The existence and uniqueness of the solution for nonlinear elliptic equations in Hilbert spaces

Jianjun Zhou

doi:10.1186/s13660-015-0764-7

Abstract

The existence and uniqueness of the mild solution for semilinear elliptic partial differential equations associated with stochastic delay evolution equations are obtained by means of infinite horizon backward stochastic differential equations. Applications to optimal control for an infinite horizon are also given.

Highlights

In this article we consider infinite horizon stochastic delay evolution equations of the form ⎧⎨dX(s) = AX(s) ds + F(Xs) ds + G(Xs) dW (s), s ≥, ⎩X = x, ( . ) whereXs(θ ) = X(s + θ ), θ ∈ [–τ, ] and x ∈ C [–τ, ], H .W is a cylindrical Wiener process in a Hilbert space
A is the generator of a C semigroup in another Hilbert space H, and the coefficients F and G are assumed to satisfy Lipschitz conditions with respect to the appropriate norms
Our approach to optimal control problems for stochastic delay evolution equations is based on backward stochastic differential equations (BSDEs), which were first introduced by Pardoux and Peng [ ]: see [ – ], as general references

Summary

Introduction

In this article we consider infinite horizon stochastic delay evolution equations of the form. We define a deterministic function v : C → R by v(x) = Y ( , x), where C denotes the space of continuous functions from [–τ , ] to H, and it turns out that v is unique mild solution of the generalization of the nonlinear elliptic partial differential equation:. Chang et al [ ] found that the value function for an optimal control problem of a general stochastic differential equation with a bounded memory is the unique viscosity solution of the HJB equation. In Fuhrman and Tessitore [ ], the existence and uniqueness of the mild solution for semilinear elliptic differential equations in Hilbert spaces were obtained by means of infinite horizon BSDEs in Hilbert spaces and Malliavin calculus, the existence of optimal control is proved by the feedback law. For every c ∈ (a, b], β has a unique continuous bilinear extension β : [C([a, b], H) ⊕ Fc] × [C([a, b], H) ⊕ Fc] → R satisfying (W)

The forward equation

Findings

Application to optimal control