My research is in Applied Mathematics, in particular, centered around nonlinear Partial Differential Equations (PDEs). My current research involves
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Global existence and exponential stability for a nonlinear thermoelastic Kirchhoff–Love plate
with Irena Lasiecka and Michael Pokojovy, Nonlinear Analysis: Real World Applications, 38 (2017): 184-221.
Abstract
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 dimension.
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Long-Time Behavior of Quasilinear Thermoelastic Kirchhoff-Love Plates with Second Sound
with Irena Lasiecka and Michael Pokojovy, Nonlinear Analysis, 186 (2019): 219-258.
Abstract
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 well-posedness of the resulting quasilinear mixed-order hyperbolic system in a suitable solution class of smooth functions mapping into Sobolev -spaces. Exploiting the sole source of energy dissipation entering the system through the hyperbolic heat flux moment, provided the initial data are small – not in the full topology of our solution class, but in a lower topology corresponding to weak solutions we prove a nonlinear stabilizability estimate furnishing global existence & uniqueness and exponential decay of classical solutions.
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Unique Continuation Properties of Over-determined Static Boussinesq Problems with Application to Uniform Stabilization of Dynamic Boussinesq Systems
with Roberto Triggiani, Applied Mathematics & Optimization, 84 (2021), 2099–2146.
Abstract
We consider several direct and adjoint Boussinesq static problems under different types of over-determined conditions. We then conclude, in each case, that the solution pair corresponding to {fluid velocity, scalar temperature} must vanish identically on the whole domain, so that the pressure is then constant (Unique Continuation Property). In going from the direct to the adjoint problem, the coupling operators between the fluid and the thermal equations switch places. As a result, the adjoint Boussinesq system has a more favorable structure than the direct Boussinesq system and hence yields UCP results under weaker requirements; typically, a reduction by one or even two units on the number of components of the fluid vector being involved in the assumptions. To illustrate: in the key direct Boussinesq problem, over-determination consists of the additional vanishing of the solution pair in a common arbitrarily small subset of the interior. In contrast, in the corresponding adjoint Boussinesq problem, only the first (d−1) components of the d-dimensional fluid velocity vector need to be assumed as vanishing on the interior subset. These UCPs for the adjoint problem are critical ingredients in the solution of corresponding uniform stabilization problems of (direct) dynamic Boussinesq systems by suitable finite dimensional feedback controls. They allow one to verify a corresponding Kalman algebraic condition for controllability.
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The Taut String Approach to Statistical Inverse Problems: Theory and Applications
with Sangjin Kim and Michael Pokojovy, Journal of Computational and Applied Mathematics, 382 (2021): 113098.
Abstract
A novel solution approach to a class of nonlinear statistical inverse problems with finitely many observations collected over a compact interval on the real line blurred by Gaussian white noise of arbitrary intensity is presented. Exploiting the nonparametric taut string estimator, we prove the state recovery strategy is convergent to a solution of the unnoisy problem at the rate of as the number of observations grows to infinity. Illustrations of the method’s application to real-world examples from hydrology, civil & electrical engineering are given and an empirical study on the robustness of our approach is presented.
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Regularity and finite element approximation for two-dimensional elliptic equations with line Dirac sources
with Hengguang Li, Peimeng Yin, and Lewei Zhao, Journal of Computational and Applied Mathematics, 393 (2021): 113518.
Abstract
We study the elliptic equation with a line Dirac delta function as the source term subject to the Dirichlet boundary condition in a two-dimensional domain. Such a line Dirac measure causes different types of solution singularities in the neighborhood of the line fracture. We establish new regularity results for the solution in a class of weighted Sobolev spaces and propose finite element algorithms that approximate the singular solution at the optimal convergence rate. Numerical tests are presented to justify the theoretical findings.
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From Low to High – and Lower – Optimal Regularity of the SMGTJ Equation with Dirichlet and Neumann Boundary Control, and with Point Control, via Explicit Representation Formulae
with Roberto Triggiani, Evolution Equations and Control Theory, (2022) 11(6): 1967-1996.
Abstract
We consider the linear third order (in time) PDE known as the SMGTJ-equation, defined on a bounded domain, under the action of either Dirichlet or Neumann boundary control g. Optimal interior and boundary regularity results were given in [1], after [41], when g ∈ L2(0, T ; L2(Γ)) ≡ L2(Σ), which, moreover, in the canonical case γ = 0, were expressed by the well-known explicit representation formulae of the wave equation in terms of cosine/sine operators [20], [17], [25, Vol II]. The interior or boundary regularity theory is however the same, whether γ = 0 or 0 6 = γ ∈ L∞(Ω), since γ 6 = 0 is responsible only for lower order terms. Here we exploit such cosine operator based-explicit representation formulae to provide optimal interior and boundary regularity results with g “smoother” than L2(Σ), qualitatively by one unit, two units, etc. in the Dirichlet boundary case. To this end, we invoke the corresponding results for wave equations, as in [17]. Similarly for the Neumann boundary case, by invoking the corresponding results for the wave equation as in [23], [24], [37] for control smoother than L2(0, T ; L2(Γ)), and [44] for control less regular in space than L2(Γ). In addition, we provide optimal interior and boundary regularity results when the SMGTJ equation is subject to interior point control, by invoking the corresponding wave equations results [42], [25, Section 9.8.2].
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Luenberger Compensentor Theory for Heat-Kelvin-Voigt-damped-Structure Interaction Models with Interface/Boundary Feedback Controls
with Roberto Triggiani, Open Mathematics, vol. 21, no. 1, (2023), pp. 20220589.
