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Irena Lasiecka

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Irena Lasiecka

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

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Mahendra Panthee

Universidade Estadual de Campinas

Alain Miranville

Université du Havre

Mines Paristech

Università degli Studi della Tuscia

Erika Hausenblas

Giovanni Paolo Galdi

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Papers by Irena Lasiecka

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.

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.

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].

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.

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.

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

Discrete and Continuous Dynamical Systems - S

Boundary feedback stabilization of a critical third–order (in time) semilinear Jordan–Moore–Gibso... more Boundary feedback stabilization of a critical third–order (in time) semilinear Jordan–Moore–Gibson–Thompson (JMGT) is considered. The word critical here refers to the usual case where media–damping effects are non–existent or non–measurable and therefore cannot be relied upon for stabilization purposes. Motivated by modeling aspects in high-intensity focused ultrasound (HIFU) technology, the boundary feedback under consideration is supported only on a portion of the boundary. At the same time, the remaining part is undissipated and subject to Neumann/Robin boundary conditions. As such, unlike Dirichlet, it fails to satisfy the Lopatinski condition, a fact which compromises tangential regularity on the boundary [37]. In such a configuration, the analysis of uniform stabilization from the boundary becomes subtle and requires careful geometric considerations and microlocal analysis estimates. The nonlinear effects in the model demand construction of suitably small solutions which are i...

The SMGT equation from the boundary: regularity and stabilization

Applicable Analysis, 2021

We consider the third-order (in time) linear equation known as SMGT-equation, as defined on a mul... more We consider the third-order (in time) linear equation known as SMGT-equation, as defined on a multidimensional bounded domain. Part A gives optimal interior and boundary regularity results from – Dirichlet or Neumann boundary terms. Explicit representation formulas are given that can be taken to define the notion of solution in the canonical case , while the same regularity results hold for Part B considers the SMGT equation under Neumann dissipative boundary conditions and critical parameter and a.e. in Ω. We provide two results: (i) uniform stabilization under minimal checkable geometric conditions, and (ii) strong stabilization in the absence of geometrical conditions.

Optimal boundary feedback control of high intensity focused ultrasound processes

2021 60th IEEE Conference on Decision and Control (CDC), 2021

Optimal control problem governed by an acoustic equation with thermal relaxation is considered. Q... more Optimal control problem governed by an acoustic equation with thermal relaxation is considered. Questions of boundary stabilizability and feedback on line control are investigated. On line control is implemented by Gain Feedback based on nonstandard Riccati equation with unbounded coefficients. While optimal control problems with smooth controls have been considered in the recent literature, [17], our new challenge is to consider "rough controls" with L2 forcing which then lead to non-smooth optimization problem and non-standard Riccati equations.

Uniform Stabilization of 3D Navier–Stokes Equations in Low Regularity Besov Spaces with Finite Dimensional, Tangential-Like Boundary, Localized Feedback Controllers

Archive for Rational Mechanics and Analysis, 2021

The present paper provides a solution in the affirmative to a recognized open problem in the theo... more The present paper provides a solution in the affirmative to a recognized open problem in the theory of uniform stabilization of 3-dimensional Navier-Stokes equations in the vicinity of an unstable equilibrium solution, by means of a 'minimal' and 'least' invasive feedback strategy which consists of a control pair {v, u} [L-T.3]. Here v is a tangential boundary feedback control, acting on an arbitrary small part Γ of the boundary Γ; while u is a localized, interior feedback control, acting tangentially on an arbitrarily small subset ω of the interior supported by Γ. The ideal strategy of taking u = 0 on ω is not sufficient. A question left open in the literature was: Can such feedback control v of the pair {v, u} be asserted to be finite dimensional also in the dimension d = 3? We here give an affirmative answer to this question, thus establishing an optimal result. To achieve the desired finite dimensionality of the feedback tangential boundary control v, it is here then necessary to abandon the Hilbert-Sobolev functional setting of past literature and replace it with a critical Besov space setting. These spaces are 'close' to L 3 (Ω) for d = 3. This functional setting is significant. It is in line with recent critical well-posedness in the full space of the non-controlled N-S equations. A key feature of such Besov spaces with tight indices is that they do not recognize compatibility conditions. The proof is constructive and is "optimal" also regarding the "minimal" number of tangential boundary controllers needed. The new setting requires the solution of novel technical and conceptual issues. These include establishing maximal regularity in the required critical Besov setting for the overall closed-loop linearized problem with tangential feedback control applied on the boundary. This result is also a new contribution to the area of maximal regularity. It escapes perturbation theory. Finally, the minimal amount of tangential boundary action is linked to the issue of unique continuation of over-determined Oseen eigenproblems.

