Reality cannot be found except in One single source,
because of the interconnection of all things with one another. ...
I maintain also that substances, whether material or immaterial,
cannot be conceived in their bare essence without any activity,
activity being of the essence of substance in general.
(Gottfried Wilhelm Leibniz, 1646-1716)

We are interested in non-trivial dynamic processes, including but not limited to nonlocal and those that are far from their equilibrium, as well as in the associated phenomena and systems, and their modeling. Such processes occur frequently in systems ranging from semiconductor plasma to chemical and biological networks, and from tiny man-made nano-objects to living cell cycles. Furthermore, in some cases we need to optimize such processes or systems, while in other cases we need to find a control law such that a certain criterion is achieved for the process or system in question.

In this focus area, our research interests have included the following topics:

• Complex dynamics, stability, nonlocal and nonequilibrium processes;
• Optimization methodologies and control of complex systems.

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Complex dynamics, stability, nonlocal and nonequilibrium processes and phenomena

Our contributions here include the analysis of systems with complex dynamic behaviour with important examples ranging from RNA nanostructures [ EJ-137 , EJ-145 ] to Carbon Nanotube (CNT) arrays [ EJ-112 , EJ-116 , EJ-122 , EJ-131 ], to nonlocal, non-equilibrium semiconductor plasma [ EJ-20 , EJ-30 , EJ-32 , EJ-37 , EJ-51 , EJ-70 ], and to low dimensional nanostructures [ EJ-95 ]. Although these examples are from diverse areas of applications, they are unified by the fact that in all these examples we have to deal with nonlinear multiscale dynamic processes representing some of the key challenges of modern applied mathematics. The mathematical tools we develop and apply in our analysis of these processes are ranging from PDE-based coupled models (e.g., models of hydrodynamic type, drift-diffusion models, and quantum corrections of such models) to elaborate atomistic methodologies.

Dynamics in low dimensional nanostructures belongs to a class of problems of our continuing interest, in particular where we have to account for dynamic coupling of different phenomena or processes [ EJ-90 ]. Many problems in the field of nanoscience require efficient solutions of eigenvalue PDE problems which subsequently are discretized. The dynamics of coupling between spectrum and resolvent under epsilon-pertubations of operator and matrix spectra were studied both theoretically and numerically in [ EJ-34 ].  The phenomenon of non-trivial pseudospectra encountered in these dynamics was treated by relating information in the complex plane to the behaviour of operators and matrices. We are also interested in several fundamental problems for multiband effective mass theory as applied to low dimensional nanostructures [ EJ-83 , EJ-94 , EJ-133 ].

On a more classical level, we were successful in generalizing the Courant-Friedrichs-Lewy condition to coupled dynamic problems of electroelasticity; we showed for the fist time the well-posedness of such problems, analyzed regularity of their solutions, and proposed efficient variational-difference numerical approximations [ EJ-1 , EJ-11 , EJ-13 , EJ-22 , EJ-24 , EJ-26 , EJ-29 , EJ-31 , EJ-71 ]. General discrete models for dynamic coupled thermoelasticity were discussed in [ EJ-41 ] where we also developed the Cayley transform technique for the analysis of stability of numerical approximations for these coupled problems. Non-trivial dynamic phenomena such as thermal spiking and degradation were analyzed by us for the first time in the context of mixed (ODEs and PDEs) coupled systems [ EJ-45 , EJ-59 ]. Another topic of interest was the study of the effect of thermal damage induced by ultrashort pulsed lasers [ EJ-89 , EJ-96 , EJ-119 ]. We also contributed to the development of models based on the hyperbolic heat conduction where the standard Fourier law is replaced by the Cattaneo-Vernotte law in the context of extended thermodynamics [ EJ-28 , EJ-40 , EJ-49 , EJ-78 ].