Abstract
An optimal, complete, continuous theory of the Luenberger dynamic compensator (or state estimator or state observer) is obtained for the recently studied class of heat-structure interaction PDE-models, with structure subject to high Kelvin-Voigt damping, and feedback control exercised either at the interface between the two media or else at the external boundary of the physical domain in three different settings. It is a first, full investigation that opens the door to numerous and far reaching subsequent work. They will include physically relevant \emph{fluid}-structure models, with wave- or plate-structures, possibly without Kelvin-Voigt damping, as explicitly noted in the text, all the way to achieving the ultimate discrete numerical theory, so critical in applications. While the general setting is functional analytic, delicate PDE-energy estimates dictate how to define the interface/boundary feedback control in each case.
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A heat-structure interaction model with (formal) ‘square-root’ damping: analyticity and uniform stability
with Roberto Triggiani, Quantum Studies: Mathematics and Foundations 11, 117–146 (2024).
Abstract
In Part I, the present paper studies a homogeneous, uncontrolled 2D or 3D heat-structure interaction model, where the structure is modeled by an elastic system with (formally) 'square-root' damping, and where the two components are subject to high-level coupled conditions at the interface between the two media. Physically the model occupies a doughnut-like domain: the heat (fluid) occupies the exterior domain while the elastic structure occupies an interior subdomain. The novelty over past literature is the (formal) 'square root' damping of the structure versus either no damping at all or else Kelvin-Voigt (viscoelastic) damping. It is shown that such homogeneous (uncontrolled) model generates a strongly continuous contraction semigroup on a natural energy space, which moreover is analytic and uniformly stable. Next, the paper provides a characterization of the domain of a fractional power related to the generator. This result is then used to study, in Part II, the corresponding non-homogeneous model subject to control action at the interface between the two media and provide for it an optimal regularity result. The choice of the heat component over the (linearized) Navier-Stokes fluid component is only a preliminary step for initial simplicity. The fluid-model introduces serious conceptual and technical difficulties. How to overcome them has been accomplished in past literature and will guide a subsequent publication.
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Development of a Universal Method for Vibrational Analysis of the Terminal Alkyne C≡C Stretch
with Kristina Streu, Sara Hunsberger, Jeanette Patel, and Clyde Daly Jr., Journal of Chemical Physics, 21 February 2024; 160 (7): 074106. doi:10.1063/5.0185580
Abstract
The terminal alkyne C≡C stretch has a large Raman scattering cross section in the “silent” region for biomolecules. This has led to many Raman tag and probe studies using this moiety to study biomolecular systems. Computational investigation of these systems is vital to aid in the interpretation of these results. In this work, we develop a method for computing terminal alkyne vibrational frequencies and isotropic transition polarizabilities which can easily and accurately be applied to any terminal alkyne molecule. We apply the discrete variable representation method to a localized version of the C≡C stretch normal mode. The errors of (1) vibrational localization to the terminal alkyne moiety, (2) anharmonic normal mode isolation, and (3) discretization of the Born-Oppenheimer potential energy surface are quantified and found to be generally small and cancel each other. This results in a method with low error compared to other anharmonic vibrational methods like VPT2 and to experiment. Several density functionals are tested using the method, and TPSS-D3, an inexpensive nonempirical density functional with dispersion corrections, is found to perform surprisingly well. Diffuse basis functions are found to be important for the accuracy of computed frequencies. Finally, the computation of vibrational properties like isotropic transition polarizabilities and the universality of the localized normal mode for terminal alkynes are demonstrated.
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Abstract representation of the SMGTJ equation under rough boundary controls: optimal interior regularity
with Irena Lasiecka and Roberto Triggiani, Math. Meth. Appl. Sci. 47 (2024), 13063–13086, DOI 10.1002/mma.8619
Abstract
We consider the linearized third order SMGTJ equation defined on a sufficiently smooth boundary domain in ℝ 3 and subject to either Dirichlet or Neumann rough boundary control. Filling a void in the literature, we present a direct general 3 × 3 system approach based on the vector state solution {position, velocity, acceleration}. It yields, in both cases, an explicit representation formula: input → solution, based on the s.c. group generator of the boundary homogeneous problem and corresponding elliptic Dirichlet or Neumann map. It is close to, but also distinctly and critically different from, the abstract variation of parameter formula that arises in more traditional boundary control problems for PDEs L-T.6. Through a duality argument based on this explicit formula, we provide a new proof of the optimal regularity theory: boundary control → {position, velocity, acceleration} with low regularity boundary control, square integrable in time and space.
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Numerical approximation of Riccati-based hyperbolic-like feedback controls
with Irena Lasiecka and Roberto Triggiani. J. Optim. Theory Appl. (2025), Accepted.
Abstract
This paper provides a (rigorous) theoretical framework for the numerical approximation of Riccati-based feedback control problems of hyperbolic-like dynamics over a finite-time horizon, with emphasis on genuine unbounded control action. Both continuous and approximation theories are illustrated by specific canonical hyperbolic-like equations with boundary control, where the abstract assumptions are actually sharp regularity properties of the hyperbolic dynamics under discussion. Assumptions are divided in two groups. A first group of dynamical assumptions (actually dynamic properties) imply some preliminary critical properties of the control problem, including the definition of the would-be Riccati operator, in terms of the original data. However, in order to guarantee that such an operator is moreover the unique solution (within a specific class) of the corresponding Differential/Integral Riccati Equation, additional smoothing assumptions on the operators defining the performance index are required. The ultimate goal is to show that the the discrete finite dimensional Riccati based feedback operator, when inserted into the original PDE dynamics, provides near optimal performance.