Papers by Günter Leugering
Asymptotic Analysis, Jan 7, 2019
We develop and investigate radiation conditions at infinity for composite piezo-elastic waveguide... more We develop and investigate radiation conditions at infinity for composite piezo-elastic waveguides. The approach is based on the Mandelstam radiation principle according to which the energy flux at infinity is directed away from the source and which implies constraints on the (sign of the) group velocities. On the other side, the Sommerfeld radiation condition implies limitations on the wave phase velocity and is, in fact, not applicable in the context of piezo-elastic wave guides. We analyze the passage to the limit when the piezo-electric moduli tend to zero in certain regions yielding purely elastic inclusions there. We provide a number of examples, e.g. elastic and acoustic waveguides as well as purely elastic insulating and conducting inclusions.
Control and Cybernetics, Jun 1, 2022
We study distributed optimal control problems, governed by space-time fractional parabolic equati... more We study distributed optimal control problems, governed by space-time fractional parabolic equations (STFPEs) involving time-fractional Caputo derivatives and spatial fractional derivatives of Sturm-Liouville type. We first prove existence and uniqueness of solutions of STFPEs on an open bounded interval and study their regularity. Then we show existence and uniqueness of solutions to a quadratic distributed optimal control problem. We derive an adjoint problem using the right-Caputo derivative in time and provide optimality conditions for the control problem. Moreover, we propose a finite difference scheme to find the approximate solution of the considered optimal control problem. In the proposed scheme, the well-known L1 method has been used to approximate the time-fractional Caputo derivative, while the spatial derivative is approximated using the Grünwald-Letnikov formula. Finally, we demonstrate the accuracy and the performance of the proposed difference scheme via examples.
arXiv (Cornell University), May 21, 2015
We study a wave equation in one space dimension with a general diffusion coefficient which degene... more 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 µa > 0. We establish observability inequalities for weakly (when µa ∈ [0, 1[) as well as strongly (when µa ∈ [1, 2[) degenerate equations. We also prove a negative result when the diffusion coefficient degenerates too violently (i.e. when µa > 2) and the blow-up of the observability time when µa converges to 2 from below. Thus, using the HUM method we deduce the exact controllability of the corresponding degenerate control problem when µa ∈ [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 [1, 2] together with the results of the previous section, to perform this stability analysis.
IFIP advances in information and communication technology, 2009
In this article, we present the Free Material Optimization (FMO) problem for plates and shells ba... more In this article, we present the Free Material Optimization (FMO) problem for plates and shells based on Naghdi's shell model. In FMO -a branch of structural optimization -we search for the ultimately best material properties in a given design domain loaded by a set of given forces. The optimization variable is the full material tensor at each point of the design domain. We give a basic formulation of the problem and prove existence of an optimal solution. Lagrange duality theory allows to identify the basic problem as the dual of an infinite-dimensional convex nonlinear semidefinite program. After discretization by the finite element method the latter problem can be solved using a nonlinear SDP code. The article is concluded by a few numerical studies.
Applied mathematics, 2017
We consider optimal control problems for the flow of gas in a pipe network. The equations of moti... more We consider optimal control problems for the flow of gas in a pipe network. The equations of motions are taken to be represented by a semi-linear model derived from the fully nonlinear isothermal Euler gas equations. We formulate an optimal control problem on a given network and introduce a time discretization thereof. We then study the well-posedness of the corresponding time-discrete optimal control problem. In order to further reduce the complexity, we consider an instantaneous control strategy. The main part of the paper is concerned with a non-overlapping domain decomposition of the semi-linear elliptic optimal control problem on the graph into local problems on a small part of the network, ultimately on a single edge.
arXiv (Cornell University), Nov 15, 2019
In this work we address the problem of boundary feedback stabilization for a geometrically exact ... more 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 +∞.