The Mathematics of Fluids and Solids

Finite-dimensional boundary uniform stabilization of the Boussinesq system in Besov spaces by critical use of Carleman estimate-based inverse theory

Journal of Inverse and Ill-posed Problems, 2021

We consider the 𝑑-dimensional Boussinesq system defined on a sufficiently smooth bounded domain a... more We consider the 𝑑-dimensional Boussinesq system defined on a sufficiently smooth bounded domain and subject to a pair { v , u } \{v,\boldsymbol{u}\} of controls localized on { Γ ~ , ω } \{\widetilde{\Gamma},\omega\} . Here, 𝑣 is a scalar Dirichlet boundary control for the thermal equation, acting on an arbitrarily small connected portion Γ ~ \widetilde{\Gamma} of the boundary Γ = ∂ ⁡ Ω \Gamma=\partial\Omega . Instead, 𝒖 is a 𝑑-dimensional internal control for the fluid equation acting on an arbitrarily small collar 𝜔 supported by Γ ~ \widetilde{\Gamma} . The initial conditions for both fluid and heat equations are taken of low regularity. We then seek to uniformly stabilize such Boussinesq system in the vicinity of an unstable equilibrium pair, in the critical setting of correspondingly low regularity spaces, by means of an explicitly constructed, finite-dimensional feedback control pair { v , u } \{v,\boldsymbol{u}\} localized on { Γ ~ , ω } \{\widetilde{\Gamma},\omega\} . In addit...

Boundary Stabilization of the Linear MGT Equation with Feedback Neumann Control

Deterministic and Stochastic Optimal Control and Inverse Problems, 2021

The Jordan-Moore-Gibson-Thompson (JMGT) equation is a well established and recently widely studie... more The Jordan-Moore-Gibson-Thompson (JMGT) equation is a well established and recently widely studied model for nonlinear acoustics (NLA). It is a third-order (in time) semilinear Partial Differential Equation (PDE) model with the distinctive feature of predicting the propagation of ultrasound waves at finite speed due to heat phenomenon know as second sound which leads the hyperbolic character of heat propagation. In this paper we consider the problem of stabilizability of the linear (known as) MGT-equation. We consider a special geometry which is suitable for studying the problem of controlling (from the boundary) the acoustic pressure involved in medical treatments like lithotripsy, thermotherapy, sonochemistry or any other procedures using High Intensity Focused Ultrasound (HIFU).

Maximal L-regularity for an abstract evolution equation with applications to closed-loop boundary feedback control problems

Journal of Differential Equations, 2021

In this paper we present an abstract maximal L p-regularity result up to T = ∞, that is tuned to ... more In this paper we present an abstract maximal L p-regularity result up to T = ∞, that is tuned to capture (linear) Partial Differential Equations of parabolic type, defined on a bounded domain and subject to finite dimensional, stabilizing, feedback controls acting on (a portion of) the boundary. Illustrations include, beside a more classical boundary parabolic example, two more recent settings: (i) the 3d-Navier-Stokes equations with finite dimensional, localized, boundary tangential feedback stabilizing controls as well as Boussinesq systems with finite dimensional, localized, feedback, stabilizing, Dirichlet boundary control for the thermal equation.

$Research paper thumbnail of Uniform stabilization of Boussinesq systems in critical \begin{document}$ \mathbf{L}^q $\end{document}-based Sobolev and Besov spaces by finite dimensional interior localized feedback controls$

Uniform stabilization of Boussinesq systems in critical \begin{document}$ \mathbf{L}^q $\end{document}-based Sobolev and Besov spaces by finite dimensional interior localized feedback controls

Discrete & Continuous Dynamical Systems - B, 2020

HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific re... more HAL is a multidisciplinary 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.

