On ergodic problem for Hamilton-Jacobi-Isaacs equations

Abstract

Abstract. We study the asymptotic behavior of λv λ as λ → 0 + , where v λ is the viscosity solution of the following Hamilton-Jacobi-Isaacs equation (infinite horizon case)We discuss the cases in which the state of the system is required to stay in an n-dimensional torus, called periodic boundary conditions, or in the closure of a bounded connected domain Ω ⊂ R n with sufficiently smooth boundary. As far as the latter is concerned, we treat both the case of the Neumann boundary conditions (reflection on the boundary) and the case of state constraints boundary conditions. Under the uniform approximate controllability assumption of one player, we extend the uniform convergence result of the value function to a constant as λ → 0 + to differential games. As far as state constraints boundary conditions are concerned, we give an example where the value function is Hölder continuous.Mathematics Subject Classification. 35B40, 49L25, 49N70.

Some Recent Aspects of Differential Game Theory

“…Such kind of results has been proved by Lions, Papanicoulaou and Varadhan [95] for first order equations (i.e., deterministic differential games). For second order equations, the result has been obtained by Alvarez and Bardi in [3], where the authors combine fundamental contributions of Evans [61,62] and of Arisawa and Lions [7] (see also Alvarez and Bardi [4,5], Bettiol [24], Ghosh and Rao [70]). For deterministic differential games (i.e., σ ≡ 0), the coercivity condition (2) is not very natural: Indeed, it means that one of the players is much more powerful than the other one.…”

“…This proof will allow to show with which efficiency the BSDE method, introduced by Shige Peng in the framework of stochastic control, works also here in the context of stochastic differential games. Concerning the uniqueness of the viscosity solutions W and V , it is a direct consequence of the comparison principle which we can formulate for (24) and (25). The reader interested in the proof or in more details is referred to [35].…”

“…Under our standard assumptions on the coefficients b, σ, f and Φ, we have the following theorem. (24), while the upper value function U is a viscosity solution of (25). Moreover, both solutions are unique in the class Θ of continuous functions V : [0, T ] × R d → R which are such that, for some constant C (which, of course, may depend on V ) the following growth condition is satisfied:…”

A PDE Approach to Finite Time Indicators in Ergodic Theory

“…This fact shows the plain advantage of F µ with respect to other popular approximated first integrals like G 1/µ (see (2.2)), whose Lie derivative depends both on x and on the flow φ 1/µ X . We stress that in this paper we are concerned with fixed µ > 0, while the classical limit µ → 0 + of both F µ and G 1/µ already appeared in the general context of Tauberian integrals [29] and -more recently -in stochastic applications, see [1,8]. Specifically, from [28], the above limit reads:…”

“…We remark that in this paper we are concerned with fixed finite µ > 0, while the classical limit µ → 0 + of both F µ and G 1/µ already appeared in the general context of Tauberian integrals [29] and -more recently -in stochastic applications, see [1,8]. In Sec.…”