Irena Lasiecka

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Irena Lasiecka

Distinguished University professor, University of memphis and Commonwealth Professor, University of Virginia

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Papers by Irena Lasiecka

Boundary Stabilization for a Heat-Kelvin-Voigt Unstable Interaction Model, with Control and Partial Observation Localized at the Interface Only

Journal of optimization theory and applications, Jul 9, 2024

Global existence and exponential stability for a nonlinear thermoelastic Kirchhoff–Love plate

Nonlinear Analysis-real World Applications, Dec 1, 2017

We study an initial-boundary-value problem for a quasilinear thermoelastic plate of Kirchhoff & L... more We study an initial-boundary-value problem for a quasilinear thermoelastic plate of Kirchhoff & Love-type with parabolic heat conduction due to Fourier, mechanically simply supported and held at the reference temperature on the boundary. For this problem, we show the short-time existence and uniqueness of classical solutions under appropriate regularity and compatibility assumptions on the data. Further, we use barrier techniques to prove the global existence and exponential stability of solutions under a smallness condition on the initial data. It is the first result of this kind established for a quasilinear non-parabolic thermoelastic Kirchhoff & Love plate in multiple dimensions.

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Weak and strong semigroups in structural acoustic Kirchhoff-Boussinesq interactions with boundary feedback

Journal of Differential Equations

Abstract We consider a structural-acoustic wall problem in three dimensions, in which the structu... more Abstract We consider a structural-acoustic wall problem in three dimensions, in which the structural wall is modeled by a 2D Kirchhoff-Boussinesq plate and the acoustic medium is subject to boundary damping. For this model we study the existence of a continuous nonlinear semigroup associated with the model in the finite energy space. We show that strong/weak continuity of the semigroups depends on the support of the boundary damping. The complications are related to supercritical nonlinearity exhibited by the plate along with the compromised boundary regularity of the acoustic waves. Compensated compactness methods along with a hidden boundary regularity of hyperbolic traces are exploited in order to establish weak (resp. strong) generation of a nonlinear semigroup subjected to feedback forces placed on the boundary of the acoustic medium.

Regularization and finite element approximation of the wave equation with Dirichlet boundary data

Banach Center Publications, 1990

A numerical method for solving the wave equation with nonhomogenuous, nonsmooth Dirichlet boundar... more A numerical method for solving the wave equation with nonhomogenuous, nonsmooth Dirichlet boundary condition is proposed. Convergence of the method is proved and some error estimates are derived [L-S-2]. The method is based on the regularization technique [L-1], [L-S-1] of the wave equation with Dirichlet boundary data. Several numerical results are provided in two dimensional case.

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A note on the Moore–Gibson–Thompson equation with memory of type II

Journal of Evolution Equations, 2019

We consider the Moore–Gibson–Thompson equation with memory of type II $$\begin{aligned} \partial ... more We consider the Moore–Gibson–Thompson equation with memory of type II $$\begin{aligned} \partial _{ttt} u(t) + \alpha \partial _{tt} u(t) + \beta A \partial _t u(t) + \gamma Au(t)-\int _0^t g(t-s) A \partial _t u(s)\mathrm{d}s=0 \end{aligned}$$ where A is a strictly positive selfadjoint linear operator (bounded or unbounded) and $\alpha ,\beta ,\gamma >0$ satisfy the relation $\gamma \le \alpha \beta $. First, we prove well-posedness of finite energy solutions, without requiring any restriction on the total mass $\varrho $ of g. This extends previous results in the literature, where such a restriction was imposed. Second, we address an open question within the context of longtime behavior of solutions. We show that an “overdamping” in the memory term can destabilize the originally stable dynamics. In fact, it is always possible to find memory kernels g, complying with the usual mass restriction $\varrho <\beta $, such that the equation admits solutions with energy growing exponentially fast, even in the regime $\gamma < \alpha \beta $ where the corresponding model without memory is exponentially stable. In particular, this provides an answer to a question recently raised in the literature.

