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Register a new userOn ergodic control problem for viscous Hamilton--Jacobi equations for weakly coupled elliptic systems
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In this article we study ergodic problems in the whole space $mathbb{R}^N$ for a weakly coupled systems of viscous Hamilton-Jacobi equations with coercive right-hand sides. The Hamiltonians are assumed to have a fairly general structure and the switching rates need not be constant. We prove the existence of a critical value $lambda^*$ such that the ergodic eigenvalue problem has a solution for every $lambdaleqlambda^*$ and no solution for $lambda>lambda^*$. Moreover, the existence and uniqueness of non-negative solutions corresponding to the value $lambda^*$ are also established. We also exhibit the implication of these results to the ergodic optimal control problems of controlled switching diffusions.
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The large time behavior of solutions to Cauchy problem for viscous Hamilton-Jacobi equation is classified. The large time asymptotics are given by very singular self-similar solutions on one hand and by self-similar viscosity solutions on the other hand
The well known phenomenon of exponential contraction for solutions to the viscous Hamilton-Jacobi equation in the space-periodic setting is based on the Markov mechanism. However, the corresponding Lyapunov exponent $lambda( u)$ characterizing the exponential rate of contraction depends on the viscosity $ u$. The Markov mechanism provides only a lower bound for $lambda( u)$ which vanishes in the limit $ u to 0$. At the same time, in the inviscid case $ u=0$ one also has exponential contraction based on a completely different dynamical mechanism. This mechanism is based on hyperbolicity of action-minimizing orbits for the related Lagrangian variational problem. In this paper we consider the discrete time case (kicked forcing), and establish a uniform lower bound for $lambda( u)$ which is valid for all $ ugeq 0$. The proof is based on a nontrivial interplay between the dynamical and Markov mechanisms for exponential contraction. We combine PDE methods with the ideas from the Weak KAM theory.
This paper is a review of results on Optimisation which are perhaps not so standard in the PDE realm. To this end, we consider the problem of deriving the PDEs associated to the optimal control of a system of either ODEs or SDEs with respect to a vector-valued cost functional. Optimisation is considered with respect to a partial ordering generated by a given cone. Since in the vector case minima may not exist, we define vectorial value functions as (Pareto) minimals of the ordering. Our main objective is the derivation of the model PDEs which turn out to be parametric families of HJB single equations instead of systems of PDEs. However, this allows the use of the theory of Viscosity Solutions.
Let $(Omega, mu)$ be a probability space endowed with an ergodic action, $tau$ of $( {mathbb R} ^n, +)$. Let $H(x,p; omega)=H_omega(x,p)$ be a smooth Hamiltonian on $T^* {mathbb R} ^n$ parametrized by $omegain Omega$ and such that $ H(a+x,p;tau_aomega)=H(x,p;omega)$. We consider for an initial condition $fin C^0 ( {mathbb R}^n)$, the family of variational solutions of the stochastic Hamilton-Jacobi equations $$left{ begin{aligned} frac{partial u^{ varepsilon }}{partial t}(t,x;omega)+Hleft (frac{x}{ varepsilon } , frac{partial u^varepsilon }{partial x}(t,x;omega);omega right )=0 & u^varepsilon (0,x;omega)=f(x)& end{aligned} right .$$ Under some coercivity assumptions on $p$ -- but without any convexity assumption -- we prove that for a.e. $omega in Omega$ we have $C^0-lim u^{varepsilon}(t,x;omega)=v(t,x)$ where $v$ is the variational solution of the homogenized equation $$left{ begin{aligned} frac{partial v}{partial t}(x)+{overline H}left (frac{partial v }{partial x}(x) right )=0 & v (0,x)=f(x)& end{aligned} right.$$
The long-time average behaviour of the value function in the calculus of variations, where both the Lagrangian and Hamiltonian are Tonelli, is known to be connected to the existence of the limit of the corresponding Abel means as the discount factor goes to zero. Still in the Tonelli case, such a limit is in turn related to the existence of solutions of the critical (or, ergodic) Hamilton-Jacobi equation. The goal of this paper is to address similar issues when the Hamiltonian fails to be Tonelli: in particular, for control systems that can be associated with a family of vector fields which satisfies the Lie Algebra rank condition. First, following a dynamical approach we characterise the unique constant for which the ergodic equation admits solutions. Then, we construct a critical solution which coincides with its Lax-Oleinik evolution.
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