Output feedback stabilization of the Korteweg-de Vries equation

International audienceThis paper presents an output feedback control law for the Korteweg-de Vries equation. The control design is based on the backstepping method and the introduction of an appropriate observer. The local exponential stability of the closed-loop system is proven. Some numerical simulations are shown to illustrate this theoretical result

Applications of Mathematical Models in Engineering

The most influential research topic in the twenty-first century seems to be mathematics, as it generates innovation in a wide range of research fields. It supports all engineering fields, but also areas such as medicine, healthcare, business, etc. Therefore, the intention of this Special Issue is to deal with mathematical works related to engineering and multidisciplinary problems. Modern developments in theoretical and applied science have widely depended our knowledge of the derivatives and integrals of the fractional order appearing in engineering practices. Therefore, one goal of this Special Issue is to focus on recent achievements and future challenges in the theory and applications of fractional calculus in engineering sciences. The special issue included some original research articles that address significant issues and contribute towards the development of new concepts, methodologies, applications, trends and knowledge in mathematics. Potential topics include, but are not limited to, the following: Fractional mathematical models; Computational methods for the fractional PDEs in engineering; New mathematical approaches, innovations and challenges in biotechnologies and biomedicine; Applied mathematics; Engineering research based on advanced mathematical tools

Reduced order framework for optimal control of nonlinear partial differential equations: ROM-based optimal flow control

A variety of partial differential equations (PDE) can govern the spatial and time evolution of fluid flows; however, direct numerical simulation (DNS) of the Euler or Navier-Stokes equation or other traditional computational fluid dynamics (CFD) models can be computationally expensive and intractable. An alternative is to use model order reduction techniques, e.g., reduced order models (ROM) via proper orthogonal decomposition (POD) or dynamic mode decomposition (DMD), to reduce the dimensionality of these nonlinear dynamical systems while still retaining the essential physics. The objective of this work is to design a reduced order numerical framework for effective simulation and control of complex flow phenomena. To build our computational method with this philosophy, we first simulate the 1D Burgers' equation ut + uux ? ?uxx = f(x, t), a well-known PDE modeling nonlinear advection-diffusion flow physics and shock waves, as a full order high resolution benchmark. We then apply canonical reduction approaches incorporating Fourier and POD modes with a Galerkin projection to approximate the solution to the posed initial boundary value problem. The control objective is simple: we seek the optimal (pointwise) input into the system that forces the spatial evolution of the PDE solution to converge to a preselected target state uT(x) at some final time T > 0. To implement an iterative control loop, we parametrize the unknown control function as a truncated Fourier series defined via a set of finite parameters. The performance of the POD ROM is compared to that of the Fourier ROM and full order model for six numerical experiments