In recent years my research work has been concentrated on two main directions. The first of them deals with linear and nonlinear partial differential equations arising in mathematical control theory. The second direction is concerned with the study of hydrodynamic equations (Navier-Stokes and Euler) in connection with problems arising in atmospheric science: tornado models, airflows containing dust particles, stability of air and fluid flows. My earlier research was devoted to attractors of nonlinear evolution equations, Hausdorff and fractal dimension of subsets of Hilbert spaces, hyperbolic harmonic maps, infinite systems of ordinary differential equations on Riemannian manifolds and evolution of probability measures governed by such systems (problems arising in nonequilibrium statistical mechanics of infinite constrained particle systems), soliton theory, and Lie group representation theory (symplectic quantization). More detailed comments on my recent work and earlier research are given below.
During the past seven years I have worked on several research projects. A significant part of my research has been related to problems in the control of linear and nonlinear distributed parameter systems. The long term goal of this joint research effort with D.S. Gilliam (Texas Tech University) and C.I. Byrnes (Washington University, St. Louis) is concerned with the development of a systematic methodology for the design of feedback control laws capable of shaping the response of linear and nonlinear infinite dimensional systems. A series of joint works in this direction is related to the well known problem of output regulation (see [1, 12-14, 17, 19, 21, 22, 25, 26] in CV). We have attracted to this research a Ph.D. student I. Laukó , who received his degree in May 1997. This work has been supported by grants from the Air Force Office of Scientific Research through Washington University. The problem of output regulation has a long history in control engineering and, for finite dimensional linear systems, it was studied extensively during the 1970's and 80's. More recently, it has been extended to nonlinear finite dimensional systems in a joint work by C.I. Byrnes and A. Isidori. One of the main tools in this geometric approach to output regulation is the center manifold theorem for finite dimensional systems. The extension of this theory to linear and nonlinear infinite dimensional systems forms a main part of current research interests in this area. In our paper [4] we were able to obtain such generalization for a wide class of infinite dimensional linear systems with bounded inputs and outputs. In the works [17, 19, 21, 25, 26] we have obtained the solution of the output regulation problem for several specific systems governed by partial differential equations. Recently we have submitted for publication a joint work with G. Weiss of Imperial College, UK on output regulation for infinite dimensional linear systems with unbounded inputs and outputs. In this work we use the theory of regular linear systems first studied by G. Weiss. The theory of regular systems is purely abstract and an important open question is to provide nontrivial examples of such systems. In a recent joint work with D.S. Gilliam and G. Weiss we were able to show that a very general class of systems governed by linear parabolic partial differential equations on bounded domains in $\bbr^n$ with (unbounded) boundary inputs and outputs consist of regular systems. The central part of the proof deals with the analysis of the limiting behavior of the solution of a certain elliptic problem when the spectral parameter tends to plus infinity along the real axis.
Another series of joint works with C.I. Byrnes and D.S. Gilliam deals with distributed parameter systems governed by nonlinear partial differential equations with feedback control laws implemented through the boundary conditions [10, 15, 16, 18, 20, 27, 30-37]. Namely in these papers we analyzed several aspects of control problems related to a viscous Burgers' equation. Though the equation itself appears to be quite simple, the dynamics it generates in the presence of nontrivial boundary conditions, modeling a proportional error boundary feedback control law, is not simple and, as yet, not completely understood. In the paper [10], joint with D.S. Gilliam and our Ph.D. student A. Balogh (degree awarded in May 1997), we have demonstrated that the closed loop Burgers' system may contain one, two or three stationary solutions depending on the values of the gain parameter in the boundary conditions. In other words we have shown the existence of a bifurcation of the set of stationary solutions. This suggests that the global locally compact attractor, whose existence was established in [13], may be nontrivial. In a joint effort with D.S. Gilliam and C.I. Byrnes we are currently investigating the complete structure of the global attractor. In the work [18] we have also established existence, uniqueness and regularity of solutions for all square integrable initial data. In [15] we have investigated the limiting behavior of the solutions of the controlled Burgers' equation when the gain parameters tend to infinity. It was shown in [15] that under such a limit the solutions converge to the trajectories of the zero dynamics system. Using this result we were able to demonstrate that our feedback control law provides a semiglobal stabilization of the original (asymptotically) unstable uncontrolled system.
