1-d Wave Equations Coupled via Viscoelastic Springs and Masses: Boundary Controllability of a Quasilinear and Exponential Stabilizability of a Linear Model - Leugering, Günter; Li, Tatsien; Wang, Yue (Springer INdAM Series, 2019)
We consider the out-of-the-plane displacements of nonlinear elastic strings which are coupled through point masses attached to the ends and viscoelastic springs. We provide the modeling, the well-posedness in the sense of classical semi-global C2 -solutions together with some extra regularity at the masses and then prove exact boundary controllability and velocity-feedback stabilizability, where controls act on both sides of the mass-spring-coupling.
CRIS-Team Scopus Importer:2019-07-19
Keywords: Coupled system of quasilinear wave equations; Dynamical boundary condition; Exact boundary controllability; Visoelastic springs

A NOTE on WEIGHTED SOBOLEV SPACES RELATED to WEAKLY and STRONGLY DEGENERATE DIFFERENTIAL OPERATORS - Kogut, Peter I.; Kupenko, Olha P.; Leugering, Günter; Wang, Yue (Journal of Optimization, Differential Equations and Their Applications, 2019)
In this paper we discuss some issues related to Poincaré’s inequality for a special class of weighted Sobolev spaces. A common feature of these spaces is that they can be naturally associated with differential operators with variable diffusion coefficients that are not uniformly elliptic. We give a classification of these spaces in the 1-D case bases on a measure of degeneracy of the corresponding weight coefficient and study their key properties.
CRIS-Team Scopus Importer:2021-04-20
Keywords: Degenerate equation; Poincaré’s inequality; Weighted Sobolev spaces

Boundary controllability of a system modelling a partially immersed obstacle - Vergara-Hermosilla, G.; Leugering, Günter; Wang, Yue (Esaim-Control Optimisation and Calculus of Variations, 2021)
In this paper, we address the problem of boundary controllability for the one-dimensional nonlinear shallow water system, describing the free surface flow of water as well as the flow under a fixed gate structure. The system of differential equations considered can be interpreted as a simplified model of a particular type of wave energy device converter called oscillating water column. The physical requirements naturally lead to the problem of exact controllability in a prescribed region. In particular, we use the concept of nodal profile controllability in which at a given point (the node) time-dependent profiles for the states are required to be reachable by boundary controls. By rewriting the system into a hyperbolic system with nonlocal boundary conditions, we at first establish the semi-global classical solutions of the system, then get the local controllability and nodal profile using a constructive method. In addition, based on this constructive process, we provide an algorithmic concept to calculate the required boundary control function for generating a solution for solving these control problem.
CRIS-Team WoS Importer:2021-08-06

Controllability properties of a hyperbolic system with dynamic boundary conditions - Leugering, Günter; Micu, Sorin; Roventa, Ionel; Wang, Yue (Journal of Evolution Equations, 2022)
In this article, we consider a system consisting of two elastic strings with attached tip masses coupled through an elastic spring. Our aim is to analyze its exact boundary controllability properties and to characterize the spaces of controllable initial data depending on the number of controls acting on the boundaries of the strings. We show that singularities in waves are "smoothed by three orders" as they crass a point mass. Consequently, when only one control acts on the extremity of the first string, the space of controlled initial data is asymmetric, the components corresponding to the second string having to be more regular than those corresponding to the first one. Roughly speaking, if the initial data for the string which is directly controlled can be in L-2 x H-1, they should be at least in H-3 x H-2 for the second string, located on the other part of the masses.
CRIS-Team WoS Importer:2022-08-05

Exact boundary controllability and its applications for a coupled system of quasilinear wave equations with dynamical boundary conditions - Wang, Yue; Leugering, Günter; Li, Tatsien (Nonlinear Analysis-Real World Applications, 2019)
A constructive method with modular structure is used to obtain the local exact boundary controllability for a coupled system of quasilinear wave equations with local or non-local dynamical boundary conditions. Some applications are given for the chain-like or star-like system of strings coupled via elastic springs, and of strings coupled via Maxwell-type and Kelvin-type viscoelastic springs, respectively.
CRIS-Team Scopus Importer:2019-05-09
Keywords: Constructive method; Coupled system of quasilinear wave equations; Dynamical boundary condition; Exact boundary controllability; Hidden regularity

