Browsing by Subject "Stability analysis"

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Stability analyses of auroral substorm onset and solar wind
(2019-12) Derr, Jason Robert; Breizman, Boris N.; Horton, C.W. (Claude Wendell), 1942-; Hazeltine, Richard D.; Shapiro, Paul R.
Pertaining to the stability analysis of auroral substorm onset, a geometric wedge model of the near-earth nightside plasma sheet is used to derive a wave equation for low frequency shear flow-interchange waves which transmit E x B sheared zonal flows along magnetic flux tubes towards the ionosphere. Discrepancies with the wave equation result used in Kalmoni et al. (2015) for shear flow-ballooning instability are discussed. The wedge wave equation is used to compute rough expressions for dispersion relations and local growth rates in the midnight region of the nightside magnetotail where the instability develops, forming the auroral beads characteristic of geomagnetic substorm onset. Stability analysis for the shear flow-interchange modes demonstrates that nonlinear analysis is necessary for quantitatively accurate results and determines the spatial scale on which the instability varies. The Rice Convection Model-Equilibrium (RCM-E) is used to provide background fields for a global magnetospheric wedge wave equation, from which the growth rates and dispersion relations can be calculated for the shear flow-interchange instability. Mapping of this growing traveling wave back to the magnetosphere yields the auroral bead projections of the instability. The cause of magnetic substorm onset by comparison with the beads, and its location in the magnetotail, is determinable once more suitable simulation run is performed. The linear stage of the marginally stable instability is discussed in detail. Subsequent nonlinear relaxation properties of the auroral arc, including saturation value of instability amplitudes, are determined. Shear flow-interchange instability appears to cause magnetic substorm onset, insofar as auroral beads are its signature. Pertaining to the stability analysis of jet microstreams, fast solar wind streams are known to be dominated by Alfvénic turbulence, i.e. large amplitude magnetic field and quasi-incompressible velocity fluctuations with a correlation corresponding to waves propagating away from the Sun. In addition, the Ulysses spacecraft has shown that microstreams, persistent long period (1/2-2 days) fluctuations in the radial velocity field are ubiquitous in the fast wind. This contribution explores the possible causal relation between microstreams and Alfvénic turbulence. We carry out a parametric study setup for the linear and nonlinear stability of the microstream jets to Kelvin-Helmholtz (KH) instabilities: starting from the profiles of density, radial speed and magnetic field observed in the solar wind, we aim to investigate both at what distance from the Sun KH instabilities may be triggered and the nature of the ensuing nonlinear dynamics.
Stability analysis and optimal control of large-scale stochastic systems
(2022-08-11) Hmedi, Hassan; Shakkottai, Sanjay; Caramanis, Constantine; Pang, Guodong; De Veciana, Gustavo; Zitkovic, Gordan
In the past years, large-scale stochastic networks have been an intense subject of study due to their use in modelling a variety of systems including telecommunications, service and data centers, patient flows, etc. The optimal control of such systems has found numerous applications such as, but not limited to, finance and cognitive neuroscience. This thesis focuses on the stability analysis and optimal control of stochastic systems. In particular, we study: (1) the ergodic properties of multiclass multi-pool networks in the Halfin-Whitt regime; and (2) the optimal control of stochastic networks assuming a structural property relating the running cost to the solution of the Hamilton-Jacobi-Bellman (HJB) equation. In the first part of this thesis, we introduce a "system-wide safety staffing" (SWSS) parameter for multiclass multi-pool networks in the Halfin-Whitt regime which have any tree topology. This parameter can be regarded as the optimal reallocation of the capacity fluctuations (positive or negative) when each server pool employs a square-root staffing rule. First, we provide an explicit form of the SWSS as a function of the system parameters, which is derived using a graph theoretic approach based on Gaussian elimination. In addition, we give an equivalent characterization of the SWSS parameter via the drift parameters of the limiting diffusion. Then, we show that if the SWSS parameter is negative, the limiting diffusion and the diffusion-scaled queueing processes are transient under any Markov control, and cannot have a stationary distribution when this parameter is zero. If it is positive, we show that the diffusion-scaled queueing processes are stabilizable, that is, there exists a scheduling policy under which the stationary distributions of the controlled processes are tight over the size of the network. Finally, we show that there exists a control under which the limiting controlled diffusion is exponentially ergodic. Thus, we have identified a necessary and sufficient condition for the stabilizability of such networks in the Halfin-Whitt regime. In the second part of this thesis, we examine two problems related to the general topic of optimal control of stochastic systems. In the first problem, we consider a linear system with Gaussian noise observed by multiple sensors which transmit measurements over a dynamic lossy network. We assume that the system is stabilizable, that is, there exists a control such that all states variables are bounded during system's behavior. First, we characterize the stationary optimal sensor scheduling policy for the finite horizon, discounted, and long-term average cost problems. Then, we show that there exists a structural property relating the running cost to the value function which is the solution of the average cost problem. In addition, we show that the value iteration algorithm converges to this solution. Further, we show that the suboptimal policies provided by the rolling horizon truncation of the value iteration also guarantee stability and provide near-optimal average cost. Lastly, we provide qualitative characterizations of the multidimensional set of measurement loss rates for which the system is stabilizable for a static network, thus extending earlier results on intermittent observations. In the second problem and motivated by the results from the previous problem, a multiplicative relative value iteration algorithm (RVI) for infinite-horizon risk-sensitive control of controlled diffusions in [doublestruck R][superscript d] is studied. We assume that the running cost is near-monotone and that it is related to the solution of the multiplicative HJB equation through a structural assumption. We show that this structural assumption implies the existence of a control under which the ground state diffusion is exponentially ergodic. In addition, we show that the multiplicative RVI algorithm converges globally to the solution of the multiplicative dynamic programming equation starting from any positive initial condition; thus extending upon the results in the literature.