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Global-in-time optimal control of stochastic third-grade fluids with additive noise

Kush Kinra, Fernanda Cipriano

Abstract

In this article, we address the velocity tracking control problem for a class of stochastic non-Newtonian fluids. More precisely, we consider the stochastic third-grade fluid equation perturbed by infinite-dimensional additive white noise and defined on the two-dimensional torus $\mathbb{T}^2$. The control acts as a distributed random external force. Taking an \emph{infinite-dimensional Ornstein-Uhlenbeck process}, the stochastic system is converted into an equivalent pathwise deterministic one, which allows to show the well-posedness of the original stochastic system globally in time. The state being a stochastic process with sample paths in $\mathrm{L}^\infty(0,T;\mathbb{H}^3(\mathbb{T}^2))$ and finite moments can be controlled in an optimal way. Namely, we establish the existence and uniqueness of solutions to the corresponding linearized state and adjoint equations. Furthermore, we derive an appropriate stability result for the state equation and verify that the Gâteaux derivative of the control-to-state mapping coincides with the solution of the linearized state equation. Finally, we establish the first-order optimality conditions and prove the existence of an optimal solution.

Global-in-time optimal control of stochastic third-grade fluids with additive noise

Abstract

In this article, we address the velocity tracking control problem for a class of stochastic non-Newtonian fluids. More precisely, we consider the stochastic third-grade fluid equation perturbed by infinite-dimensional additive white noise and defined on the two-dimensional torus . The control acts as a distributed random external force. Taking an \emph{infinite-dimensional Ornstein-Uhlenbeck process}, the stochastic system is converted into an equivalent pathwise deterministic one, which allows to show the well-posedness of the original stochastic system globally in time. The state being a stochastic process with sample paths in and finite moments can be controlled in an optimal way. Namely, we establish the existence and uniqueness of solutions to the corresponding linearized state and adjoint equations. Furthermore, we derive an appropriate stability result for the state equation and verify that the Gâteaux derivative of the control-to-state mapping coincides with the solution of the linearized state equation. Finally, we establish the first-order optimality conditions and prove the existence of an optimal solution.

Paper Structure

This paper contains 24 sections, 11 theorems, 185 equations.

Table of Contents

  1. Introduction
  2. Notations and auxiliary results
  3. Function spaces
  4. Linear and nonlinear operators
  5. Linear stochastic equation
  6. Control problem
  7. Main results
  8. Existence and uniqueness of global solutions of state equation
  9. Proof of Theorem \ref{['Wellposedness-state']}
  10. Finite-dimensional approximation
  11. A priori estimates
  12. , weak and strong limits.
  13. Passing in .
  14. Uniqueness
  15. More regular a priori estimate
  16. ...and 9 more sections

Key Result

Proposition 2.2

Suppose that $\mathrm{Tr}\left((-\Delta)^{3-2\gamma} GG^{\ast}\right)<\infty$ for $\gamma\in(0,\frac{1}{2})$. Then for $p\geq2$ where $L\geq1$ depends on $\mathrm{Tr}\left((-\Delta)^{3-2\gamma} GG^{\ast}\right)$, $\gamma$ and $\theta$, and is independent of $p$. In addition, for any given $\hat{c}>0$, we can choose where $\| GG^{\ast}\|_{Op}$ is the operator norm of operator $GG^*$ from $\mathbb{

Theorems & Definitions (25)

  • Remark 2.1
  • Proposition 2.2
  • Remark 2.3
  • proof : Proof of Proposition \ref{['thm_z_alpha']}
  • Remark 2.4
  • Theorem 2.5
  • Definition 3.1
  • Theorem 3.2
  • Remark 3.3
  • Definition 3.4
  • ...and 15 more