Mathematical Control & Related Fields, 2022
In the present paper we deal with parabolic fractional initial-boundary value problems of Sturm–L... more In the present paper we deal with parabolic fractional initial-boundary value problems of Sturm–Liouville type in an interval and in a general star graph. We first give several existence, uniqueness and regularity results of weak and very-weak solutions. We prove the existence and uniqueness of solutions to a quadratic boundary optimal control problem and provide a characterization of the optimal contol via the Euler–Lagrange first order optimality conditions. We then investigate the analogous problems for a fractional Sturm–Liouville problem in a general star graph with mixed Dirichlet and Neumann boundary controls. The existence and uniqueness of minimizers, and the characterization of the first order optimality conditions are obtained in a general star graph by using the method of Lagrange multipliers.
SIAM Journal on Control and Optimization, 2020
In this work we address the problem of boundary feedback stabilization for a geometrically exact ... more 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 +∞.
SIAM Journal on Control and Optimization, 2021
. In this article, we extend the time-domain decomposition method described by Lagnese and Leuger... more . In this article, we extend the time-domain decomposition method described by Lagnese and Leugering to semilinear optimal control problems for hyperbolic balance laws with spatio-temporal varying coe cients. We provide the design of the iterative method applied to the global rst-order optimality system, prove its convergence, and derive an a posteriori error estimate. The analysis is done entirely on the continuous level. A distinguishing feature of the method is that the decomposed optimality system can be interpreted as an optimality system of a local "virtual" optimal control problem. Thus, the iterative time-domain decomposition of the optimality system can be interpreted as an iterative parallel scheme for virtual optimal control problems on the subintervals. A typical example and further comments are given to show the range of potential applications. Moreover, we provide some numerical experiments to give a rst interpretation of the role of the parameters involved in the iterative process.
Springer Optimization and Its Applications, 2016
HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific r... more HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
SIAM Journal on Control and Optimization, 2020
HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific r... more HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Computational Optimization and Applications, 2017
We study the transient optimization of gas transport networks including both discrete controls du... more We study the transient optimization of gas transport networks including both discrete controls due to switching of controllable elements and nonlinear fluid dynamics described by the system of isothermal Euler equations, which are partial differential equations in time and 1-dimensional space. This combination leads to mixed-integer optimization problems subject to nonlinear hyperbolic partial differential equations on a graph. We propose an instantaneous control approach in which suitable Euler discretizations yield systems of ordinary differential equations on a graph. This networked system of ordinary differential equations is shown to be well-posed and affine-linear solutions of these systems are derived analytically. As a consequence, finitedimensional mixed-integer linear optimization problems are obtained for every time step that can be solved to global optimality using general-purpose solvers. We illustrate our approach in practice by presenting numerical results on a realistic gas transport network.
Mathematical Control & Related Fields, 2017
For a system that is governed by the isothermal Euler equations with friction for ideal gas, the ... more 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.
SIAM Journal on Control and Optimization, 2017
We study a wave equation in one space dimension with a general diffusion coefficient which degene... more 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 µa > 0. We establish observability inequalities for weakly (when µa ∈ [0, 1[) as well as strongly (when µa ∈ [1, 2[) degenerate equations. We also prove a negative result when the diffusion coefficient degenerates too violently (i.e. when µa > 2) and the blow-up of the observability time when µa converges to 2 from below. Thus, using the HUM method we deduce the exact controllability of the corresponding degenerate control problem when µa ∈ [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 [1, 2] together with the results of the previous section, to perform this stability analysis.
Networks & Heterogeneous Media, 2015
We consider a system of scalar nonlocal conservation laws on networks that model a highly re-entr... more We consider a system of scalar nonlocal conservation laws on networks that model a highly re-entrant multi-commodity manufacturing system as encountered in semi-conductor production. Every single commodity is modeled by a nonlocal conservation law, and the corresponding PDEs are coupled via a collective load, the work in progress. We illustrate the dynamics for two commodities. In the applications, directed acyclic networks naturally occur, therefore this type of networks is considered. On every edge of the network we have a system of coupled conservation laws with nonlocal velocity. At the junctions the right hand side boundary data of the foregoing edges is passed as left hand side boundary data to the following edges and PDEs. For distributing junctions, where we have more than one outgoing edge, we impose time dependent distribution functions that guarantee conservation of mass. We provide results of regularity, existence and well-posedness of the multicommodity network model for L p -, BV -and W 1,p -data. Moreover, we define an L 2 -tracking type objective and show the existence of minimizers that solve the corresponding optimal control problem.
