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Optimal control of third grade fluids with multiplicative noise

Yassine Tahraoui, Fernanda Cipriano

TL;DR

This work addresses the stochastic optimal control of velocity fields in 2D/3D bounded domains for non-Newtonian third-grade fluids driven by multiplicative noise. It develops a variational framework: the forward problem is the stochastic third-grade fluid equation, the control enters as a distributed force, and the cost includes a stopping time. The authors establish the existence of an optimal control, prove that the forward state admits a Gâteaux derivative given by the linearized equation, and derive a backward stochastic adjoint equation (uniqueness in 2D) along with a duality relation that yields a necessary optimality condition. The results extend to cost functionals that depend on derivatives of the velocity, providing a rigorous route to track velocity fields and derivative-based quantities in stochastic non-Newtonian fluids, with potential applications to turbulence control.

Abstract

This work aims to control the dynamics of certain non-Newtonian fluids in a bounded domain of $\mathbb{R}^d$, $d=2,3$ perturbed by a multiplicative Wiener noise, the control acts as a predictable distributed random force, and the goal is to achieve a predefined velocity profile under a minimal cost. Due to the strong nonlinearity of the stochastic state equations, strong solutions are available just locally in time, and the cost functional includes an appropriate stopping time. First, we show the existence of an optimal pair. Then,we show that the solution of the stochastic forward linearized equation coincides with the Gâteaux derivative of the control-to-state mapping, after establishing some stability results. Next, we analyse the backward stochastic adjoint equation; where the uniqueness of solution holds only when $d=2$. Finally, we establish a duality relation and deduce the necessary optimality conditions.

Optimal control of third grade fluids with multiplicative noise

TL;DR

This work addresses the stochastic optimal control of velocity fields in 2D/3D bounded domains for non-Newtonian third-grade fluids driven by multiplicative noise. It develops a variational framework: the forward problem is the stochastic third-grade fluid equation, the control enters as a distributed force, and the cost includes a stopping time. The authors establish the existence of an optimal control, prove that the forward state admits a Gâteaux derivative given by the linearized equation, and derive a backward stochastic adjoint equation (uniqueness in 2D) along with a duality relation that yields a necessary optimality condition. The results extend to cost functionals that depend on derivatives of the velocity, providing a rigorous route to track velocity fields and derivative-based quantities in stochastic non-Newtonian fluids, with potential applications to turbulence control.

Abstract

This work aims to control the dynamics of certain non-Newtonian fluids in a bounded domain of , perturbed by a multiplicative Wiener noise, the control acts as a predictable distributed random force, and the goal is to achieve a predefined velocity profile under a minimal cost. Due to the strong nonlinearity of the stochastic state equations, strong solutions are available just locally in time, and the cost functional includes an appropriate stopping time. First, we show the existence of an optimal pair. Then,we show that the solution of the stochastic forward linearized equation coincides with the Gâteaux derivative of the control-to-state mapping, after establishing some stability results. Next, we analyse the backward stochastic adjoint equation; where the uniqueness of solution holds only when . Finally, we establish a duality relation and deduce the necessary optimality conditions.
Paper Structure (33 sections, 26 theorems, 181 equations)

This paper contains 33 sections, 26 theorems, 181 equations.

Table of Contents

  1. Introduction
  2. Content of the study
  3. Functional spaces and notations
  4. The stochastic setting
  5. Definition of the diffusion coefficient and assumptions
  6. Assumptions on the data.
  7. Existence of local strong solution
  8. Stability results
  9. The optimal control problem and main results
  10. Cost functional
  11. Main results
  12. Existence and uniqueness of optimal control
  13. Linearized state equation
  14. Approximation
  15. Uniform estimates
  16. ...and 18 more sections

Key Result

Theorem 2.1

(Busuioc) Suppose that $f \in (H^m(D))^d,\, m \in \mathbb{N}$. Then there exists a unique (up to a constant for $\textbf{P}$) solution $(h,\textbf{P}) \in (H^{m+2}(D))^d\times H^{m+1}(D)$ of the Stokes problem Stokes such that

Theorems & Definitions (51)

  • Theorem 2.1
  • Remark 2.1
  • Remark 2.2
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.2
  • Remark 2.3
  • Proposition 2.3
  • Lemma 3.1
  • proof
  • ...and 41 more