Uniform Stabilization of Navier–Stokes Equations in Critical $$L^q$$-Based Sobolev and Besov Spaces by Finite Dimensional Interior Localized Feedback Controls

Applied Mathematics & Optimization, 2019

We consider 2-or 3-dimensional incompressible Navier-Stokes equations defined on a bounded domain... more We consider 2-or 3-dimensional incompressible Navier-Stokes equations defined on a bounded domain Ω, with no-slip boundary conditions and subject to an external force, assumed to cause instability. We then seek to uniformly stabilize such N-S system, in the vicinity of an unstable equilibrium solution, in critical L q-based Sobolev and Besov spaces, by finite dimensional feedback controls. These spaces are 'close' to L 3 (Ω) for d = 3. This functional setting is significant. In fact, in the case of the uncontrolled N-S dynamics, extensive research efforts have recently lead to the space L 3 (R 3) as being a critical space for the issue of well-posedness in the full space. Thus, our present work manages to solve the stated uniform stabilization problem for the controlled N-S dynamics in a correspondingly related function space setting. In this paper, the feedback controls are localized on an arbitrarily small open interior subdomain ω of Ω. In addition to providing a solution of the uniform stabilization problem in such critical function space setting, this paper manages also to much improve and simplify, at both the conceptual and computational level, the solution given in the more restrictive Hilbert space setting in the literature. Moreover, such treatment sets the foundation for the authors' final goal in a subsequent paper. Based critically on said low functional level where compatibility conditions are not recognized, the subsequent paper solves in the affirmative a presently open problem: whether uniform stabilization by localized tangential boundary feedback controls, which-in addition-are finite dimensional, is also possible in dim Ω = 3.

Stabilization of a nonlinear flow-plate interaction via component-wise decomposition

Bulletin of the Brazilian Mathematical Society, New Series, 2016

Asymptotic-in-time interior feedback control of a panel interacting with an inviscid, subsonic fl... more Asymptotic-in-time interior feedback control of a panel interacting with an inviscid, subsonic flow is considered. The classical model [8] is given by a clamped nonlinear plate strongly coupled to a convected wave equation on the half space. In the absence of energy dissipation the plate dynamics converge to a compact and finite dimensional set [6, 7]. With a sufficiently large velocity feedback control on the structure we show that the full flow-plate system exhibits strong convergence to the set of stationary states in the natural energy topology. We show a decomposition of the dynamics into “smooth” component and global-in-timeHadamard continuous component, thus permitting approximation by smooth data. That the flows are subsonic is critical for our approach. Our result implies that flutter (a periodic or chaotic end behavior) is not present in subsonic flows with sufficient viscous damping in the structure.

Preface: In Memory of A.V. Balakrishnan

Applied Mathematics & Optimization, 2016

Mathematical Theory of Evolutionary Fluid-Flow Structure Interactions

Springer eBooks, 2018

Optimal Control Problems and Riccati Equations for Systems with Unbounded Controls and Partially Analytic Generators-Applications to Boundary and Point Control Problems

Functional Analytic Methods for Evolution Equations, 2004

ABSTRACT 1. Introduction I. Abstract Theory 2. Mathematical Setting and Formulation of the Contro... more ABSTRACT 1. Introduction I. Abstract Theory 2. Mathematical Setting and Formulation of the Control Problem 2.1. Dynamics and Related Control Problem 2.2. Singular Estimate Assumptions 2.3. Examples of Point and Boundary Control Models 3. Abstract Results for Control Problems with Singular Estimate 3.1. Finite Horizon Control Problem - Differential Riccati Equation (DRE) 3.2. Infinite Horizon Control Problem - Algebraic Riccati Equations (ARE) 4. Finite Horizon Problem - Proofs 4.1. Characterization of Optimal Solution 4.2. Continuity in Time of Optimal Solutions 4.3. Regularity of Evolution F(t,s)\Phi (t,s) and of Riccati Operator P(t) 4.4. Singular Estimate and Left-differentiability for the Evolution Operator 4.5. Differential Riccati Equation - DRE 4.6. The Function r(t) II. Applications to Point and Boundary Control Problems 5. Boundary Control Problems for Thermoelastic Plates 5.1. PDE Model 5.2. Semigroup Formulation 5.3. Main Result 5.4. Proof of Theorem 5.1 6. Composite Beam Models with Boundary Control 6.1. PDE Model and Main Results 6.2. Transformations of the Model 6.3. Semigroup Formulation 6.4. Singular Estimate and Completion and Proper Proof of Theorem 6.1 7. Point and Boundary Control Problems in Acoustic-structure Interactions 7.1. Description of the Model 7.2. Semigroup Formulation 7.3. Finite Horizon Control Problem 7.4. Infinite Horizon Control Problem References

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.

