Keywords

Some Fluid Models

We consider a fluid flow occupying a bounded domain \(\Omega _{F} \subset\) \(\mathbb{R}^{N}\), with N = 2 or N = 3, at the initial time t = 0, and a domain \(\Omega _{F}\)(t) at time t > 0. Let us denote by ρ\((x,t) \in \mathbb{R}^{+}\) the density of the fluid at time t at the point \(x\ \in \Omega _{F}\)(t) and by u(x, t)  ∈  \(\mathbb{R}^{N}\) its velocity. The fluid flow equations are derived by writing the mass conservation

$$\displaystyle{ \frac{\partial \rho } {\partial t} + \mbox{ div}(\rho u) = 0\quad \mbox{ in}\;\,\Omega _{F}(t),\quad \mbox{ for}\;\,t> 0, }$$
(1)

and the balance of momentum

$$\displaystyle{ \begin{array}{l} \rho \left (\frac{\partial u} {\partial t} + (u \cdot \nabla )u\right ) = \mbox{ div}\,\sigma \,\mbox{ +}\,\rho \,f\mbox{ } \\ \mbox{ in}\,\;\Omega _{F}(t),\quad \mbox{ for}\,t> 0\\ \end{array} }$$
(2)

where σ is the so-called constraint tensor and f represents a volumic force. For an isothermal fluid, there is no need to complete the system by the balance of energy. The physical nature of the fluid flow is taken into account in the choice of the constraint tensor σ. When the volume is preserved by the fluid flow transport, the fluid is called incompressible. The incompressibility condition reads as div u = 0 in \(\Omega _{F}(t)\). The incompressible Navier-Stokes equations are the classical model to describe the evolution of isothermal incompressible and Newtonian fluid flows. When in addition the density of the fluid is assumed to be constant, ρ(x, t) = ρ0, the equations reduce to

$$\displaystyle\begin{array}{rcl} & & \mbox{ div}\,u = 0, \\ & & \rho _{0}\left (\frac{\partial u} {\partial t} + (u \cdot \nabla )u\right ) =\nu \Delta u -\nabla p +\rho _{0}\,f \\ & & \mbox{ in}\,\;\Omega _{F}(t),\quad t> 0, {}\end{array}$$
(3)

which are obtained by setting

$$\displaystyle{ \sigma =\nu \left (\nabla u + \left (\nabla u\right )^{T}\right ) + \left (\mu -\frac{2\nu } {3}\right )\mbox{ div}\,u\,I - pI, }$$
(4)

in Eq. (2). When div u = 0, the expression of σ simplifies. The coefficients ν  >  0 and μ  >  0 are the viscosity coefficients of the fluid, and p(x, t) its pressure at the point \(x \in \Omega _{F}(t)\) and at time t > 0.

This model has to be completed with boundary conditions on \(\partial \Omega _{F}(t)\) and an initial condition at time t = 0.

The incompressible Euler equations with constant density are obtained by setting ν = 0 in the above system.

The compressible Navier-Stokes system is obtained by coupling the equation of conservation of mass Eq. (1) with the balance of momentum Eq. (2), where the tensor σ is defined by Eq. (4), and by completing the system with a constitutive law for the pressure.

Control Issues

There are unstable steady states of the Navier-Stokes equations which give rise to interesting control problems (e.g., to maximize the ratio “lift over drag”), but which cannot be observed in real life because of their unstable nature. In such situations, we would like to maintain the physical model close to an unstable steady state by the action of a control expressed in feedback form, that is, as a function either depending on an estimation of the velocity or depending on the velocity itself. The estimation of the velocity of the fluid may be recovered by using some real-time measurements. In that case, we speak of a feedback stabilization problem with partial information. Otherwise, when the control is expressed in terms of the velocity itself, we speak of a feedback stabilization problem with full information.

Another interesting issue is to maintain a fluid flow (described by the Navier-Stokes equations) in the neighborhood of a nominal trajectory (not necessarily a steady state) in the presence of perturbations. This is a much more complicated issue which is not yet solved.

In the case of a perturbation in the initial condition of the system (the initial condition at time t = 0 is different from the nominal velocity held at time t = 0), the exact controllability to the nominal trajectory consists in looking for controls driving the system in finite time to the desired trajectory.

Thus, control issues for fluid flows are those encountered in other fields. However there are specific difficulties which make the corresponding problems challenging. When we deal with the incompressible Navier-Stokes system, the pressure plays the role of a Lagrange multiplier associated with the incompressibility condition. Thus, we have to deal with an infinite-dimensional nonlinear differential algebraic system. In the case of a Dirichlet boundary control, the elimination of the pressure, by using the so-called Leray or Helmholtz projector, leads to an unusual form of the corresponding control operator; see Raymond (2006). In the case of an internal control, the estimation of the pressure to prove observability inequalities is also quite tricky; see Fernandez-Cara et al. (2004). From the numerical viewpoint, the approximation of feedback control laws leads to very large-size problems, and new strategies have to be found for tackling these issues.

