1,986 research outputs found
Exponential convergence rate of ruin probabilities for level-dependent L\'evy-driven risk processes
We explicitly find the rate of exponential long-term convergence for the ruin
probability in a level-dependent L\'evy-driven risk model, as time goes to
infinity. Siegmund duality allows to reduce the pro blem to long-term
convergence of a reflected jump-diffusion to its stationary distribution, which
is handled via Lyapunov functions.Comment: 20 pages, 5 figure
Convex computation of the region of attraction of polynomial control systems
We address the long-standing problem of computing the region of attraction
(ROA) of a target set (e.g., a neighborhood of an equilibrium point) of a
controlled nonlinear system with polynomial dynamics and semialgebraic state
and input constraints. We show that the ROA can be computed by solving an
infinite-dimensional convex linear programming (LP) problem over the space of
measures. In turn, this problem can be solved approximately via a classical
converging hierarchy of convex finite-dimensional linear matrix inequalities
(LMIs). Our approach is genuinely primal in the sense that convexity of the
problem of computing the ROA is an outcome of optimizing directly over system
trajectories. The dual infinite-dimensional LP on nonnegative continuous
functions (approximated by polynomial sum-of-squares) allows us to generate a
hierarchy of semialgebraic outer approximations of the ROA at the price of
solving a sequence of LMI problems with asymptotically vanishing conservatism.
This sharply contrasts with the existing literature which follows an
exclusively dual Lyapunov approach yielding either nonconvex bilinear matrix
inequalities or conservative LMI conditions. The approach is simple and readily
applicable as the outer approximations are the outcome of a single semidefinite
program with no additional data required besides the problem description
Optimal Stabilization using Lyapunov Measures
Numerical solutions for the optimal feedback stabilization of discrete time
dynamical systems is the focus of this paper. Set-theoretic notion of almost
everywhere stability introduced by the Lyapunov measure, weaker than
conventional Lyapunov function-based stabilization methods, is used for optimal
stabilization. The linear Perron-Frobenius transfer operator is used to pose
the optimal stabilization problem as an infinite dimensional linear program.
Set-oriented numerical methods are used to obtain the finite dimensional
approximation of the linear program. We provide conditions for the existence of
stabilizing feedback controls and show the optimal stabilizing feedback control
can be obtained as a solution of a finite dimensional linear program. The
approach is demonstrated on stabilization of period two orbit in a controlled
standard map
On Duality for Lyapunov Functions of Nonstrict Convex Processes
This paper provides a novel definition for Lyapunov functions for difference
inclusions defined by convex processes. It is shown that this definition
reflects stability properties of nonstrict convex processes better than
previously used definitions. In addition the paper presents conditions under
which a weak Lyapunov function for a convex process yields a strong Lyapunov
function for the dual of the convex process
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