Advances in Control and Numerical Methods for Hyperbolic Systems

The recent developments in the research area of hyperbolic systems and their control, discretization, and numerical solutions have shown significant advancements. The field is moving towards more flexible and robust control strategies, particularly through the introduction of dynamic extensions in backstepping control. These extensions enable the design of state feedback controllers that can achieve goals beyond static feedback, such as complete input-output decoupling and internal stability. Additionally, there is a growing focus on developing monolithic limiting techniques for enriched Galerkin methods to enforce nonlinear stability constraints, ensuring local mass conservation and entropy stability. The use of mapped coercivity in analyzing nonlinear operators and finite element discretizations is also emerging as a robust tool, offering a unifying framework for stability and existence proofs in Banach spaces. Furthermore, the integration of Virtual Element Method (VEM) concepts with hyperbolic problems, such as the PAMPA algorithm, demonstrates the potential for high-order accuracy and robustness in solving complex problems on general polygonal grids. The introduction of cycle-free mesh sweeping algorithms for Boltzmann Transport and the theoretical study of Rusanov-type schemes' wave-speed estimates, monotonicity, and stability are also notable contributions. These developments collectively push the boundaries of what can be achieved in the control and numerical solution of hyperbolic systems, emphasizing innovation and robustness in computational methods.

Noteworthy papers include:

  • The paper on dynamic extensions for backstepping control introduces a modular design that allows for straightforward transfer of results to hyperbolic PDE-ODE systems.
  • The study on bound-preserving and entropy stable enriched Galerkin methods presents a novel approach to enforcing nonlinear stability constraints, with demonstrated effectiveness in numerical experiments.
  • The work on mapped coercivity provides a robust framework for analyzing nonlinear PDEs and their discretizations, bypassing traditional decompositions.

Sources

Using dynamic extensions for the backstepping control of hyperbolic systems

Bound-preserving and entropy stable enriched Galerkin methods for nonlinear hyperbolic equations

Mapped coercivity for the stationary Navier-Stokes equations and their finite element approximations

Virtual finite element and hyperbolic problems: the PAMPA algorithm

Cycle-Free Polytopal Mesh Sweeping for Boltzmann Transport

Rusanov-type Schemes for Hyperbolic Equations: Wave-Speed Estimates, Monotonicity and Stability

A Novel and Simple Invariant-Domain-Preserving Framework for PAMPA Scheme: 1D Case

A Space-Time Discontinuous Petrov-Galerkin Finite Element Formulation for a Modified Schr\"odinger Equation for Laser Pulse Propagation in Waveguides

Pathwise uniform convergence of numerical approximations for a two-dimensional stochastic Navier-Stokes equation with no-slip boundary conditions

A numerical method for solving the generalized tangent vector of hyperbolic systems

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