Optimal Control in a Free Boundary Fluid Elasticity Interaction

We establish existence of an optimal control for the problem of minimizing flow turbulence in the case of a nonlinear fluid-structure interaction model in the framework of the known local well-posedness theory. If the initial configuration is regular, in an appropriate sense, then a class of sufficiently smooth control inputs contains an element that minimizes, within the control class, the vorticity of the fluid flow around a moving and deforming elastic solid.


Fluid
The fluid occupies domain Ω f (t) = Ω \ Ω e (t), which is moving with time, and has smooth boundary Γ(t) ∪ Γ f Assume the fluid is Newtonian, viscous, and incompressible.Its behavior is described by its velocity u and pressure p The viscosity of the fluid is ν > 0 The fluid strain tensor is given by The fluid state satisfies the following Navier-Stokes equations with given forcing term F : The fluid equations have homogeneous Dirichlet boundary conditions on the boundary Γ f , that is,

Mapping
The evolution of the fluid domain Ω f (t) is induced by the structural deformation through the common interface Γ(t) Ω f (t) is described according to a map acting in a fixed reference domain The material reference configuration for the solid is Ω e 0 := Ω e (0) ⊂ Ω with boundary Γ 0 := Γ(0), and Ω f 0 := Ω f (0) = Ω \ Ω e 0 is the reference fluid configuration The volume Ω is described by a smooth, injective map: is the position at time t ≥ 0 of the material point x ϕ preserves the boundary Γ f , i.e. ϕ = I Γ f on Γ f J(ϕ) := det(Dϕ) defines the Jacobian of the deformation.We take ϕ to be orientation-preserving so

Solid
Assume a St. Venant-Kirchoff material, which assumes large displacement and small deformation Elasticity evolves according to the response function for the second Piola-Kirchoff stress tensor, Σ = λTr(σ(ϕ))I + µ(σ(ϕ)) is the Green-St.Venant nonlinear strain tensor and λ > 0 and µ > 0 are the Lamé constants of the material

Solid
The equation of elastodynamics is written on the reference configuration Ω e 0 in terms of the response function for the first Piola-Kirchoff stress tensor Then ϕ satisfies the nonlinear elastodynamics equations: The interaction takes place on the interface and is realized via suitable transmission boundary conditions by requiring continuity of both fluid and boundary velocities, and of the normal stress tensors across Γ: where σ(p, u) = −pI + 2νε(u) and n = n(t) is the unit outer normal vector along Γ(t) with respect to Ω e (t).
To summarize, the PDE model in the mixed Eulerian-Lagrangian formulation is with initial conditions

Control
In many applications, a central goal is the optimization or optimal control of a considered process optimize fluid velocity or pressure optimize the deformation of the structure minimize wall shear stresses We focus on minimization of turbulence in the fluid The optimization addressed from the point of view of optimal control determining the optimal action upon the system in order to minimize vorticity of the flow.

Minimization Problem
The minimization problem: for an admissible set of controls U, minimize min Goals: 1 Existence of optimal control acting inside the fluid domain.
This will show that the turbulence inside the flow can be reduced by applying a body force on the fluid.

First order necessary conditions of optimality
Finding a suitable adjoint problem and using it to explicitly compute the gradient of the functional J.
Characterization of the optimal control, paving the way for a numerical study of the problem.

Lagrangian variables
We move the system to the Lagrangian frame in the variables v , q, and f representing the velocity, pressure, and forcing term, respectively, in reference configuration We require the following functional framework.Define

Existence of Unique Solution
Existence Result [Coutand and Shkoller (2006)] Let Ω ⊂ R 3 be a bounded domain of class H 4 .Let T > 0, and let ν > 0, λ > 0, µ > 0 be given.Let Ω e (t) ⊂ Ω be the closure of the open set representing the solid body at each time t ∈ [0, T ].For f ∈ E with f (0) ∈ H 4 (Ω), f t (0) ∈ H 4 (Ω), and necessary compatibility conditions on initial data, there exists T ∈ (0, T ) depending on u 0 , f , and Ω f 0 , such that there exists a unique solution (v , q) ∈ W T × Z T of the NS-elasticity coupling.
Theorem 1 (Bociu, Castle, M., Toundykov) Given initial data satisfying the regularity and compatibility conditions as in the existence result of Coutand and Shkoller, there exists a solution to the minimization problem (P).
That is, there exists f for some fixed R > 0, and (v , q) ∈ W T × Z T such that the functional J(f ) attains a minimum at f , and (v , q) is the solution of the NS-elasticity coupling with forcing term f .

Summary of the proof:
Let {f n } ∈ Q ad be a minimizing sequence for J (v n , q n ) = (v fn , q fn ), where the latter is the associated solution of the NS-elasticity coupling with right hand side f n By the coercivity of J, we know that {f n } is a bounded sequence in E .Now we use the following estimate given in [3]: ( ) ∀n, where C δ 0 is a constant, N 0 (u 0 , (w i ) 3 i=1 ) and N((q i ) 2 i=0 ) are generic smooth functions depending only on Then there exists a pair ( f , (v , q)) and a subsequence, still denoted by (f n , (v n , q n )) such that By lower-semicontinuity, we have that Similarly we obtain that Thus q = ∂ tt q a.e. in (0, T ) × Ω f 0 .Consequently the weak limit q is in Z T .
Taken together, we have (v , q) ∈ W T × Z T .
First Order Necessary Conditions of Optimality

Revisiting Goals 1 Existence of optimal control
Showing that we can pass with the limit follows from Lions-Aubin compactness arguments due to high regularity on velocity and pressure [6].
2 First order necessary conditions of optimality Finding a suitable adjoint problem and using it to explicitly compute the gradient of the functional J.A linearization was obtained by Bociu and Zolésio [7].

Challenges and Ongoing Work
The linearization shows that the boundary and its curvatures play an essential role in the final dynamical linearized systems around some equilibrium and cannot be neglected.
Simplification and well-posedness of the adjoint system for the steady state problem.
Finding a suitable adjoint system in the dynamical case and using it to explicitly compute the gradient of the corresponding cost functional.
In both cases, characterizing the optimal control via the first order necessary conditions of optimality.