Hyperbolic systems with vanishing eigenvalues can also be stabilized

Significance 

Differential equations are important in engineering and science. They describe various natural or social phenomena, are mathematical formulations of physical laws governing the phenomena, and summarize our understanding of the phenomena. With differential equations, the control of many phenomena becomes feasible and workable. For many years, control has been one of fundamental issues in the mathematical study of differential equations.

Among various differential equations, hyperbolic systems of first-order partial differential equations with relaxation (also called hyperbolic systems of balance laws) are of extreme importance. They model many different non-equilibrium processes ranging from the axonal transport in neuroscience to the motion of dissipative relativistic fluids.

Recently, mathematical and engineering works have majored mainly on the control of hyperbolic partial differential equations. This can be attributed to its wide applications in areas such as heat exchanges and traffic flows. Several boundary feedback controls have been designed for certain hyperbolic systems of balance laws. These published works show that the controllability can be realized through boundary controls. Unfortunately, all these earlier works assume that the hyperbolic systems under consideration have no vanishing eigenvalues. In other words, the existing results are valid only for non-characteristic boundaries. On the other hand, characteristic boundaries are of more frequent occurrence and do arise in some applications such as the transport of neurofilaments in axons. Therefore, finding control laws at characteristic boundaries for hyperbolic systems with relaxation is highly desirable.

To this note, Professor Wen-An Yong from Tsinghua University investigated the boundary control of a class of one-dimensional linear hyperbolic systems of balance laws with characteristic boundaries. The class is characterized with a structural stability condition found by the author in the beginning of 1990’s. In the past, the stability condition has been shown to be satisfied by many many well-validated partial differential equations from various different fields in mathematical physics. It roots deeply in non-equilibrium thermodynamics. The main aim of the present work was to explicitly use the structural stability condition to prove exponential stability. The research work is currently published in the journal, Automatica.

In brief, the author assumed that the hyperbolic systems fulfill his structural stability condition. This assumption is meaning because of the relevance of the stability condition in significantly characterizing the common properties of various classical models. Under this assumption, he introduced a new and simple Lyapunov function which enables him to establish the exponential stability for the case with characteristic boundaries. The results show that the system with vanishing eigenvalues could also be stabilized by using the feedback boundary controls.

Professor Wen-An Yong is the first to utilize the structural boundary condition to prove the exponential stability for the case with characteristic boundaries. To actualize the study, he applied the obtained results to an earlier developed model for the transport of neurofilaments in axons. Interestingly, it proved suitable for exploring the various connections between the brain and the associated nerve cells. This was attribute to the fact that the model consists of five first-order partial differential equation in one dimension together with three vanishing eigenvalues and satisfies the structural stability condition. The structural stability condition will be of great importance in studying fundamental properties of various classical models.

About the author

Professor Wen-An Yong received his Dr. rer. nat. and Habilitation in mathematics, both from University of Heidelberg, Germany. He is presently a professor in the Department of Mathematical Sciences at Tsinghua University, China. His main research interests include hyperbolic PDEs and non-equilibrium thermodynamics/mathematical modeling with applications to diverse scientific disciplines: kinetic theory, biochemical kinetics, computational fluid dynamics, geophysics, chemical engineering, viscoelastic fluids, and boundary control of hyperbolic PDEs.

Professor YONG was known to the international hyperbolic PDE community for two of his works. One is his simple proof of the Glimm’s interaction estimates. The proof has been adopted in several books published by Springer Verlag. Another is his “systematic description of the fundamental properties” of hyperbolic relaxation systems via four types of conditions: Relaxation Criterion, Structural Stability Condition, Entropy-production Inequality, and Conservation-dissipation Principle. Under the conditions, he systematically studied the stability of zero-relaxation limit, global existence and long-time behaviors of smooth solutions, existence of shock structures, validity of the Chapman-Enskog expansion, and feedback boundary control, clarified the formulation of boundary conditions and derived the corresponding reduced boundary conditions.

A few years ago, he realized that the four types of conditions above could be well regarded as stability criteria for non-equilibrium thermodynamics and as a common basis of many different fields. Indeed, the Conservation-dissipation Principle can be understood as a nonlinearization of the celebrated Onsager’s reciprocal relation in modern thermodynamics. Specifically, he proved the stability (and convergence) of the lattice Boltzmann method (LBM), together with his young collaborators he proposed a conservation-dissipation formalism (CDF) for non-equilibrium thermodynamics, and a system of PDEs he proposed for compressible viscoelastic fluid flows has been verified to be the unique correct model among the available ones.

Reference

Yong, W. (2019). Boundary stabilization of hyperbolic balance laws with characteristic boundaries. Automatica, 101, 252-257.

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