Progress in Computational Physics (PiCP) Vol 1: Wave Propagation in Periodic Media, 2012
This chapter focuses on acoustic, electromagnetic, elastic and piezo-electric wave propagation th... more This chapter focuses on acoustic, electromagnetic, elastic and piezo-electric wave propagation through heterogenous layers. The motivation is provided by the demand for a better understanding of meta-materials and their possible construction. We stress the analogies between the mathematical treatment of phononic, photonic and elastic meta-materials. Moreover, we treat the cloaking problem in more detail from an analytical and simulation oriented point of view. The novelty in the approach presented here is with the interlinked homogenization-and optimization procedure.
Mathematics and Mechanics of Solids, 2013
The paper concerns the analysis of equilibrium problems for 2D elastic bodies with thin inclusion... more The paper concerns the analysis of equilibrium problems for 2D elastic bodies with thin inclusions modeled in the framework of Timoshenko beams. The first focus is on the well-posedness of the model problem in a variational setting. Then delaminations of the inclusions are considered, forming a crack between the elastic body and the inclusion. Nonlinear boundary conditions at the crack faces are considered to prevent a mutual penetration between the faces. The corresponding variational formulations together with weak and strong solutions are discussed. The model contains various physical parameters characterizing the mechanical properties of the inclusion, such as flexural and shear stiffness. The paper provides an asymptotic analysis of such parameters. It is proved that in the limit cases corresponding to infinite and zero rigidity, we obtain rigid inclusions and cracks with the non-penetration conditions, respectively. Finally, exemplary networks of Timoshenko beams are considere...
SIAM Journal on Control and Optimization, 2014
In this contribution the optimal boundary control problem for a first order nonlinear, nonlocal h... more In this contribution the optimal boundary control problem for a first order nonlinear, nonlocal hyperbolic pde is studied. Motivated by various applications ranging from re-entrant manufacturing systems to particle synthesis processes, we establish the regularity of solutions for W 1,p -data. Based on a general L 2 tracking type cost functional, the existence, uniqueness, and regularity of the adjoint system in W 1,p is derived using the special structure induced from the nonlocal flux function of the state equation. The assumption of W 1,p -and not L p -regularity comes thereby due to the fact that the adjoint equation asks for more regularity to be well defined. This problem is discussed in detail and we give a solution by defining a special type of cost functional, such that the corresponding optimality system is well defined.
Optimal Control Problems for Partial Differential Equations on Reticulated Domains, 2011
This chapter is intended to provide various facts, notions, and concepts which play a fundamental... more This chapter is intended to provide various facts, notions, and concepts which play a fundamental role in modern asymptotic analysis of optimization problems. We recall some main concepts and basic results of measure theory, Sobolev spaces, and boundary value problems which are used later. We include proofs only if the line of arguments is of importance for the understanding of subsequent remarks. For a deeper insight in the subject, we refer to the books of Adams [2], Bucur and Buttazzo [38], Evans and Gariepy [106], Kantorovich and Akilov [128], Lions and Magenes [173], Maz'ya [185], Yosida [251], Ziemer [267], and so on.
Mathematics and Mechanics of Complex Systems, 2014
We propose a model for a two-dimensional elastic body with a thin elastic inclusion modeled by a ... more We propose a model for a two-dimensional elastic body with a thin elastic inclusion modeled by a beam equation. Moreover, we assume that a delamination of the inclusion may take place resulting in a crack. Nonlinear boundary conditions are imposed at the crack faces to prevent mutual penetration between the faces. Both variational and differential problem formulations are considered, and existence of solutions is established. Furthermore, we study the dependence of the solution on the rigidity of the embedded beam. It is proved that in the limit cases corresponding to infinite and zero rigidity, we obtain a rigid beam inclusion and cracks with nonpenetration conditions, respectively. Anisotropic behavior of the beam is also analyzed. MSC2010: 74-XX.
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Papers by Günter Leugering