We are interested in transport phenomena and dynamic problems not only in the context of physical and engineering systems, but also in the context of biological and socio-economic systems, including finance. Our contributions to the latter can be found here. In addition to RNA nanostructures [ EJ-137 , EJ-145 ] , in the context of the former we are also interested in RNA silencing phenomena [ EP-67 ], DNA (and other complex molecules) relaxation dynamics [ EJ-105 , EJ-109 ], modelling genetic networks as a multiscale process [ EP-37 ] (as well as in modeling other networks arising in physical, biological, and engineering sciences), and the dynamics of cell cycles [ EJ -130 ].

In dealing with a number of problems listed above, in some cases we encounter highly oscillatory functions characterizing certain processes or phenomena, but given with incomplete information. The development of efficient formulae for numerical integration of such functions is a well-known challenge, and we developed and theoretically justified several classes of optimal formulae for numerical integration of highly oscillatory functions in both one- and multi-dimensional cases [ EJ-27 , EJ-43 , EJ-48 , EJ-66 ]. We have also been working on geometric and variational approaches in the analysis of complex dynamics.

Nonequilibrium Molecular Dynamics (NMD) is a subject that goes back to works by Fermi. By using NMD, we explored geometry and temperature dependent properties of materials at different scales, including the analysis of such properties in low dimensional nanostructures. Our interests include structural, thermodynamic, mechanical, transport, and optoelectronic properties, as well as relative and phase stability of materials at the nanoscale. Other tools that we are developing in these studies are based on first principles calculations, nonequilibrium Green's function techniques and density functional theory  [ EJ-123 , EJ-125 , EJ-134 , EJ-139 , EJ-140 , EJ-144 ].

Optimization methodologies and control of complex systems

We are interested in optimization of complex systems such as multicomponent, multiphase materials and systems, polymers, composites.

For example, in solving such problems with atomistic and Molecular Dynamics (MD) tools, an initial topological optimization can lead to a substantial speed up of subsequent computations. New efficient algorithms for such an optimization were discussed in [ EJ-58 ] based on several distance geometry ideas. Another example includes multiphase materials where we proposed and applied a new hybrid optimization procedure in the context of shape memory materials [ EJ-118 ].

We are also interested in control of complex systems. In particular, in [ EJ-60 ] in the context of control theory we analyzed in detail a theoretical framework for deterministic and stochastic dynamics with hyperbolic Hamilton-Jacobi-Bellman equations. It was proposed to approximate the original controlled dynamics with a sequence of PDEs. By applying the Steklov-Poincare operator technique the general form of equations in such a sequence was established. In [ EJ-21 ] we dealt with nonsmooth optimal control problems in the case when the control was allowed to be a discontinuous function. We analysed smoothness assumptions on an adjoint process in deterministic and stochastic cases. Possibilities of steep generalized space-gradients of the adjoint function implied the necessity of an approximation of the Hamiltonian. The key question of such an approximation is a relationship between the control and the value function. Under quite general assumptions it was proved that the performance measure for the original process is determined by the control function with possible discontinuities. Many complex systems operate under the presence of uncertainty which may include human factors. It is known that with the increasing complexity of technological systems that operate in dynamically changing environments and require human supervision or a human operator, the relative share of human errors is increasing across all modern applications. This indicates that in the analysis and control of such systems, human factors should not be eliminated by conventional formal mathematical methodologies. Instead, they must be incorporated into the modelling framework. These issues were addressed in [ EJ-129 ]. We demonstrated that the problem can be reduced to a set of Hamilton-Jacobi-Bellman equations where human factors are incorporated via estimations of the system Hamiltonian. The proposed methodology provides a way to integrate human factors into the solving process of the models for complex dynamic systems.

The topic of particular interest is control of low dimensional nanostructures and other physical and biological systems at the nanoscale. In the context of low dimensional nanostructures, our control parameters may include strain, thermal, piezoelectric, and magnetic fields, depending on the types of problems at hand [ EJ-76 , EJ-90 , EJ-94 , EJ-135 , EJ-138 ].

Time and space and gravitation have no separate existence from matter.
(Albert Einstein , 1879-1955)

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