In another research effort [24], joint with A. Balogh and D.S. Gilliam, we have announced results (and outlined their proofs) on the feedback regularization of $3$-dimensional Navier-Stokes equations on a bounded domain. The paper containing full proofs is currently in preparation.
A series of works [23, 28, 29] joint with H.T. Banks (North Carolina State
University) and D.S. Gilliam is devoted to global solvability of damped
hyperbolic systems containing strong nonlinearity. Such systems model the
time evolution of elastomers. This work was continued by (our student joint
with D.S. Gilliam) G.A. Pinter (degree awarded May 1997) who proved that
under certain additional assumptions these systems have global attractors
(paper accepted in {\em Nonlinear Analysis}).
Within the past three years I have also begun a joint research work
with D.S. Gilliam on mathematical analysis of tornado dynamics. This work
has been supported by a grant from the Texas Advanced Research Program.
During the Summer and Fall of 1998 I presented a series of lectures on
tornado dynamics to a audience consisting of graduate students and faculty
members from the Department of Mathematics and Statistics and the Departments
of Geosciences and Civil Engineering. These lectures contained a
detailed review of known tornado models with full mathematical derivations
and, also, the necessary material from hydrodynamics and thermodynamics
of a fluid/gas flow. Currently I am working jointly with D.S. Gilliam on
a book on tornado dynamics, which is under consideration by the SIAM Frontiers
in Science.
While working on the tornado project, at the suggestion of Dr. R. Peterson
(Department of Geosciences), we began to study the system of
hydrodynamic equations that govern an airflow containing dust particles.
The paper containing full proofs is in preparation. In this paper we have
considered the Navier-Stokes and Euler equations coupled through the Stokes'
drag law. We have established the existence of a unique strong solution
on a finite time interval and investigated the limiting behavior of this
solution when the dust becomes very fine. We were able to rigorously
justify the fact (suggested by G. Saffman in 1961) that such a limit results
in a reduction of the effective viscosity of the flow.
I have also collaborated in joint research with Professor C.F. Martin
(Texas Tech University) on problems in the theory of polynomial and exponential
polynomial approximation of functions [38, 39]. We have a joint paper on applications of stochastic differential equations to the observability problem for linear and nonlinear diffusion equations
[40].
A significant part of my early research was related to the study of
nonlinear partial differential equations and infinite systems of ordinary
differential equations describing functions which take their values in
Riemannian manifolds. As a result of these interests I have published a
joint work with O.A. Ladyzhenskaya on global existence of hyperbolic harmonic
maps from a plane to an arbitrary complete Riemannian manifold [49].
In addition to this work, I have also published several papers on
the global existence of solutions for infinite systems of ordinary differential
equations on Riemannian manifolds[43-48, 51]. The main thrust of these works is the study of the
dynamics for geometric models of nonequilibrium statistical mechanics.
I have also done some work concerning global existence of solutions
for infinite systems of integro-differential equations (BBGKY hierarchy)
describing the time evolution of probability measures on infinite dimensional
phase spaces for these systems [45]. I have also carried out
research on soliton theory [52] and an application of Hamiltonian
mechanics methods in Lie group representation theory (sympletic quantization)
[53]. One other aspect
of my research interests has been the study of long-time behavior of solutions
of nonlinear evolution equations and attractor theory [41]. This includes
the study of subsets of infinite dimensional spaces having finite Hausdorff
or fractal dimensions and the connection between these results and hydrodynamics
[42].