A looped-functional approach for robust stability analysis of linear impulsive systems

A new functional-based approach is developed for the stability analysis of linear impulsive systems. The new method, which introduces looped-functionals, considers non-monotonic Lyapunov functions and leads to LMIs conditions devoid of exponential terms. This allows one to easily formulate dwell-times results, for both certain and uncertain systems. It is also shown that this approach may be applied to a wider class of impulsive systems than existing methods. Some examples, notably on sampled-data systems, illustrate the efficiency of the approach.Comment: 13 pages, 2 figures, Accepted at Systems & Control Letter

Another Approach to Consensus and Maximally Informed Opinions with Increasing Evidence

Merging of opinions results underwrite Bayesian rejoinders to complaints about the subjective nature of personal probability. Such results establish that sufficiently similar priors achieve consensus in the long run when fed the same increasing stream of evidence. Initial subjectivity, the line goes, is of mere transient significance, giving way to intersubjective agreement eventually. Here, we establish a merging result for sets of probability measures that are updated by Jeffrey conditioning. This generalizes a number of different merging results in the literature. We also show that such sets converge to a shared, maximally informed opinion. Convergence to a maximally informed opinion is a (weak) Jeffrey conditioning analogue of Bayesian “convergence to the truth” for conditional probabilities. Finally, we demonstrate the philosophical significance of our study by detailing applications to the topics of dynamic coherence, imprecise probabilities, and probabilistic opinion pooling

Data-driven satisficing measure and ranking

We propose an computational framework for real-time risk assessment and prioritizing for random outcomes without prior information on probability distributions. The basic model is built based on satisficing measure (SM) which yields a single index for risk comparison. Since SM is a dual representation for a family of risk measures, we consider problems constrained by general convex risk measures and specifically by Conditional value-at-risk. Starting from offline optimization, we apply sample average approximation technique and argue the convergence rate and validation of optimal solutions. In online stochastic optimization case, we develop primal-dual stochastic approximation algorithms respectively for general risk constrained problems, and derive their regret bounds. For both offline and online cases, we illustrate the relationship between risk ranking accuracy with sample size (or iterations).Comment: 26 Pages, 6 Figure

Feynman-Kac representation of fully nonlinear PDEs and applications

The classical Feynman-Kac formula states the connection between linear parabolic partial differential equations (PDEs), like the heat equation, and expectation of stochastic processes driven by Brownian motion. It gives then a method for solving linear PDEs by Monte Carlo simulations of random processes. The extension to (fully)nonlinear PDEs led in the recent years to important developments in stochastic analysis and the emergence of the theory of backward stochastic differential equations (BSDEs), which can be viewed as nonlinear Feynman-Kac formulas. We review in this paper the main ideas and results in this area, and present implications of these probabilistic representations for the numerical resolution of nonlinear PDEs, together with some applications to stochastic control problems and model uncertainty in finance

Combinatorial Voting

We study elections that simultaneously decide multiple issues, where voters have independent private values over bundles of issues. The innovation is in considering nonseparable preferences, where issues may be complements or substitutes. Voters face a political exposure problem: the optimal vote for a particular issue will depend on the resolution of the other issues. Moreover, the probabilities that the other issues will pass should be conditioned on being pivotal. We prove that equilibrium exists when distributions over values have full support or when issues are complements. We then study large elections with two issues. There exists a nonempty open set of distributions where the probability of either issue passing fails to converge to either 1 or 0 for all limit equilibria. Thus, the outcomes of large elections are not generically predictable with independent private values, despite the fact that there is no aggregate uncertainty regarding fundamentals. While the Condorcet winner is not necessarily the outcome of a multi-issue election, we provide sufficient conditions that guarantee the implementation of the Condorcet winner. © 2012 The Econometric Society

Robust ℋ2 Performance: Guaranteeing Margins for LQG Regulators

This paper shows that ℋ2 (LQG) performance specifications can be combined with structured uncertainty in the system, yielding robustness analysis conditions of the same nature and computational complexity as the corresponding conditions for ℋ∞ performance. These conditions are convex feasibility tests in terms of Linear Matrix Inequalities, and can be proven to be necessary and sufficient under the same conditions as in the ℋ∞ case. With these results, the tools of robust control can be viewed as coming full circle to treat the problem where it all began: guaranteeing margins for LQG regulators