Boundary stabilization for a Heat-Kelvin-Voigt unstable interaction model, with control and partial observation localized on the interface only

Research Square (Research Square), Mar 26, 2024

A prototype model for a Fluid-Structure interaction is considered. We aim to stabilize [enhance s... more A prototype model for a Fluid-Structure interaction is considered. We aim to stabilize [enhance stability] of the model by having access only to a portion of the state. Toward this goal we shall construct a compensator-based Luenberger design, with the following two goals: (1) reconstruct the original system asymptotically-by tracking partial information about the full state, (2) stabilize the original unstable system by feeding an admissible control based on a system which is obtained from the compensator. The ultimate result is boundary control/stabilization of partially observed and originally unstable fluid-structure interaction with restricted information on the current state and without any knowledge of the initial condition. This prevents applicability of known methods in either open-loop or closed loop stabilization/control.

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Long-Time Behavior of Quasilinear Thermoelastic Kirchhoff-Love Plates with Second Sound

arXiv (Cornell University), Nov 2, 2018

We consider an initial-boundary-value problem for a thermoelastic Kirchhoff & Love plate, thermal... more We consider an initial-boundary-value problem for a thermoelastic Kirchhoff & Love plate, thermally insulated and simply supported on the boundary, incorporating rotational inertia and a quasilinear hypoelastic response, while the heat effects are modeled using the hyperbolic Maxwell-Cattaneo-Vernotte law giving rise to a 'second sound' effect. We study the local wellposedness of the resulting quasilinear mixed-order hyperbolic system in a suitable solution class of smooth functions mapping into Sobolev H k -spaces. Exploiting the sole source of energy dissipation entering the system through the hyperbolic heat flux moment, provided the initial data are small in a lower topology (basic energy level corresponding to weak solutions), we prove a nonlinear stabilizability estimate furnishing global existence & uniqueness and exponential decay of classical solutions.

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Moore–Gibson–Thompson equation with memory, part I: exponential decay of energy

Zeitschrift für Angewandte Mathematik und Physik, Apr 1, 2016

We are interested in the Moore-Gibson-Thompson(MGT) equation with memory We first classify the me... more

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Feedback control of the acoustic pressure in ultrasonic wave propagation

arXiv (Cornell University), Mar 5, 2018

Classical models for the propagation of ultrasound waves are the Westervelt equation, the Kuznets... more Classical models for the propagation of ultrasound waves are the Westervelt equation, the Kuznetsov and the Khokhlov-Zabolotskaya-Kuznetsov equations. The Jordan-Moore-Gibson-Thompson equation is a prominent example of a Partial Differential Equation (PDE) model which describes the acoustic velocity potential in ultrasound wave propagation, where the paradox of infinite speed of propagation of thermal signals is eliminated; the use of the constitutive Cattaneo law for the heat flux, in place of the Fourier law, accounts for its being of third order in time. Aiming at the understanding of the fully quasilinear PDE, a great deal of attention has been recently devoted to its linearization -referred to in the literature as the Moore-Gibson-Thompson equation -whose mathematical analysis is also of independent interest, posing already several questions and challenges. In this work we consider and solve a quadratic control problem associated with the linear equation, formulated consistently with the goal of keeping the acoustic pressure close to a reference pressure during ultrasound excitation, as required in medical and industrial applications. While optimal control problems with smooth controls have been considered in the recent literature, we aim at relying on controls which are just L 2 in time; this leads to a singular control problem and to non-standard Riccati equations. In spite of the unfavourable combination of the semigroup describing the free dynamics that is not analytic, with the challenging pattern displayed by the dynamics subject to boundary control, a feedback synthesis of the optimal control as well as well-posedness of operator Riccati equations are established.

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Global Existence and Exponential Stability for a Nonlinear Thermoelastic Kirchhoff-Love Plate

arXiv (Cornell University), Dec 2, 2016

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Energy Decay in H 2 × L 2 for Semilinear Plates with Nonlinear Boundary Dissipation Acting via Moments Only

Applied Optimization, 1998

ABSTRACT

Control problems in industry. Proceedings from the SIAM symposium on control problems, San Diego, California, USA, July 22-23, 1994

This volume contains articles presented at the SIAM Symposium on Control Problems in Industry, he... more This volume contains articles presented at the SIAM Symposium on Control Problems in Industry, held from July 22-23, 1994, in San Diego. At this symposium, experts from both academia and industry were brought together to share their experiences, and present new results and trends in modern control theory, with a focus on real industrial applications. The aim of this volume is to provide a survey of control theory applications at the frontiers of modern technology, in particular on practical problems. It looks at a range of control theoretic results, including new models, new constructions and new algorithms, leading to solutions of industrial problems.

