arXiv subjects

Günter Leugering

Publications and source records attributed to Günter Leugering.

9 recordsLinked to original sources

arXiv2025

Exact boundary controllability and stabilizability of a degenerated Timoshenko beam

This paper investigates the boundary controllability and stabilizability of a Timoshenko beam subject to degeneracy at one end, while control is applied at the opposite boundary. Degeneracy in this context is measured by the real parameters for $μ_a\in [0,2)$ for $a\in\{K,EI\}$, where $K(x)$ denotes shear stiffness and $EI(x)$ bending stiffness. We differentiate between weak degeneracy $μ_a\in [0,1)$ and strong degeneracy $μ_a\in [1,2)$, which may occur independently in shear and bending. Our study establishes observability inequalities for both weakly and strongly degenerate equations under Dirichlet, Robin, and Neumann boundary conditions. Using energy multiplier techniques and the Hilbert Uniqueness Method (HUM), we derive conditions for exact boundary controllability and show that appropriate boundary state and velocity feedback controls at the non-degenerate end can stabilize the system exponentially. Extending results previously obtained for the 1-dimensional wave equation in \cite{AlabauCannarsaLeugering2017}, this study highlights new control strategies and stabilization effects specific to the degenerate Timoshenko beam system, addressing challenges pertinent to real-world structural damping and control applications.

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arXiv2025

Asymmetric exact controllability for networks of spatial elastic strings, springs and masses

We consider networks of elastic strings with end masses, where the coupling is modeled via elastic springs. The model is representative of a network of nonlinear strings, where the strings are coupled to elastic bodies. The coupled system converges to the classical string network model with Kirchhoff and continuity transmission conditions as the spring stiffness terms approach infinity and the masses at the nodes vanish. Due to the presence of point masses at the nodes, the boundary conditions become dynamic, and consequently, the corresponding first-order system of quasilinear balance laws exhibits nonlocal boundary conditions. We demonstrate well-posedness in the sense of semi-global classical solutions \cite{li} (i.e., for arbitrarily large time intervals provided that the initial and boundary data are small enough) and observe extra regularity at the masses as in \cite{WangLeugeringLi2017,WangLeugeringLi2019}. We prove local and global-local exact boundary controllability of a star-like network when control is active at the endpoints of the string-spring-mass system, except for one clamped end. In this case, at multiple nodes, a complex smoothing pattern appears, leading to asymmetric control spaces when the springs and masses are present. Furthermore, the rank of the Laplacian matrix at the junction is crucial for the controllability property, particularly in models containing wave equations with degeneration at dynamic boundaries, which can be interpreted as damage in mechanical vibration systems where parts of the springs are missing.

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arXiv2021

Optimisation of Region of Attraction Estimates for the Exponential Stabilisation of the Intrinsic Geometrically Exact Beam Model

A systematic approach to maximise estimates on the region of attraction in the exponential stabilisation of geometrically exact (nonlinear) beam models via boundary feedback is presented. Starting from recently established stability results based on Lyapunov arguments, the main contribution of the presented work is to maximise the analytically found bounds on the initial datum, for which local exponential stability is guaranteed, via search of (optimal) polynomial Lyapunov functionals using an iterative semi-definite programming approach.

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arXiv2021

Nodal profile control for networks of geometrically exact beams

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 [Exact boundary controllability of nodal profile for Saint-Venant system on a network with loops, in J. Math. Pures Appl., 2018]. 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.

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arXiv2021

On Boundary Exact Controllability of One-Dimensional Wave Equations with Weak and Strong Interior Degeneration

In this paper we study exact boundary controllability for a linear wave equation with strong and weak interior degeneration of the coefficient in the principle part of the elliptic operator. The objective is to provide a well-posedness analysis of the corresponding system and derive conditions for its controllability through boundary actions. Passing to a relaxed version of the original problem, we discuss existence and uniqueness of solutions, and using the HUM method we derive conditions on the rate of degeneracy for both exact boundary controllability and the lack thereof.

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arXiv2020

Boundary feedback stabilization for the intrinsic geometrically exact beam model

In this work we address the problem of boundary feedback stabilization for a geometrically exact shearable beam, allowing for large deflections and rotations and small strains. The corresponding mathematical model may be written in terms of displacements and rotations (GEB), or intrinsic variables (IGEB). A nonlinear transformation relates both models, allowing to take advantage of the fact that the latter model is a one-dimensional first-order semilinear hyperbolic system, and deduce stability properties for both models. By applying boundary feedback controls at one end of the beam, while the other end is clamped, we show that the zero steady state of IGEB is locally exponentially stable for the $H^1$ and $H^2$ norms. The proof rests on the construction of a Lyapunov function, where the theory of Coron \& Bastin '16 plays a crucial role. The major difficulty in applying this theory stems from the complicated nature of the nonlinearity and lower order term where no smallness arguments apply. Using the relationship between both models, we deduce the existence of a unique solution to the GEB model, and properties of this solution as time goes to $+\infty$.

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arXiv2016

Neumann boundary feedback stabilization for a nonlinear wave equation: A strict $H^2$-Lyapunov function

For a system that is governed by the isothermal Euler equations with friction for ideal gas, the corresponding field of characteristic curves is determined by the velocity of the flow. This velocity is determined by a second-order quasilinear hyperbolic equation. For the corresponding initial-boundary value problem with Neumann-boundary feedback, we consider non-stationary solutions locally around a stationary state on a finite time interval and discuss the well-posedness of this kind of problem. We introduce a strict $H^2$-Lyapunov function and show that the boundary feedback constant can be chosen such that the $H^2$-Lyapunov function and hence also the $H^2$-norm of the difference between the non-stationary and the stationary state decays exponentially with time.

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arXiv2016

A shape-topological control problem for nonlinear crack - defect interaction: the anti-plane variational model

We consider the shape-topological control of a singularly perturbed variational inequality. The geometry-dependent state problem that we address in this paper concerns a heterogeneous medium with a micro-object (defect) and a macro-object (crack) modeled in 2d. The corresponding nonlinear optimization problem subject to inequality constraints at the crack is considered within a general variational framework. For the reason of asymptotic analysis, singular perturbation theory is applied resulting in the topological sensitivity of an objective function representing the release rate of the strain energy. In the vicinity of the nonlinear crack, the anti-plane strain energy release rate is expressed by means of the mode-III stress intensity factor, that is examined with respect to small defects like micro-cracks, holes, and inclusions of varying stiffness. The result of shape-topological control is useful either for arrests or rise of crack growth.

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arXiv2015

Control and stabilization of degenerate wave equations

We study a wave equation in one space dimension with a general diffusion coefficient which degenerates on part of the boundary. Degeneracy is measured by a real parameter $μ_a>0$. We establish observability inequalities for weakly (when $μ_a \in [0,1[$) as well as strongly (when $μ_a \in [1,2[$) degenerate equations. We also prove a negative result when the diffusion coefficient degenerates too violently (i.e. when $μ_a>2$) and the blow-up of the observability time when $μ_a$ converges to $2$ from below. Thus, using the HUM method we deduce the exact controllability of the corresponding degenerate control problem when $μ_a \in [0,2[$. We conclude the paper by studying the boundary stabilization of the degenerate linearly damped wave equation and show that a suitable boundary feedback stabilizes the system exponentially. We extend this stability analysis to the degenerate nonlinearly boundary damped wave equation, for an arbitrarily growing nonlinear feedback close to the origin. This analysis proves that the degeneracy does not affect the optimal energy decay rates at large time. We apply the optimal-weight convexity method of \cite{alaamo2005, alajde2010} together with the results of the previous section, to perform this stability analysis.

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