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.

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].

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.

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.

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

Discrete and Continuous Dynamical Systems - S

Boundary feedback stabilization of a critical third–order (in time) semilinear Jordan–Moore–Gibso... more Boundary feedback stabilization of a critical third–order (in time) semilinear Jordan–Moore–Gibson–Thompson (JMGT) is considered. The word critical here refers to the usual case where media–damping effects are non–existent or non–measurable and therefore cannot be relied upon for stabilization purposes. Motivated by modeling aspects in high-intensity focused ultrasound (HIFU) technology, the boundary feedback under consideration is supported only on a portion of the boundary. At the same time, the remaining part is undissipated and subject to Neumann/Robin boundary conditions. As such, unlike Dirichlet, it fails to satisfy the Lopatinski condition, a fact which compromises tangential regularity on the boundary [37]. In such a configuration, the analysis of uniform stabilization from the boundary becomes subtle and requires careful geometric considerations and microlocal analysis estimates. The nonlinear effects in the model demand construction of suitably small solutions which are i...

The SMGT equation from the boundary: regularity and stabilization

Applicable Analysis, 2021

We consider the third-order (in time) linear equation known as SMGT-equation, as defined on a mul... more We consider the third-order (in time) linear equation known as SMGT-equation, as defined on a multidimensional bounded domain. Part A gives optimal interior and boundary regularity results from – Dirichlet or Neumann boundary terms. Explicit representation formulas are given that can be taken to define the notion of solution in the canonical case , while the same regularity results hold for Part B considers the SMGT equation under Neumann dissipative boundary conditions and critical parameter and a.e. in Ω. We provide two results: (i) uniform stabilization under minimal checkable geometric conditions, and (ii) strong stabilization in the absence of geometrical conditions.

Optimal boundary feedback control of high intensity focused ultrasound processes

2021 60th IEEE Conference on Decision and Control (CDC), 2021

Optimal control problem governed by an acoustic equation with thermal relaxation is considered. Q... more Optimal control problem governed by an acoustic equation with thermal relaxation is considered. Questions of boundary stabilizability and feedback on line control are investigated. On line control is implemented by Gain Feedback based on nonstandard Riccati equation with unbounded coefficients. While optimal control problems with smooth controls have been considered in the recent literature, [17], our new challenge is to consider "rough controls" with L2 forcing which then lead to non-smooth optimization problem and non-standard Riccati equations.

Uniform Stabilization of 3D Navier–Stokes Equations in Low Regularity Besov Spaces with Finite Dimensional, Tangential-Like Boundary, Localized Feedback Controllers

Archive for Rational Mechanics and Analysis, 2021

The present paper provides a solution in the affirmative to a recognized open problem in the theo... more The present paper provides a solution in the affirmative to a recognized open problem in the theory of uniform stabilization of 3-dimensional Navier-Stokes equations in the vicinity of an unstable equilibrium solution, by means of a 'minimal' and 'least' invasive feedback strategy which consists of a control pair {v, u} [L-T.3]. Here v is a tangential boundary feedback control, acting on an arbitrary small part Γ of the boundary Γ; while u is a localized, interior feedback control, acting tangentially on an arbitrarily small subset ω of the interior supported by Γ. The ideal strategy of taking u = 0 on ω is not sufficient. A question left open in the literature was: Can such feedback control v of the pair {v, u} be asserted to be finite dimensional also in the dimension d = 3? We here give an affirmative answer to this question, thus establishing an optimal result. To achieve the desired finite dimensionality of the feedback tangential boundary control v, it is here then necessary to abandon the Hilbert-Sobolev functional setting of past literature and replace it with a critical Besov space setting. These spaces are 'close' to L 3 (Ω) for d = 3. This functional setting is significant. It is in line with recent critical well-posedness in the full space of the non-controlled N-S equations. A key feature of such Besov spaces with tight indices is that they do not recognize compatibility conditions. The proof is constructive and is "optimal" also regarding the "minimal" number of tangential boundary controllers needed. The new setting requires the solution of novel technical and conceptual issues. These include establishing maximal regularity in the required critical Besov setting for the overall closed-loop linearized problem with tangential feedback control applied on the boundary. This result is also a new contribution to the area of maximal regularity. It escapes perturbation theory. Finally, the minimal amount of tangential boundary action is linked to the issue of unique continuation of over-determined Oseen eigenproblems.