Thermal boundary control of a thermoelastic system preprint

Flutter Stabilization for an Unstable, Hyperbolic Flow-Plate Interaction

Advances in mathematical fluid mechanics, Nov 20, 2023

Vanishing relaxation time dynamics of the Jordan Moore-Gibson-Thompson equation arising in nonlinear acoustics

arXiv (Cornell University), Nov 22, 2020

The (third-order in time) JMGT equation [15, 7] is a nonlinear (quasi-linear) Partial Differentia... more The (third-order in time) JMGT equation [15, 7] is a nonlinear (quasi-linear) Partial Differential Equation (PDE) model introduced to describe a nonlinear propagation of sound in an acoustic medium. The important feature is that the model avoids the infinite speed of propagation paradox associated with a classical second order in time equation referred to as Westervelt equation. Replacing Fourier's law by Maxwell-Cattaneo's law gives rise to the third order in time derivative scaled by a small parameter τ > 0, the latter represents the thermal relaxation time parameter and is intrinsic to the medium where the dynamics occur. In this paper we provide an asymptotic analysis of the third order model when τ → 0. It is shown that the corresponding solutions converge in a strong topology of the phase space to a limit which is the solution of Westervelt equation. In addition, rate of convergence is provided for solutions displaying higher order regularity. This addresses an open question raised in [1], where a related JMGT equation has been studied and weak star convergence of the solutions when τ → 0 has been established. Thus, our main contribution is showing strong convergence on infinite time horizon, along with related rates of convergence valid on a finite time horizon. The key to unlocking the difficulty owns to a tight control and propagation of the "smallness" of the initial data in carrying the estimates at three different topological levels. The rate of convergence allows one then to estimate the relaxation time needed for the signal to reach the target. The interest in studying this type of problems is motivated by a large array of applications arising in engineering and medical sciences.

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Boundary feedback stabilization of a critical nonlinear JMGT equation with Neumann-undissipated part of the boundary

arXiv (Cornell University), Dec 31, 2021

Boundary feedback stabilization of a critical, nonlinear Jordan-Moore-Gibson-Thompson (JMGT) equa... more Boundary feedback stabilization of a critical, nonlinear Jordan-Moore-Gibson-Thompson (JMGT) equation is considered. JMGT arises in modeling of acoustic waves involved in medical/engineering treatments like lithotripsy, thermotherapy, sonochemistry, or any other procedures using High Intensity Focused Ultrasound (HIFU). It is a well-established and recently widely studied model for nonlinear acoustics (NLA): a third-order (in time) semilinear Partial Differential Equation (PDE) with the distinctive feature of predicting the propagation of ultrasound waves at finite speed due to heat phenomenon know as second sound which leads to the hyperbolic character of heat propagation. In practice, the JMGT dynamics is largely used for modeling the evolution of the acoustic velocity and, most importantly, the acoustic pressure as sound waves propagate through certain media. In this work, critical refers to (usual) case where mediadamping effects are non-existent or non-measurable and therefore cannot be relied upon for stabilization purposes. In this paper the issue of boundary stabilizability of originally unstable (JMGT) equation is resolved. Motivated by modeling aspects in HIFU technology, boundary feedback is supported only on a portion of the boundary, while the remaining part of the boundary is left free (available to control actions). Since the boundary conditions imposed on the "free" part of the boundary fail to satisfy Lopatinski condition (unlike Dirichlet boundary conditions), the analysis of uniform stabilization from the boundary becomes very subtle and requires careful geometric considerations.

