4,424 research outputs found

    Solution of Evolutionary Games via Hamilton-Jacobi-Bellman Equations

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    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

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    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

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    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

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    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

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    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 1/N1/\sqrt{N}, where NN 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 λ\lambda, starting from a known quasi-potential for λ=0\lambda=0. 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

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    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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