1,721,016 research outputs found
A converse Lyapunov theorem for almost sure stabilizability
No full textWe prove a converse Lyapunov theorem for almost sure stabilizability and almost sure asymptotic stabilizability of controlled diffusions:
given a stochastic system a.s. stochastic open-loop stabilizable at the origin, we construct a lower semicontinuous positive definite function
whose level sets form a local basis of viable neighborhoods of the equilibrium. This result provides, with the direct Lyapunov theorems proved
in a companion paper, a complete Lyapunov-like characterization of the a.s. stabilizabilit
Lyapunov stabilizability of controlled diffusions via a superoptimality principle for viscosity solutions
Full text linkWe prove optimality principles for semicontinuous bounded viscosity solutions of Hamilton-Jacobi-Bellman equations. In particular, we provide a representation formula for viscosity supersolutions as value functions of suitable obstacle control problems. This result is applied to extend the Lyapunov direct method for stability to controlled Ito stochastic differential equations. We define the appropriate concept of the Lyapunov function to study stochastic open loop stabilizability in probability and local and global asymptotic stabilizability (or asymptotic controllability). Finally, we illustrate the theory with some examples
Stationary equilibria and their stability in a Kuramoto MFG with strong interaction
No full textRecently, R. Carmona, Q. Cormier, and M. Soner proposed a Mean Field Game (MFG) version of the classical Kuramoto model, which describes synchronization phenomena in a large population of "rational" interacting oscillators. The MFG model exhibits several stationary equilibria, but the characterization of these equilibria and their ability to capture dynamic equilibria in long time remains largely open. In this paper, we demonstrate that, up to a phase translation, there are only two possible stationary equilibria: the incoherent equilibrium and the self-organizing equilibrium, given that the interaction parameter is sufficiently large. Furthermore, we present some local stability properties of the self-organizing equilibrium
Concentration of ground states in stationary mean-field games systems
Full text linkWe provide the existence of classical solutions to stationary mean-field game systems in the whole space R-N, with coercive potential and aggregating local coupling, under general conditions on the Hamiltonian. The only structural assumption we make is on the growth at infinity of the coupling term in terms of the growth of the Hamiltonian. This result is obtained using a variational approach based on the analysis of the nonconvex energy associated to the system. Finally, we show that in the vanishing viscosity limit, mass concentrates around the flattest minima of the potential. We also describe the asymptotic shape of the rescaled solutions in the vanishing viscosity limit, in particular proving the existence of ground states, i.e., classical solutions to mean-field game systems in the whole space without potential, and with aggregating coupling
Volume constrained minimizers of the fractional perimeter with a potential energy
Full text linkWe consider volume-constrained minimizers of the fractional perimeter with the addition of a potential energy in the form of a volume inte- gral. Such minimizers are solutions of the prescribed fractional curvature problem. We prove existence and regularity of minimizers under suitable assumptions on the potential energy, which cover the periodic case. In the small volume regime we show that minimizers are close to balls, with a quantitative estimate
Liouville properties and critical value of fully nonlinear elliptic operators
Full text linkWe prove some Liouville properties for sub- and supersolutions of fully nonlinear degenerate elliptic equations in the whole space. Our assumptions allow the coefficients of the first order terms to be large at infinity, provided they have an appropriate sign, as in Ornstein-Uhlenbeck operators. We give two applications. The first is a stabilization property for large times of solutions to fully nonlinear parabolic equations. The second is the solvability of an ergodic Hamilton-Jacobi-Bellman equation that identifies a unique critical value of the operator
Almost sure stabilizability of controlled degenerate diffusions
Full text linkWe develop a direct Lyapunov method for the almost sure open-loop stabilizability and asymptotic stabilizability of controlled degenerate diffusion processes. The infinitesimal decrease condition for a Lyapunov function is a new form of Hamilton–Jacobi–Bellman partial differential inequality of second order. We give local and global versions of the first and second Lyapunov theorems, assuming the existence of a lower semicontinuous Lyapunov function satisfying such an inequality in the viscosity sense. An explicit formula for a stabilizing feedback is provided for affine systems with smooth Lyapunov function. Several examples illustrate the theory
Brake orbits and heteroclinic connections for first order mean field games
Full text linkWe consider first order variational mean field games (MFG) in the whole space, with aggregative interactions and density constraints, having stationary equilibria consisting of two disjoint compact sets of distributions with finite quadratic moments. Under general assumptions on the interaction potential, we provide a method for the construction of periodic in time solutions to the MFG, which oscillate between the two sets of static equilibria, for arbitrarily large periods. Moreover, as the period increases to infinity, we show that these periodic solutions converge, in a suitable sense, to heteroclinic connections. As a model example, we consider a MFG system where the interactions are of (aggregative) Riesz-type
The isoperimetric problem for nonlocal perimeters
Full text linkWe consider a class of nonlocal generalized perimeters which includes fractional perimeters and Riesz type potentials. We prove a general isoperimetric inequality for such functionals, and we discuss some applications. In particular we prove existence of an isoperimetric profile, under suitable assumptions on the interaction kernel
Homogenization of a mean field game system in the small noise limit
Full text linkThis paper concerns the simultaneous effect of homogenization and of the
small noise limit for a order mean field games (MFG) system
with local coupling and quadratic Hamiltonian. We show under some additional
assumptions that the solutions of our system converge to a solution of an
effective order system whose effective operators are defined
through a cell problem which is a order system of ergodic
MFG type. We provide several properties of the effective operators and we show
that in general the effective system looses the MFG structure.Comment: Corrected version, we eliminate a Proposition (7.5) which was false,
without affecting the final resul
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