This monograph explores the modeling of conservation and balance laws of one-dimensional hyperbolic systems using partial differential equations. It presents typical examples of hyperbolic systems for a wide range of physical engineering applications, allowing readers to understand the concepts in whichever setting is most familiar to them. With these examples, it also illustrates how control boundary conditions may be defined for the most commonly used control devices.
The authors begin with the simple case of systems of two linear conservation laws and then consider the stability of systems under more general boundary conditions that may be differential, nonlinear, or switching. They then extend their discussion to the case of nonlinear conservation laws and demonstrate the use of Lyapunov functions in this type of analysis. Systems of balance laws are considered next, starting with the linear variety before they move on to more general cases of nonlinear ones. They go on to show how the problem of boundary stabilization of systems of two balance laws by both full-state and dynamic output feedback in observer-controller form is solved by using a “backstepping” method, in which the gains of the feedback laws are solutions of an associated system of linear hyperbolic PDEs. The final chapter presents a case study on the control of navigable rivers to emphasize the main technological features that may occur in real live applications of boundary feedback control.
Stability and Boundary Stabilization of 1-D Hyperbolic Systems will be of interest to graduate students and researchers in applied mathematics and control engineering. The wide range of applications it discusses will help it to have as broad an appeal within these groups as possible.

Stability and Boundary Stabilization of 1-D Hyperbolic Systems

This study presents a fair performance comparison of the continuous finite element method, the symmetric interior penalty discontinuous Galerkin method, and the hybridized discontinuous Galerkin (H...

A Performance Comparison of Continuous and Discontinuous Galerkin Methods with Fast Multigrid Solvers

We present an algorithmic framework for matrix-free evaluation of discontinuous Galerkin finite element operators based on sum factorization on quadrilateral and hexahedral meshes. We identify a set of kernels for fast quadrature on cells and faces targeting a wide class of weak forms originating from linear and nonlinear partial differential equations. Different algorithms and data structures for the implementation of operator evaluation are compared in an in-depth performance analysis. The sum factorization kernels are optimized by vectorization over several cells and faces and an even-odd decomposition of the one-dimensional compute kernels. In isolation our implementation then reaches up to 60\% of arithmetic peak on Intel Haswell and Broadwell processors and up to 50\% of arithmetic peak on Intel Knights Landing. The full operator evaluation reaches only about half that throughput due to memory bandwidth limitations from loading the input and output vectors, MPI ghost exchange, as well as handling variable coefficients and the geometry. Our performance analysis shows that the results are often within 10\% of the available memory bandwidth for the proposed implementation, with the exception of the Cartesian mesh case where the cost of gather operations and MPI communication are more substantial.

Fast matrix-free evaluation of discontinuous Galerkin finite element operators

In this paper, we use swapping design filters to bring systems of $n+1$ partial differential equations of the hyperbolic type to static form. Standard parameter identification laws can then be applied to estimate unknown parameters in the boundary conditions. Proof of boundedness of the adaptive laws are offered, and the results are demonstrated in simulations.

An Adaptive Observer Design for $n+1$ Coupled Linear Hyperbolic PDEs Based on Swapping

We present an algorithmic framework for matrix-free evaluation of discontinuous Galerkin finite element operators. It relies on fast quadrature with sum factorization on quadrilateral and hexahedral meshes, targeting general weak forms of linear and nonlinear partial differential equations. Different algorithms and data structures are compared in an in-depth performance analysis. The implementations of the local integrals are optimized by vectorization over several cells and faces and an even-odd decomposition of the one-dimensional interpolations. Up to 60% of the arithmetic peak on Intel Haswell, Broadwell, and Knights Landing processors is reached when running from caches and up to 40% of peak when also considering the access to vectors from main memory. On 2×14 Broadwell cores, the throughput is up to 2.2 billion unknowns per second for the 3D Laplacian and up to 4 billion unknowns per second for the 3D advection on affine geometries, close to a simple copy operation at 4.7 billion unknowns per second. Our experiments show that MPI ghost exchange has a considerable impact on performance and we present strategies to mitigate this effect. Finally, various options for evaluating geometry terms and their performance are discussed. Our implementations are publicly available through the deal.II finite element library.

https://dl.acm.org/doi/pdf/10.1145/3325864

Fast Matrix-Free Evaluation of Discontinuous Galerkin Finite Element Operators

Backstepping stabilization of the linearized Saint-Venant–Exner model☆

This work presents a parallel finite element solver of incompressible two-phase flow targeting large-scale simulations of three-dimensional dynamics in high-throughput microfluidic separation devic...

A fast massively parallel two-phase flow solver for microfluidic chip simulation

Using the backstepping design, we achieve exponential stabilization of the coupled Saint-Venant-Exner (SVE) PDE model of water dynamics in a sediment-filled canal with arbitrary values of canal bottom slope, friction, porosity, and water-sediment interaction under subcritical or supercritical flow regime. The studied SVE model consists of two rightward convecting transport Partial Differential Equations (PDEs) and one leftward convecting transport PDE. A single boundary input control (with actuation located only at downstream) strategy is adopted. A full state feedback controller is firstly designed, which guarantees the exponential stability of the closed-loop control system. Then, an output feedback controller is designed based on the reconstruction of the distributed state with a backstepping observer. It also guarantees the exponential stability of the closed-loop control system. The flow regime depends on the dimensionless Froude number Fr, and both our controllers can deal with the subcritical (Fr 1) flow regime. They achieve the exponential stability results without any restrictive conditions in contrast to existing results.

Backstepping Stabilization of the Linearized Saint-Venant-Exner Model

Control of shallow waves of two unmixed fluids by backstepping

This paper presents an algebraic method to design a linear feedback control for regulating the water flow in open channels. We deal with a hyperbolic system of partial differential equations describing the behavior of the water flow and the sediment transport. By using an a priori estimation techniques and the Faedo–Galerkin method, we build a stabilizing boundary control. This control law ensures a decrease of the energy and convergence of the controlled system.

Ababacar Diagne

Papers

Backstepping stabilization of the linearized Saint-Venant–Exner model☆

A fast massively parallel two-phase flow solver for microfluidic chip simulation

Backstepping Stabilization of the Linearized Saint-Venant-Exner Model

Control of shallow waves of two unmixed fluids by backstepping

Control of shallow water and sediment continuity coupled system