteweg-de Vries equation

This paper presents the design of an output feedback control for a linear Korteweg-de Vries equation. This design is based on the backstepping method which uses a Volterra transformation. An appropriate observer is introduced and the exponential stability of the closed-loop system is proven.

Stabilization of a linear Korteweg-de Vries equation with a saturated internal control

This article deals with the design of saturated controls in the context of partial differential equations. It is focused on a linear Korteweg-de Vries equation, which is a mathematical model of waves on shallow water surfaces. In this article, we close the loop with a saturating input that renders the equation nonlinear. The well-posedness is proven thanks to the nonlinear semigroup theory. The proof of the asymptotic stability of the closed-loop system uses a Lyapunov function.

On the determination of the principal coefficient from boundary measurements in a KdV equation

Abstract This paper concerns the inverse problem of retrieving the principal coefficient in a Korteweg–de Vries (KdV) equation from boundary measurements of a single solution. The Lipschitz stability of this inverse problem is obtained using a new global Carleman estimate for the linearized KdV equation. The proof is based on the Bukhgeĭm–Klibanov method.

Global stabilization of a Korteweg-de Vries equation with saturating distributed control

This article deals with the design of saturated controls in the context of partial differential equations. It focuses on a Korteweg–de Vries equation, which is a nonlinear mathematical model of waves on shallow water surfaces. Two different types of saturated controls are considered. The well-posedness is proven applying a Banach fixed-point theorem, using some estimates of this equation and some properties of the saturation function. The proof of the asymptotic stability of the closed-loop system is separated in two cases: (i) when the control acts on all the domain, a Lyapunov function together with a sector condition describing the saturating input is used to conclude on the stability; (ii) when the control is localized, we argue by contradiction. Some numerical simulations illustrate the stability of the closed-loop nonlinear partial differential equation.

Null Controllability of the Stabilized Kuramoto--Sivashinsky System with One Distributed Control

This paper presents a control problem for a one-dimensional nonlinear parabolic system, which consists of a Kuramoto--Sivashinsky--Korteweg de Vries equation coupled to a heat equation. We address the problem of controllability by means of a control supported in an interior open subset of the domain and acting on one equation only. The local null-controllability of the system is proved. The proof is based on a Carleman estimate for the linearized system around the origin. A local inversion theorem is applied to get the result for the nonlinear system.

Backstepping observer based-control for an anti-damped boundary wave PDE in presence of in-domain viscous damping

This paper presents a backstepping control design for a one-dimensional wave PDE with in-domain viscous damping, subject to a dynamical anti-damped boundary condition. Its main contribution is the design of an observer-based control law which stabilizes the wave PDE velocity, using only boundary mesurements. Numerical simulations on an oil-inspired example show the relevance of our result and illustrate the merits of this control design.