More detailed description of research accomplishments

What follows in a description of Dr. Shubov’s research and main results in reverse chronological order starting from the most recent project.

1. Mathematical Analysis of Aircraft Wing Models
   
   As has already been mentioned, this is a joint project with FSRC at UCLA and NASA Dryden FRC (Edwards Airforce Base, CA). The research on design and experimental, numerical, and theoretical analysis of aircraft wing models has been conducted at both Centers for many years. The ultimate goal of the entire wing modeling projects is to give specific practical recommendations to aircraft industry engineers working on flutter suppression in aircraft wings. Flutter is known as a very dangerous instability in aircraft wings, tails, control and surfaces which occurs at a certain critical speed (called the flutter speed) and results in a destruction of a fluttering surface. Damage inflicted by flutter results in a significant cost to the entire aircraft industry. Experimental, computational, and numerical analysis of flutter is ranked by NASA Aeronautics among top research projects.
   
   Dr. Shubov's work on the wing project is devoted to the spectral, asymptotic, and stability analysis of a complicated aircraft wing model. This model has been developed at FSRC of USLA in collaboration with NASA Dryden. It is considered as the most physically complete and complicated of all known aircraft wing models. The model has been investigated numerically at the UCLA Center and in the Fall of 1999 it has been tested in a series of four flight experiments at Edwards Airforce Base. The flight experiment results have shown an excellent agreement with the theoretical predictions of the model, at least, for lower frequency aeroelastic modes. So, there is no doubt that the model is practically important.
   
   In a series of her most recent 14 papers [see Ref [1, 3–7, 10-15, 1*, 2*] in "Publications")  Dr. Shubov was able to carry out the first detailed rigorous mathematical analysis of the aforementioned model to appear on the literature on aeroelasticity. The results of this analysis provide new information about the dynamics of the aircraft wing in flight. This information is not available from numerical analysis. Numerous researches in the field of Aerospace Engineering expressed an opinion that her theoretical results provide new insight,  which complements the picture obtained from computer simulations.

   Here are the excerpts from the Refrees' reports on Dr. Shubov's recent paper [11] accepted to Proceedings of the Royal Society.

   Referee #1: "The paper constitutes an important progress in the mathematical analysis and understanding of aircraft using flutter. It seems that a more general qualitative analysis of the problem is not available so far..."

   Referee #2: "The main result of this paper, namely that the set of aeroelastic modes splits into two sequences, which are asymptotically close to bending and torsion modes respectively, was believed to be true for years. But neither a proof for the splitting nor a precise estimate for the asymptotic behavior was available in the literature. So the paper constitutes a substantial progress to flutter analysis. I expect that these results will also allow further investigations of the stability boundary of the mission region on the airplane..."
   
   The aforementioned model is governed by a complicated system of evolution – convolution type intergro – differential equations. The differential part of the system of equations describes the vibrations of the wing when the aircraft is on the ground. The integral convolution type part represents forces and moments exerted on the wing by the surrounding airflow. Her work presents the following results.
    

   1. Asymptotic and spectral analysis of the differential part of the system. In particular, she has derived explicit asymptotic formulae for the eigenvalues and eigenfunctions of the nonselfadjoint operator, which is the dynamics generator of the system. Secondly, she proved the Riesz basis property of the generalized eigenfunctions. This result is of primary importance for obtaining the solutions of the system in the form of eigenfunction expansions. The techniques, used in this work, were sharpened in a series of her previous papers described in the next section.
       

   2. Asymptotic and spectral analysis of the full integro – differential system. In particular, she was able to derive first in the world literature on aeroelasticity explicit asymptotic formulae for the aeroelastic modes and for the corresponding mode shapes. She was also able to prove the Riesz basis property of the mode shapes.
      Derivation of these results required some new ideas in addition to the known techniques. More specifically, she showed that the investigation of aeroelastic modes and mode shapes can be reduced to the study of a complicated operator – valued meromorphic function of the spectral parameter called the generalized resolvent. This study required a combination of asymptotic techniques with sophisticated methods of modern functional analysis such as Nagy – Foias functional model.

   
    

   3. An intensive work on the continuation of this project is under way. She is currently working on representation of the solutions of the full integro – differential system, which governs the dynamics of the wing, in the form of expansions with respect to the aeroelastic mode shapes. This result will provide a totally new tool for the theoretical and numerical investigation of flutter.

   
    

   4. The design of a new even more complete model of an aircraft wing is in progress at UCLA and NASA Dryden, which will be a primary focus of future work on flutter analysis.

   
    

2. Spectral and Asymptotic Analysis of Elastic Structures with Energy Dissipation. Applications to Control and Stabilization Problems.
   
