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New global Carleman estimates and null controllability for forward/backward semi-linear parabolic SPDEs

Lei Zhang, Fan Xu, Bin Liu

TL;DR

The paper develops global Carleman estimates for linear forward and backward parabolic SPDEs with general random coefficients and both $L^2$ and $H^{-1}$ source terms, enabling a duality/HUM-based approach to null controllability. It then proves global null controllability for linear backward and forward SPDEs with gradient terms and extends these results to semi-linear cases via a fixed-point argument in a carefully chosen weighted Banach space. The results advance the theory of stochastic controllability by handling gradient-dependent nonlinearities under general randomness and without relying on compact embeddings, significantly broadening the applicability of Carleman-based methods. These techniques have potential implications for stochastic control and filtering in systems modeled by parabolic SPDEs with gradient interactions.

Abstract

In this paper, we study the null controllability for parabolic SPDEs involving both the state and the gradient of the state. To start with, an improved global Carleman estimate for linear forward (resp. backward) parabolic SPDEs with general random coefficients and square-integrable source terms is derived. Based on this, we further develop a new global Carleman estimate for linear forward (resp. backward) parabolic SPDEs with source terms in the Sobolev space of negative order, which enables us to deal with the global null controllability for linear backward (resp. forward) parabolic SPDEs with gradient terms. As a byproduct, a special weighted energy-type estimate for the controlled system that explicitly depends on the parameters $λ,μ$ and the weighted function $θ$ is obtained, which makes it possible to extend the previous linear null controllability to semi-linear backward (resp. forward) parabolic SPDEs by applying the fixed-point argument in an appropriate Banach space.

New global Carleman estimates and null controllability for forward/backward semi-linear parabolic SPDEs

TL;DR

The paper develops global Carleman estimates for linear forward and backward parabolic SPDEs with general random coefficients and both and source terms, enabling a duality/HUM-based approach to null controllability. It then proves global null controllability for linear backward and forward SPDEs with gradient terms and extends these results to semi-linear cases via a fixed-point argument in a carefully chosen weighted Banach space. The results advance the theory of stochastic controllability by handling gradient-dependent nonlinearities under general randomness and without relying on compact embeddings, significantly broadening the applicability of Carleman-based methods. These techniques have potential implications for stochastic control and filtering in systems modeled by parabolic SPDEs with gradient interactions.

Abstract

In this paper, we study the null controllability for parabolic SPDEs involving both the state and the gradient of the state. To start with, an improved global Carleman estimate for linear forward (resp. backward) parabolic SPDEs with general random coefficients and square-integrable source terms is derived. Based on this, we further develop a new global Carleman estimate for linear forward (resp. backward) parabolic SPDEs with source terms in the Sobolev space of negative order, which enables us to deal with the global null controllability for linear backward (resp. forward) parabolic SPDEs with gradient terms. As a byproduct, a special weighted energy-type estimate for the controlled system that explicitly depends on the parameters and the weighted function is obtained, which makes it possible to extend the previous linear null controllability to semi-linear backward (resp. forward) parabolic SPDEs by applying the fixed-point argument in an appropriate Banach space.
Paper Structure (13 sections, 12 theorems, 104 equations)

This paper contains 13 sections, 12 theorems, 104 equations.

Table of Contents

  1. Introduction
  2. Comparison with related works
  3. Notations and assumptions
  4. Main results
  5. Organization of the paper
  6. An improved $L^2$-Carleman estimates
  7. Controllability of backward parabolic SPDEs
  8. A new Carleman estimate
  9. The linear controlled system
  10. The semi-linear controlled system
  11. Controllability of forward parabolic SPDEs
  12. Carleman estimates for backward system
  13. Controllability of forward semi-linear parabolic SPDEs

Key Result

Lemma 1.1

Let $\mathcal{O}_1$ be a nonempty subset of $\mathcal{O}$ such that $\mathcal{O}_1\subset\subset \mathcal{O}'$ (i.e., $\overline{\mathcal{O}}_1\subset \mathcal{O}'$), then there exists a function $\beta\in \mathcal{C}^4(\overline{\mathcal{O}};[0,1])$ such that $0< \beta (x)\leq 1~\textrm{in}~\mathca

Theorems & Definitions (29)

  • Lemma 1.1: fursikov1996controllability
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Theorem 1.8
  • Remark 1.9
  • Theorem 1.10
  • ...and 19 more