Almost Global Stochastic Stability

We develop a method to prove almost global stability of stochastic differential equations in the sense that almost every initial point (with respect to the Lebesgue measure) is asymptotically attracted to the origin with unit probability. The method can be viewed as a dual to Lyapunov's second method for stochastic differential equations and extends the deterministic result in [A. Rantzer, Syst. Contr. Lett., 42 (2001), pp. 161--168]. The result can also be used in certain cases to find stabilizing controllers for stochastic nonlinear systems using convex optimization. The main technical tool is the theory of stochastic flows of diffeomorphisms.Comment: Submitte

Improving Global Financial Stability

This report concludes that the failure of developing country governments and international financial institutions to adapt to changing markets helped trigger some of the world's financial crises. Arguing that global finance is "more susceptible to crisis than it need be," the report targets both developing and developed countries and the IMF as being in serious need of reform to prevent future breakdowns. The report endorses international standards to be adopted by developing countries, and hails private-sector participation and resources as vital to building an accepted set of best practices

Improving Global Financial Stability - Executive Summary

Executive summary of "Improving Global Financial Stability" report. Includes findings and recommendations in brief

Global imbalances and financial stability

Much of the policy debate in coming years will hinge on two questions: have global imbalances contributed to the financial crisis? Is a reduction in global imbalances a prerequisite to ensuring global financial stability? In light of available research and analysis, it is reasonable to argue that common causes likely lay behind both the crisis and global imbalances. They include heterogeneous saving preferences, asymmetric financial development across countries engaged in global financial markets, and the undersupply of liquid and safe assets at the aggregate level. Looking ahead, the international community has to strike the right balance between, on the one hand, countries’ legitimate sovereignty over monetary, capital account, and exchange rate policies and, on the other hand, intensified interdependencies, the global system’s increased complexity, and diverging economic prospects across countries. Rebalancing world demand will no doubt be a gradual, long-run process. To help foster an orderly unwinding, all countries need to ensure that their policies do not create further distortions in the global economy. Several improvements to the international monetary system could be considered to help reduce incentives for distortive policies.

Stability of Charged Global AdS4_4 Spacetimes

We study linear and nonlinear stability of asymptotically AdS$_4$ solutions in Einstein-Maxwell-scalar theory. After summarizing the set of static solutions we first examine thermodynamical stability in the grand canonical ensemble and the phase transitions that occur among them. In the second part of the paper we focus on nonlinear stability in the microcanonical ensemble by evolving radial perturbations numerically. We find hints of an instability corner for vanishingly small perturbations of the same kind as the ones present in the uncharged case. Collapses are avoided, instead, if the charge and mass of the perturbations come to close the line of solitons. Finally we examine the soliton solutions. The linear spectrum of normal modes is not resonant and instability turns on at extrema of the mass curve. Linear stability extends to nonlinear stability up to some threshold for the amplitude of the perturbation. Beyond that, the soliton is destroyed and collapses to a hairy black hole. The relative width of this stability band scales down with the charge Q, and does not survive the blow up limit to a planar geometry.Comment: 43 pg, 22 fig. Published version. Appendix adde

Smoothness dependent stability in corrosion detection

We consider the stability issue for the determination of a linear corrosion in a conductor by a single electrostatic measurement. We established a global log-log type stability when the corroded boundary is simply Lipschitz. We also improve such a result obtaining a global log stability by assuming that the damaged boundary is $C^{1,1}$-smooth

Sum-of-squares of polynomials approach to nonlinear stability of fluid flows: an example of application

With the goal of providing the first example of application of a recently proposed method, thus demonstrating its ability to give results in principle, global stability of a version of the rotating Couette flow is examined. The flow depends on the Reynolds number and a parameter characterising the magnitude of the Coriolis force. By converting the original Navier-Stokes equations to a finite-dimensional uncertain dynamical system using a partial Galerkin expansion, high-degree polynomial Lyapunov functionals were found by sum-of-squares-of-polynomials optimization. It is demonstrated that the proposed method allows obtaining the exact global stability limit for this flow in a range of values of the parameter characterising the Coriolis force. Outside this range a lower bound for the global stability limit was obtained, which is still better than the energy stability limit. In the course of the study several results meaningful in the context of the method used were also obtained. Overall, the results obtained demonstrate the applicability of the recently proposed approach to global stability of the fluid flows. To the best of our knowledge, it is the first case in which global stability of a fluid flow has been proved by a generic method for the value of a Reynolds number greater than that which could be achieved with the energy stability approach