   As mentioned earlier, this is another direction of Professor Dr. Shubov’s research (see Ref [8, 9, 16-38]). As is well known, at the present moment, there is no general theory of nonselfadjoint operators in a Hilbert space. Such theory has been constructed only for some special classes of nonselfadjoint operators. New examples of operators, for which spectral analysis can be carried out, are of great interest from the point of view of pure theory and, also, for applications to physical and engineering problems.
   
   In a series of 25 aforementioned works, Dr. Shubov developed the spectral analysis for several new classes of nonselfadjoint operators. These operators are the dynamic generators for increasingly more complicated hyperbolic differential equations and systems, which govern the dynamics of vibrating elastic structures with energy dissipation. The list of these systems include (a) spatially nonhomogeneous string subject to viscous and boundary dampings; (b) three dimensional spatially nonhomogeneous damped wave equation with spherically symmetric coefficients; (c) Timoshenko beam model with variable coefficients (the most complete physical model of a thick beam having important applications in structural engineering); (d) coupled system of Timoshenko and Euler – Bernoulli beams. This series of results has now culminated in her work on wing models. At the present, she has a detailed program of extending the asymptotic and spectral analysis to the nonselfadjoint operators, which are the dynamics generators for systems governed by three dimensional damped wave equations with spatially nonhomogeneous nonspherically symmetric coefficients on bounded domains. This program is partially based on her previous works on resonances for SchrÖdinger operators with nonspherically symmetric potentials (see Ref. [50, 51, 53]).
   
   In the work on this research project, Dr. Shubov has suggested and successfully applied a new method of spectral analysis of nonselfadjoint differential operators, the transforma-tion operator method. In [17], she introduced the notion of transformation operators for the damped wave equation. This is a generalization of the classical concept of a transformation operator originally introduced in the late fiftieth for 1–dim SchrÖdinger equation (or, equivalently, for undamped wave equation) in the works by Gelfand, Levitan, and Marchenko in connection with the inverse scattering problem. In papers [18, 22, 24] she used the transformation operator method to prove the Riesz basis property of the generalized eigenvectors for several classes of nonselfadjoint differential operators.
   
   In a series of works [16, 21, 25, 26, 28-30, 34-36] the results of spectral analysis have been applied to the solution of several controllability problems for the aforementioned systems. Dr. Shubov was able to generalize the well – known result by D.L. Russell (Virginia Tech) obtained in the sixtieth for undamped wave equation. These works have attracted great attention. In paticular, she has received a letter from world famous French mathematician, J.L.Lions, in which he praised the results.
    

3. Resonance Phenomena in Quantum Scattering Theory and Acoustics.
   Quantum Defect Method
   
   Before 1995, Dr. Shubov did considerable research on the mathematical problems of quantum scattering theory and propagation of acoustical waves (Ref. [39–55]). A significant part of this work was devoted to resonances in quantum and acoustical scattering. In these works she sharpened some of the techniques that she later successfully used in work on spectral analysis of nonselfadjoint operators. Resonances are known as long lived vibrations of the medium, which slowly decrease with time. Resonance phenomena play an important role in quantum mechanics. They are also of significant practical interest in underwater acoustics.
   
   In a series of her early works [50, 51, 53], she studied resonances for a general class of 3–dimensional SchrÖdinger operators with nonspherically symmetric potentials. She was able to prove the existence of resonances and derive explicit asymptotic formulae for resonances and resonance states. Explicit description of resonances is usually very difficult. Most of the known examples deal either with 1–dimensional systems or systems possessing spherical symmetry. To study the nonspherically symmetric problem, she suggested the method employing an analysis of infinite systems of integral equations. In this work she received a very positive response from Professor Lax (Courant Institute and New York University), a world famous mathematician, responsible for classic results in the field of resonances.
   
   A series of works [40-46] is devoted to a detailed asymptotic analysis of resonances for 3–dimensional SchrÖdingers operator with spherically symmetric slowly decreasing Coulomb – type potentials. In paper [42], she discovered the existence of the so–called low energy chain of resonances. This phenomenon never occurs for fast decreasing potentials and is related to the Coulomb term in the potential energy.
   
   Another series of works [40, 41, 43-45] is devoted to the so–called Quantum Defect Theory. This theory deals with the description of the energy spectra for hydrogen like ions. In [46] she derived an explicit and very complicated formula for the quantum defect of an ion descried by Coulomb potential perturbed by a compactly supported potential. Paper [40] is devoted to the analysis of quantum defect in the presence of an external electric field (the Stark effect).
    
   In a long paper [39], Dr. Shubov presented a detailed analysis of resonances and resonance states in the problem of scattering of acoustical waves by an inhomogeneity of the density of the medium. In this paper she showed that resonance states in acoustics, unlike in quantum mechanics, form Riesz basis in the energy space, by applying the Lax–Phillips scattering theory. Her papers [47, 48] are devoted to abstract properties of Riesz bases in a Hilbert space and to Sz. Nagy–Foias functional model of nonsedfadjoint operators.