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Boundary Stabilization of the linear MGT equation with Feedback Neumann control

arXiv (Cornell University), Dec 8, 2020

The Jordan-Moore-Gibson-Thompson (JMGT)[12, 20, 44] equation is a benchmark model describing prop... more The Jordan-Moore-Gibson-Thompson (JMGT)[12, 20, 44] equation is a benchmark model describing propagation of nonlinear acoustic waves in heterogeneous fluids at rest. This is a third order (in time) dynamics which accounts for a finite speed of propagation of heat signals (see [14, 15, 20, 21, 26]). In this paper, we study a boundary stabilization of linearized version (also known as MGT-equation) in the critical case, configuration in which the smallness of the diffusion effects leads to conservative dynamics [24]. Through a single measurement in feedback form made on a non-nempty, relatively open portion of the boundary under natural geometric conditions, we were able to obtain uniform exponential stability results that are, in addition, uniform with respect to the space-dependent viscoelasticity parameter which no longer needs to be assumed positive and in fact can be degenerate and taken to be zero on the whole domain.This result, of independent interest in the area of boundary stabilization of MGT equations, provides a necessary first step for the study of optimal boundary feedback control on infinite horizon [6].

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Long-time dynamics of a hinged-free plate driven by a non-conservative force

arXiv (Cornell University), Jul 3, 2020

A partially hinged, partially free rectangular plate is considered, with the aim to address the p... more A partially hinged, partially free rectangular plate is considered, with the aim to address the possible unstable end behaviors of a suspension bridge subject to wind. This leads to a nonlinear plate evolution equation with a nonlocal stretching active in the span-wise direction. The windflow in the chord-wise direction is modeled through a piston-theoretic approximation, which provides both weak (frictional) dissipation and non-conservative forces. The long-time behavior of solutions is analyzed from various points of view. Compact global attractors, as well as fractal exponential attractors, are constructed using the recent quasi-stability theory. The nonconservative nature of the dynamics requires the direct construction of a uniformly absorbing ball, and this relies on the superlinearity of the stretching. For some parameter ranges, the non-triviality of the attractor is shown through the spectral analysis of the stationary linearized (non self-adjoint) equation and the existence of multiple unimodal solutions is shown. Several stability results, obtained through energy estimates under various smallness conditions and/or assumptions on the equilibrium set, are also provided. Finally, the existence of a finite set of determining modes for the dynamics is demonstrated, justifying the usual modal truncation in engineering for the study of the qualitative behavior of suspension bridge dynamics.

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Elastic stabilization of an intrinsically unstable hyperbolic flow–structure interaction on the 3D half-space

Mathematical Models and Methods in Applied Sciences, Mar 1, 2023

The strong asymptotic stabilization of 3D hyperbolic dynamics is achieved by a damped 2D elastic ... more The strong asymptotic stabilization of 3D hyperbolic dynamics is achieved by a damped 2D elastic structure. The model is a Neumann wave-type equation with low regularity coupling conditions given in terms of a nonlinear von Karman plate. This problem is motivated by the elimination of aeroelastic instability (sustained oscillations of bridges, airfoils, etc.) in engineering applications. Empirical observations indicate that the subsonic wave-plate system converges to equilibria. Classical approaches which decouple the plate and wave dynamics have fallen short. Here, we operate on the model as it appears in the engineering literature with no regularization and achieve stabilization by microlocalizing the Neumann boundary data for the wave equation (given through the plate dynamics). We observe a compensation by the plate dynamics precisely where the regularity of the 3D Neumann wave is compromised (in the characteristic sector).

Uniqueness of the Riccati operator of the non-standard ARE of a third order dynamics with boundary control

Control and Cybernetics

The Moore-Gibson-Thompson [MGT] dynamics is considered. This third order in time evolution arises... more The Moore-Gibson-Thompson [MGT] dynamics is considered. This third order in time evolution arises within the context of acoustic wave propagation with applications in high frequency ultrasound technology. The optimal boundary feedback control is constructed in order to have on-line regulation. The above requires wellposedness of the associated Algebraic Riccati Equation. The paper by Lasiecka and Triggiani (2022) recently contributed a comprehensive study of the Optimal Control Problem for the MGT-third order dynamics with boundary control, over an infinite time-horizon. A critical missing point in such a study is the issue of uniqueness (within a specific class) of the corresponding highly non-standard Algebraic Riccati Equation. The present note resolves this problem in the positive, thus completing the study of Lasiecka and Triggiani (2022) with the final goal of having on line feedback control, which is also optimal.

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