Correlation effects in the trapping problem: general approach and rigorous results

The problem of Brownian survival among randomly located traps is considered with emphasis on the role of trap correlations. We proceed from the general representation of the survival probability as the expected value of the emptiness probability function applied to the Wiener sausage. Using the definition of (pure) trap attraction vs. repulsion in terms of the emptiness probability function, we prove the physical conjecture about the trapping slowdown or acceleration, according to the “sign” of correlations. Two specific models are studied along this line, in which the emptiness probability can be found explicitly; in particular, the long-time survival asymptotics is derived. A remarkable correlation effect of the survival probability dependence on the trap size in one dimension is also discussed

The discreteness of the spectrum of the Schrödinger operator equation and some properties of the s-numbers of the inverse Schrödinger operator

In this article, we investigate the discreteness and some other properties of the spectrum for the Schrödinger operator L defined by the formula LY=-d 2 y/dx 2 +A(A+I)/x 2 y+Q(x)y on the space L 2 (H, [0, ?)), where H is a Hilbert space. For the first time, an estimate is obtained for sum of the s-numbers of the inverse Schrödinger operator. The obtained results were applied to the Laplace's equation in an angular region.

Stabilizability and control co-design for discrete-time switched linear systems

International audienceIn this work we deal with the stabilizability property for discrete-time switched linear systems. First we provide a constructive necessary and sufficient condition for stabilizability based on set-theory and the characterization of a universal class of Lyapunov functions. Such a geometric condition is considered as the reference for comparing the computation-oriented sufficient conditions. The classical BMI conditions based on Lyapunov-Metzler inequalities are considered and extended. Novel LMI conditions for stabilizability, derived from the geometric ones, are presented that permit to combine generality with convexity. For the different conditions, the geometrical interpretations are provided and the induced stabilizing switching laws are given. The relations and the implications between the stabiliz-ability conditions are analyzed to infer and compare their conservatism and their complexity. The results are finally extended to the problem of the co-design of a control policy, composed by both the state feedback and the switching control law, for discrete-time switched linear systems. Constructive conditions are given in form of LMI that are necessary and sufficient for the stabilizability of systems which are periodic stabilizable