Moreover, the issues that we have described for the incompressible Navier-Stokes equations may be studied for other models like the compressible Navier-Stokes equations, the Euler equations (describing nonviscous fluid flows) both for compressible and incompressible models, or even more complicated models.

Feedback Stabilization of Fluid Flows

Let us now describe what are the known results for the incompressible Navier-Stokes equations in 2D or 3D bounded domains, with a control acting locally in a Dirichlet boundary condition. Let us consider a given steady state (u s , p s ) satisfying the equation

$$\displaystyle{\begin{array}{l} -\nu \Delta u_{s} + (u_{s} \cdot \nabla )u_{s} + \nabla p_{s} = f_{s}, \\ \,\,\,\,\mbox{ and}\,\,\,\,\mbox{ div}\,\,u_{s} = 0\,\;\,\mbox{ in}\,\;\Omega _{F},\\ \end{array} }$$

with some boundary conditions which may be of Dirichlet type or of mixed type (Dirichlet-Neumann-Navier type). For simplicity, we only deal with the case of Dirichlet boundary conditions

$$\displaystyle{u_{s} = g_{s}\quad \mbox{ on}\;\;\partial \Omega _{F},}$$

where g s andf s are time-independent functions. In the case \(\Omega _{F}(t) = \Omega _{F}\), not depending on t, the corresponding instationary model is

$$\displaystyle{ \begin{array}{l} \frac{\partial u} {\partial t} -\nu \Delta u + (u \cdot \nabla )u + \nabla p = f_{s} \\ \mbox{ and}\quad \mbox{ div}\,\;u\,\mbox{ =}\,\mbox{ 0}\,\;\mbox{ in}\;\,\Omega _{F} \times (0,\,\infty ), \\ u = g_{s} +\sum \nolimits _{ i=1}^{N_{c}}f_{i}(t)g_{i},\,\,\partial \Omega _{F} \times (0,\,\infty ) \\ u(0) = u_{0}\quad \mbox{ on}\,\;\Omega _{F}\mbox{.}\\ \end{array} }$$
(5)

In this model, we assume that u0  ≠  u s , g i are given functions with localized supports in \(\partial \Omega _{F}\) and \(f(t) = (f_{1}(t),\,.\,.\,.\,,\,f_{N_{c}}(t))\) is a finite-dimensional control. Due to the incompressibility condition, the functions g i have to satisfy

$$\displaystyle{\int _{\partial \Omega _{F}}g_{i} \cdot n = 0,}$$

where n is the unit normal to \(\partial \Omega _{F}\), outward \(\Omega _{F}\).

The stabilization problem, with a prescribed decay rate −α < 0, consists in looking for a control f in feedback form, that is, of the form

$$\displaystyle{ f(t) = K(u(t) - u_{s}), }$$
(6)

such that the solution to the Navier-Stokes system Eq. (5), with f defined by Eq. (6), obeys

$$\displaystyle{\left \|e^{\alpha t}(u(t) - u_{ s})\right \|_{z} \leq \varphi \left (\left \|u_{0} - u_{s}\right \|_{z}\right ),}$$

for some norm Z, provided \(\left \|u_{0} - u_{s}\right \|_{z}\) is small enough and where \(\varphi\) is a nondecreasing function. The mapping K, called the feedback gain, may be chosen linear.

The usual procedure to solve this stabilization problem consists in writing the system satisfied by u − u s , in linearizing this system, and in looking for a feedback control stabilizing this linearized model. The issue is first to study the stabilizability of the linearized model and, when it is stabilizable, to find a stabilizing feedback gain. Among the feedback gains that stabilize the linearized model, we have to find one able to stabilize, at least locally, the nonlinear system too.

The linearized controlled system associated with Eq. (5) is

$$\displaystyle{ \begin{array}{l} \frac{\partial v}{\partial t} -\nu \Delta v + (u_{s} \cdot \nabla )v + (v \cdot \nabla )u_{s} + \nabla q = 0 \\ \mbox{ and}\quad \mbox{ div}\;\,v\,\mbox{ =}\,\mbox{ 0}\,\,\mbox{ in}\;\,\Omega _{F} \times (0,\,\infty ), \\ v =\sum \nolimits _{ i=1}^{N_{c}}f_{i}(t)g_{i}\quad \mbox{ on}\;\;\partial \Omega _{F} \times (0,\,\infty ), \\ v(0)\, =\, v_{0}\quad \mbox{ on}\;\,\Omega _{F}.\\ \end{array} }$$
(7)

