Rajeev Rajaram

Modeling complex systems macroscopically: Case/agent-based modeling, synergetics, and the continuity equation

Recently, the continuity equation (also known as the advection equation) has been used to study stability properties of dynamical systems, where a linear transfer operator approach was used to examine the stability of a nonlinear equation both in continuous and discrete time (Vaidya and Mehta, IEEE Trans Autom Control 2008, 53, 307–323; Rajaram et al., J Math Anal Appl 2010, 368, 144–156). Our study, which conducts a series of simulations on residential patterns, demonstrates that this usage of the continuity equation can advance Hakens synergetic approach to modeling certain types of complex, self-organizing social systems macroscopically. The key to this advancement comes from employing a case-based approach that (1) treats complex systems as a set of cases and (2) treats cases as dynamical vsystems which, at the microscopic level, can be conceptualized as k dimensional row vectors; and, at the macroscopic level, as vectors with magnitude and direction, which can be modeled as population densities. Our case-based employment of the continuity equation has four benefits for agent-based and case-based modeling and, more broadly, the social scientific study of complex systems where transport or spatial mobility issues are of interest: it (1) links microscopic (agent-based) and macroscopic (structural) modeling; (2) transforms the dynamics of highly nonlinear vector fields into the linear motion of densities; (3) allows predictions to be made about future states of a complex system; and (4) mathematically formalizes the structural dynamics of these types of complex social systems.

Exact boundary controllability results for a Rao–Nakra sandwich beam

Abstract The Rao–Nakra model of a three layer sandwich beam is analyzed for exact boundary controllability. The damped and undamped cases are considered. The multiplier method is used to obtain required observability inequalities that imply controllability. It is shown that if the control time T is large enough, under certain other parametric restrictions, the system is exactly controllable.

Connection between almost everywhere stability of an ODE and advection PDE

A result on the necessary and sufficient conditions for almost everywhere stability of an invariant set in continuous-time dynamical systems is presented. It is shown that the existence of a Lyapunov density is equivalent to the almost everywhere stability of an invariant set. Furthermore, such a density can be obtained as the positive solution of a linear partial differential equation analogous to the positive solution of Lyapunov equation for stable linear systems.

Cases, Clusters, Densities: Modeling the Nonlinear Dynamics of Complex Health Trajectories

In the health informatics era, modeling longitudinal data remains problematic. The issue is method: health data are highly nonlinear and dynamic, multilevel and multidimensional, comprised of multiple major/minor trends, and causally complex – making curve fitting, modeling and prediction difficult. The current study is fourth in a series exploring a case-based density (CBD) approach for modeling complex trajectories; which has the following advantages: it can (1) convert databases into sets of cases (k dimensional row vectors; i.e., rows containing k elements); (2) compute the trajectory (velocity vector) for each case based on (3) a set of bio-social variables called traces; (4) construct a theoretical map to explain these traces; (5) use vector quantization (i.e., k-means, topographical neural nets) to longitudinally cluster case trajectories into major/minor trends; (6) employ genetic algorithms and ordinary di.erential equations to create a microscopic (vector field) model (the inverse problem) of these trajectories; (7) look for complex steady-state behaviors (e.g., spiraling sources, etc) in the microscopic model; (8) draw from thermodynamics, synergetics and transport theory to translate the vector field (microscopic model) into the linear movement of macroscopic densities; (9) use the macroscopic model to simulate known and novel case-based scenarios (the forward problem); and (10) construct multiple accounts of the data by linking the theoretical map and k dimensional profile with the macroscopic, microscopic and cluster models. Given the utility of this approach, our purpose here is to organize our method (as applied to recent research) so it can be employed by others.

The utility of nonequilibrium statistical mechanics, specifically transport theory, for modeling cohort data

This article introduces a new case-based density approach to modeling big data longitudinally, which uses ordinary differential equations and the linear advection partial differential equations PDE to treat macroscopic, dynamical change as a transport issue of aggregate cases across continuous time. The novelty of this approach comes from its unique data-driven treatment of cases: which are K dimensional vectors; where the velocity vector for each case is computed according to its particular measurements on some set of empirically defined social, psychological, or biological variables. The three main strengths of this approach are its ability to: 1 translate the data driven, nonlinear trajectories of microscopic constituents cases into the linear movement of macroscopic trajectories, which take the form of densities; 2 detect the presence of multiple, complex steady state behaviors, including sinks, spiraling sources, saddles, periodic orbits, and attractor points; and 3 predict the motion of novel cases and time instances. To demonstrate the utility of this approach, we used it to model a recognized cohort dynamic: the longitudinal relationship between a countrys per capita gross domestic product GDP and its longevity rates. Data for the model came from the widely used Gapminder dataset. Empirical results, including the strength of the models fit and the novelty of its results particularly on a topic of such extensive study support the utility of our new approach.

Simultaneous boundary control of a Rao-Nakra sandwich beam

We consider the problem of boundary control of a system of three coupled partial differential equations that describe a three layer (Rao-Nakra type) sandwich beam with damping proportional to shear included in the core layer. In the case where one control is applied to each equation, we obtain exact controllablity modulo a finite dimensional quotient in a time determined by the three wave speeds. We show that in a longer time, under some mild conditions on the parameters, we can recover a similar exact controllability result using only two, or possibly even one appropriately chosen control function.

Actuator and sensor placement in linear advection PDE

We study the problem of actuator and sensor placement in a linear advection partial differential equation (PDE). The problem is motivated by its application to actuator and sensor placement in building systems for the control and detection of a scalar quantity such as temperature and contaminants. We propose a gramian based approach to the problem of actuator and sensor placement. The special structure of the advection PDE is exploited to provide an explicit formula for the controllability and observability gramian in the form of a multiplication operator. The explicit formula for the gramian, as a function of actuator and sensor location, is used to provide test criteria for the suitability of a given sensor and actuator location. Furthermore, the solution obtained using gramian based criteria is interpreted in terms of the flow of the advective vector field. In particular, the almost everywhere uniform stability property and ergodic properties of the advective vector field are shown to play a crucial role in deciding the location of actuators and sensors. Simulation results are performed to support the main results of this paper.

Exact controllability of a system of coupled strings in parallel

We consider a system of parallel identical strings coupled by distributed springs and dampers. It is shown that despite the distributed damping, there exists a branch of eigenvalues that is purely imaginary leading to sustained oscillations. It is also proved that when Dirichlet boundary controls are applied and the control time is greater than or equal to , where c is the wave speed, the system is exactly controllable in an appropriate state space.

Experimental study of stochastic resonance in atomic force microscopes

Stochastic resonance is an interesting phenomenon which can occur in bistable systems subject to periodic and random forcing. This effect produces an improvement in the sensitivity of the bistable system to the periodic signal. In this paper, stochastic resonance for atomic force microscope (AFM) is studied. The experimental results indicate that the AFM can be modeled as a bistable system similar to the Schmitt trigger, for which stochastic resonance has been well studied. The results indicate that stochastic resonance in AFM can be applied in many technological contexts as, for example, in the analysis of the effects of thermal noise in order to optimize the achievable resolution for imaging.

Lyapunov density for coupled systems

We prove a necessary and sufficient condition for the existence of Lyapunov density for a system of coupled autonomous ordinary differential equations. In particular, we characterize the kinds of couplings that preserve almost everywhere uniform stability of the origin provided the isolated systems have an almost everywhere uniformly stable equilibrium point at the origin.