Exact boundary controllability for 1-D quasilinear wave equations with dynamical boundary conditions - Wang, Yue; Leugering, Günter; Li, Tatsien (Mathematical Methods in the Applied Sciences, 2016)
By equivalently replacing the dynamical boundary condition by a kind of nonlocal boundary conditions, and noting a hidden regularity of solution on the boundary with a dynamical boundary condition, a constructive method with modular structure is used to get the local exact boundary controllability for 1-D quasilinear wave equations with dynamical boundary condition
Keywords: Dynamical boundary condition; Exact boundary controllability; Quasilinear wave equation; Semi-global C2 solution

Exact Boundary Controllability for the Spatial Vibration of String with Dynamical Boundary Conditions - Wang, Yue; Leugering, Günter; Li, Tatsien (Chinese Annals of Mathematics Series B, 2020)
This paper deals with the spatial vibration of an elastic string with masses at the endpoints. The authors derive the corresponding quasilinear wave equation with dynamical boundary conditions, and prove the exact boundary controllability of this system by means of a constructive method with modular structure.
CRIS-Team Scopus Importer:2020-05-22
Keywords: 35L05; 35L72; 93B05; Dynamical boundary condition; Exact boundary controllability; Spatial vibration of a string

Exact boundary controllability of partial nodal profile for network of strings - Wang, Yue; Li, Tatsien (Nonlinear Analysis-Real World Applications, 2021)
Based on the theory of exact boundary controllability of nodal profile for hyperbolic systems, the authors propose the concept of exact boundary controllability of partial nodal profile to expand the scope of applications. With the new concept, we can shorten the controllability time, save the number of controls, and increase the number of charged nodes with given nodal profiles. Furthermore, we introduce the concept of in-situ controlled node to deal with a new situation that one node can be charged and controlled simultaneously. The minimum number of boundary controls on the entire tree-like network is determined by using the concept of ‘degree of freedom of charged nodes’ introduced. And the concept of ‘control path’ is introduced to appropriately divide the network, so that we can determine the infimum of controllability time. General frameworks of constructive proof are given on a single interval, a star-like network, a chain-like network and a planar tree-like network for linear wave equation(s) with Dirichlet, Neumann, Robin and dissipative boundary conditions to establish a complete theory on the exact boundary controllability of partial nodal profile.
CRIS-Team Scopus Importer:2021-07-30
Keywords: Control path; Exact boundary controllability; Partial nodal profile; Piecewise C; Planar tree-like network; The degree of freedom of charged node

Nodal profile control for networks of geometrically exact beams - Leugering, Günter; Rodriguez, Charlotte; Wang, Yue (Journal De Mathematiques Pures Et Appliquees, 2021)
In this work, we consider networks of so-called geometrically exact beams, namely, shearable beams that may undergo large motions. The corresponding mathematical model, commonly written in terms of displacements and rotations expressed in a fixed basis (Geometrically Exact Beam model, or GEB), has a quasilinear governing system. However, the model may also be written in terms of intrinsic variables expressed in a moving basis attached to the beam (Intrinsic GEB model, or IGEB) and while the number of equations is then doubled, the latter model has the advantage of being of first-order, hyperbolic and only semilinear. First, for any network, we show the existence and uniqueness of semi-global in time classical solutions to the IGEB model (i.e., for arbitrarily large time intervals, provided that the data are small enough). Then, for a specific network containing a cycle, we address the problem of local exact controllability of nodal profiles for the IGEB model – we steer the solution to satisfy given profiles at one of the multiple nodes by means of controls applied at the simple nodes – by using the constructive method of Zhuang, Leugering and Li (2018) [52]. Afterwards, for any network, we show that the existence of a unique classical solution to the IGEB network implies the same for the corresponding GEB network, by using that these two models are related by a nonlinear transformation. In particular, this allows us to give corresponding existence, uniqueness and controllability results for the GEB network.
CRIS-Team Scopus Importer:2021-08-13
Keywords: Geometrically exact beam; Networks; Nodal profile controllability; Well-posedness