4,424 research outputs found
Solution of Evolutionary Games via Hamilton-Jacobi-Bellman Equations
This poster is focused on construction of solutions for bimatrix evolutionary games based on methods of the theory of optimal control and generalized solutions of Hamilton-Jacobi-Bellman equations. It is assumed that the evolutionary dynamics describe interactions of agents in large population groups in biological and social models or interactions of investors in financial markets.
Interactions of agents are subject to the dynamic process which provides the possibility to control flows between different types of behavior or investments. It is worth noting that the dynamics of interactions can be interpreted as the system of Kolmogorov’s type differential equations. Parameters of the dynamics are not fixed a priori and can be treated as controls constructed either as time programs or on the feedback principle.
Payoff functionals in the evolutionary game of two coalitions are determined by the limit of average matrix gains on an infinite horizon. The notion of a dynamical Nash equilibrium is introduced in the class of control feedbacks within Krasovskii’s theory of differential games.
Elements of a dynamical Nash equilibrium are based on guaranteed feedbacks constructed within the framework of the theory of generalized solutions of Hamilton-Jacobi-Bellman equations. The value functions for the series of differential games are constructed analytically and their stability properties are verified using the technique of conjugate derivatives.
The equilibrium trajectories are generated on the basis of positive feedbacks originated by value functions. It is shown that the proposed approach provides new qualitative results for the equilibrium trajectories in evolutionary games and ensures better results for payoff functionals than replicator dynamics in evolutionary games or Nash values in static bimatrix games.
The efficiency of the proposed approach is demonstrated by applications to construction of equilibrium dynamics for agents’ interactions in financial markets
A new method for large time behavior of degenerate viscous Hamilton--Jacobi equations with convex Hamiltonians
We investigate large-time asymptotics for viscous Hamilton--Jacobi equations with possibly degenerate diffusion terms. We establish new results on the convergence, which are the first general ones concerning equations which are neither uniformly parabolic nor first order. Our method is based on the nonlinear adjoint method and the derivation of new estimates on long time averaging effects. It also extends to the case of weakly coupled systems
Best-fit quasi-equilibrium ensembles: a general approach to statistical closure of underresolved Hamiltonian dynamics
A new method of deriving reduced models of Hamiltonian dynamical systems is
developed using techniques from optimization and statistical estimation. Given
a set of resolved variables that define a model reduction, the
quasi-equilibrium ensembles associated with the resolved variables are employed
as a family of trial probability densities on phase space. The residual that
results from submitting these trial densities to the Liouville equation is
quantified by an ensemble-averaged cost function related to the information
loss rate of the reduction. From an initial nonequilibrium state, the
statistical state of the system at any later time is estimated by minimizing
the time integral of the cost function over paths of trial densities.
Statistical closure of the underresolved dynamics is obtained at the level of
the value function, which equals the optimal cost of reduction with respect to
the resolved variables, and the evolution of the estimated statistical state is
deduced from the Hamilton-Jacobi equation satisfied by the value function. In
the near-equilibrium regime, or under a local quadratic approximation in the
far-from-equilibrium regime, this best-fit closure is governed by a
differential equation for the estimated state vector coupled to a Riccati
differential equation for the Hessian matrix of the value function. Since
memory effects are not explicitly included in the trial densities, a single
adjustable parameter is introduced into the cost function to capture a
time-scale ratio between resolved and unresolved motions. Apart from this
parameter, the closed equations for the resolved variables are completely
determined by the underlying deterministic dynamics
On the Large Time Behavior of Solutions of Hamilton-Jacobi Equations Associated with Nonlinear Boundary Conditions
In this article, we study the large time behavior of solutions of first-order
Hamilton-Jacobi Equations, set in a bounded domain with nonlinear Neumann
boundary conditions, including the case of dynamical boundary conditions. We
establish general convergence results for viscosity solutions of these
Cauchy-Neumann problems by using two fairly different methods : the first one
relies only on partial differential equations methods, which provides results
even when the Hamiltonians are not convex, and the second one is an optimal
control/dynamical system approach, named the "weak KAM approach" which requires
the convexity of Hamiltonians and gives formulas for asymptotic solutions based
on Aubry-Mather sets
Perturbative calculation of quasi-potential in non-equilibrium diffusions: a mean-field example
In stochastic systems with weak noise, the logarithm of the stationary
distribution becomes proportional to a large deviation rate function called the
quasi-potential. The quasi-potential, and its characterization through a
variational problem, lies at the core of the Freidlin-Wentzell large deviations
theory%.~\cite{freidlin1984}.In many interacting particle systems, the particle
density is described by fluctuating hydrodynamics governed by Macroscopic
Fluctuation Theory%, ~\cite{bertini2014},which formally fits within
Freidlin-Wentzell's framework with a weak noise proportional to ,
where is the number of particles. The quasi-potential then appears as a
natural generalization of the equilibrium free energy to non-equilibrium
particle systems. A key physical and practical issue is to actually compute
quasi-potentials from their variational characterization for non-equilibrium
systems for which detailed balance does not hold. We discuss how to perform
such a computation perturbatively in an external parameter , starting
from a known quasi-potential for . In a general setup, explicit
iterative formulae for all terms of the power-series expansion of the
quasi-potential are given for the first time. The key point is a proof of
solvability conditions that assure the existence of the perturbation expansion
to all orders. We apply the perturbative approach to diffusive particles
interacting through a mean-field potential. For such systems, the variational
characterization of the quasi-potential was proven by Dawson and Gartner%.
~\cite{dawson1987,dawson1987b}. Our perturbative analysis provides new explicit
results about the quasi-potential and about fluctuations of one-particle
observables in a simple example of mean field diffusions: the
Shinomoto-Kuramoto model of coupled rotators%. ~\cite{shinomoto1986}. This is
one of few systems for which non-equilibrium free energies can be computed and
analyzed in an effective way, at least perturbatively
A PDE approach to large-time asymptotics for boundary-value problems for nonconvex Hamilton-Jacobi Equations
We investigate the large-time behavior of three types of initial-boundary
value problems for Hamilton-Jacobi Equations with nonconvex Hamiltonians. We
consider the Neumann or oblique boundary condition, the state constraint
boundary condition and Dirichlet boundary condition. We establish general
convergence results for viscosity solutions to asymptotic solutions as time
goes to infinity via an approach based on PDE techniques. These results are
obtained not only under general conditions on the Hamiltonians but also under
weak conditions on the domain and the oblique direction of reflection in the
Neumann case
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