The Mathematics of Fluids and Solids

Finite-dimensional boundary uniform stabilization of the Boussinesq system in Besov spaces by critical use of Carleman estimate-based inverse theory

Journal of Inverse and Ill-posed Problems, 2021

We consider the 𝑑-dimensional Boussinesq system defined on a sufficiently smooth bounded domain a... more We consider the 𝑑-dimensional Boussinesq system defined on a sufficiently smooth bounded domain and subject to a pair { v , u } \{v,\boldsymbol{u}\} of controls localized on { Γ ~ , ω } \{\widetilde{\Gamma},\omega\} . Here, 𝑣 is a scalar Dirichlet boundary control for the thermal equation, acting on an arbitrarily small connected portion Γ ~ \widetilde{\Gamma} of the boundary Γ = ∂ ⁡ Ω \Gamma=\partial\Omega . Instead, 𝒖 is a 𝑑-dimensional internal control for the fluid equation acting on an arbitrarily small collar 𝜔 supported by Γ ~ \widetilde{\Gamma} . The initial conditions for both fluid and heat equations are taken of low regularity. We then seek to uniformly stabilize such Boussinesq system in the vicinity of an unstable equilibrium pair, in the critical setting of correspondingly low regularity spaces, by means of an explicitly constructed, finite-dimensional feedback control pair { v , u } \{v,\boldsymbol{u}\} localized on { Γ ~ , ω } \{\widetilde{\Gamma},\omega\} . In addit...

Boundary Stabilization of the Linear MGT Equation with Feedback Neumann Control

Deterministic and Stochastic Optimal Control and Inverse Problems, 2021

The Jordan-Moore-Gibson-Thompson (JMGT) equation is a well established and recently widely studie... more The Jordan-Moore-Gibson-Thompson (JMGT) equation is a well established and recently widely studied model for nonlinear acoustics (NLA). It is a third-order (in time) semilinear Partial Differential Equation (PDE) model with the distinctive feature of predicting the propagation of ultrasound waves at finite speed due to heat phenomenon know as second sound which leads the hyperbolic character of heat propagation. In this paper we consider the problem of stabilizability of the linear (known as) MGT-equation. We consider a special geometry which is suitable for studying the problem of controlling (from the boundary) the acoustic pressure involved in medical treatments like lithotripsy, thermotherapy, sonochemistry or any other procedures using High Intensity Focused Ultrasound (HIFU).

Maximal L-regularity for an abstract evolution equation with applications to closed-loop boundary feedback control problems

Journal of Differential Equations, 2021

In this paper we present an abstract maximal L p-regularity result up to T = ∞, that is tuned to ... more In this paper we present an abstract maximal L p-regularity result up to T = ∞, that is tuned to capture (linear) Partial Differential Equations of parabolic type, defined on a bounded domain and subject to finite dimensional, stabilizing, feedback controls acting on (a portion of) the boundary. Illustrations include, beside a more classical boundary parabolic example, two more recent settings: (i) the 3d-Navier-Stokes equations with finite dimensional, localized, boundary tangential feedback stabilizing controls as well as Boussinesq systems with finite dimensional, localized, feedback, stabilizing, Dirichlet boundary control for the thermal equation.

$Research paper thumbnail of Uniform stabilization of Boussinesq systems in critical \begin{document}$ \mathbf{L}^q $\end{document}-based Sobolev and Besov spaces by finite dimensional interior localized feedback controls$

Uniform stabilization of Boussinesq systems in critical \begin{document}$ \mathbf{L}^q $\end{document}-based Sobolev and Besov spaces by finite dimensional interior localized feedback controls

Discrete & Continuous Dynamical Systems - B, 2020

HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific re... more HAL is a multidisciplinary 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.

Uniform Stabilization of Navier–Stokes Equations in Critical $$L^q$$-Based Sobolev and Besov Spaces by Finite Dimensional Interior Localized Feedback Controls

Applied Mathematics & Optimization, 2019

We consider 2-or 3-dimensional incompressible Navier-Stokes equations defined on a bounded domain... more We consider 2-or 3-dimensional incompressible Navier-Stokes equations defined on a bounded domain Ω, with no-slip boundary conditions and subject to an external force, assumed to cause instability. We then seek to uniformly stabilize such N-S system, in the vicinity of an unstable equilibrium solution, in critical L q-based Sobolev and Besov spaces, by finite dimensional feedback controls. These spaces are 'close' to L 3 (Ω) for d = 3. This functional setting is significant. In fact, in the case of the uncontrolled N-S dynamics, extensive research efforts have recently lead to the space L 3 (R 3) as being a critical space for the issue of well-posedness in the full space. Thus, our present work manages to solve the stated uniform stabilization problem for the controlled N-S dynamics in a correspondingly related function space setting. In this paper, the feedback controls are localized on an arbitrarily small open interior subdomain ω of Ω. In addition to providing a solution of the uniform stabilization problem in such critical function space setting, this paper manages also to much improve and simplify, at both the conceptual and computational level, the solution given in the more restrictive Hilbert space setting in the literature. Moreover, such treatment sets the foundation for the authors' final goal in a subsequent paper. Based critically on said low functional level where compatibility conditions are not recognized, the subsequent paper solves in the affirmative a presently open problem: whether uniform stabilization by localized tangential boundary feedback controls, which-in addition-are finite dimensional, is also possible in dim Ω = 3.

Stabilization of a nonlinear flow-plate interaction via component-wise decomposition

Bulletin of the Brazilian Mathematical Society, New Series, 2016

Asymptotic-in-time interior feedback control of a panel interacting with an inviscid, subsonic fl... more Asymptotic-in-time interior feedback control of a panel interacting with an inviscid, subsonic flow is considered. The classical model [8] is given by a clamped nonlinear plate strongly coupled to a convected wave equation on the half space. In the absence of energy dissipation the plate dynamics converge to a compact and finite dimensional set [6, 7]. With a sufficiently large velocity feedback control on the structure we show that the full flow-plate system exhibits strong convergence to the set of stationary states in the natural energy topology. We show a decomposition of the dynamics into “smooth” component and global-in-timeHadamard continuous component, thus permitting approximation by smooth data. That the flows are subsonic is critical for our approach. Our result implies that flutter (a periodic or chaotic end behavior) is not present in subsonic flows with sufficient viscous damping in the structure.

Preface: In Memory of A.V. Balakrishnan

Applied Mathematics & Optimization, 2016

Mathematical Theory of Evolutionary Fluid-Flow Structure Interactions

Springer eBooks, 2018

Optimal Control Problems and Riccati Equations for Systems with Unbounded Controls and Partially Analytic Generators-Applications to Boundary and Point Control Problems

Functional Analytic Methods for Evolution Equations, 2004

ABSTRACT 1. Introduction I. Abstract Theory 2. Mathematical Setting and Formulation of the Contro... more ABSTRACT 1. Introduction I. Abstract Theory 2. Mathematical Setting and Formulation of the Control Problem 2.1. Dynamics and Related Control Problem 2.2. Singular Estimate Assumptions 2.3. Examples of Point and Boundary Control Models 3. Abstract Results for Control Problems with Singular Estimate 3.1. Finite Horizon Control Problem - Differential Riccati Equation (DRE) 3.2. Infinite Horizon Control Problem - Algebraic Riccati Equations (ARE) 4. Finite Horizon Problem - Proofs 4.1. Characterization of Optimal Solution 4.2. Continuity in Time of Optimal Solutions 4.3. Regularity of Evolution F(t,s)\Phi (t,s) and of Riccati Operator P(t) 4.4. Singular Estimate and Left-differentiability for the Evolution Operator 4.5. Differential Riccati Equation - DRE 4.6. The Function r(t) II. Applications to Point and Boundary Control Problems 5. Boundary Control Problems for Thermoelastic Plates 5.1. PDE Model 5.2. Semigroup Formulation 5.3. Main Result 5.4. Proof of Theorem 5.1 6. Composite Beam Models with Boundary Control 6.1. PDE Model and Main Results 6.2. Transformations of the Model 6.3. Semigroup Formulation 6.4. Singular Estimate and Completion and Proper Proof of Theorem 6.1 7. Point and Boundary Control Problems in Acoustic-structure Interactions 7.1. Description of the Model 7.2. Semigroup Formulation 7.3. Finite Horizon Control Problem 7.4. Infinite Horizon Control Problem References