The easiest way for proving the stabilizability of the controlled system Eq. (7) is to verify the Hautus criterion. It consists in proving the following unique continuation result. If \((\phi _{j},\psi _{j},\lambda _{j})\) is the solution to the eigenvalue problem

$$\displaystyle\begin{array}{rcl} \lambda _{j}\phi _{j}& -& \nu \Delta \phi _{j} - (u_{s} \cdot \nabla )\phi _{j} + (\nabla u_{s})^{T}\phi _{ j} \\ & & +\nabla \psi _{j} = 0\quad \mbox{ and}\quad \mbox{ div}\phi _{j} = 0\,\,\mbox{ in}\,\;\,\Omega _{F}, \\ & & \phi _{j} = 0\,\,\,\,\mbox{ on}\;\,\,\partial \Omega _{F},\quad \mathrm{Re}\;\lambda _{j} \geq -\alpha, {}\end{array}$$
(8)

and if in addition (ϕ j , ψ j ) satisfies

$$\displaystyle{\int _{\partial \Omega _{F}}g_{i} \cdot \sigma (\phi _{j},\,\psi _{j})n = 0\,\quad \mbox{ for}\,\,\mbox{ all}\,\;1\, \leq i \leq N_{c},}$$

then (\(\phi _{j},\psi _{j}) = 0\). By using a unique continuation theorem due to Fabre and Lebeau (1996), we can explicitly determine the functions g i so that this condition is satisfied; see Raymond and Thevenet (2010). For feedback stabilization results of the Navier-Stokes equations in two or three dimensions, we refer to Fursikov (2004), Raymond (2006), Barbu et al. (2006), Raymond (2007), Badra (2009), and Vazquez and Krstic (2008).

Controllability to Trajectories of Fluid Flows

If \(\left (\tilde{u}\left (t\right ),\tilde{p}\left (t\right )\right )_{0\leq t<\infty }\) is a solution to the Navier-Stokes system, the controllability problem to the trajectory \(\left (\tilde{u}\left (t\right ),\tilde{p}\left (t\right )\right )_{0\leq t<\infty }\), in time T  >  0, may be rewritten as a null controllability problem satisfied by \((v,\,q) = (u -\tilde{ u},\)\(p -\tilde{ p})\). The local null controllability in time T > 0 follows from the null controllability of the linearized system and from a fixed point argument. The linearized controlled system is

$$\displaystyle{ \begin{array}{l} \frac{\partial v}{\partial t}-\nu \Delta v + \left (\tilde{u}\left (t\right ) \cdot \nabla \right )v+(v \cdot \nabla )\tilde{u}\left (t\right ) + \nabla q = 0 \\ \,\,\mbox{ and}\quad \mbox{ div}\,v\,\mbox{ =}\,\mbox{ 0}\,\;\mbox{ in}\;\,\Omega _{F} \times (0,\,T), \\ v = m_{c}\,f\quad \mbox{ on}\,\,\,\partial \Omega _{F} \times (0,\,T), \\ v(0) = v_{0} \in L^{2}(\Omega _{F}; \mathbb{R}^{N}),\quad \mbox{ div}\,\,v_{\mathrm{0}}\mbox{ =}\,\mbox{ 0.}\\ \end{array} }$$
(9)

The nonnegative function m c is used to localize the boundary control f. The control f is assumed to satisfy

$$\displaystyle{ \int _{\partial \Omega _{F}}m_{c}\,f \cdot n = 0. }$$
(10)

As for general linear dynamical systems, the null controllability of the linearized system follows from an observability inequality for the solutions to the following adjoint system

$$\displaystyle{ \begin{array}{l} - \frac{\partial \phi } {\partial t} -\nu \Delta \phi -\left (\tilde{u}\left (t\right ) \cdot \nabla \right )\phi + \left (\nabla \tilde{u}\left (t\right )\right )^{T}\phi + \nabla \psi = 0 \\ \mbox{ and}\,\,\,\mbox{ div}\,\;\phi \,\mbox{ =}\,\mbox{ 0}\,\,\mbox{ in}\,\;\,\Omega _{F} \times (0,T), \\ \phi = 0\quad \mbox{ on}\,\,\;\partial \Omega _{F} \times (0,T), \\ \phi (T) \in L^{2}(\Omega _{F}; \mathbb{R}^{N}),\quad \mbox{ div}\;\,\phi \mbox{ (}T\mbox{ )}\,\mbox{ =}\,\mbox{ 0.}\\ \end{array} }$$
(11)

Contrary to the stabilization problem, the null controllability by a control of finite dimension seems to be out of reach and it will be impossible in general. We look for a control \(f \in L^{2}(\partial \Omega _{F}\); \(\mathbb{R}^{N})\), satisfying Eq. (10), driving the solution to system Eq. (9) in time T to zero, that is, such that the solution \(v_{v_{0},\,f}\) obeys \(v_{v_{0},\,f}(T) = 0\). The linearized system Eq. (9) is null controllable in time T  >  0 by a boundary control \(f \in L^{2}(\partial \Omega _{F}\); \(\mathbb{R}^{N})\) obeying Eq. (10), if and only if there exists C > 0 such that

$$\displaystyle{ \int _{\Omega _{F}}\left \vert \phi (0)\right \vert ^{2}dx \leq C\int _{ \partial \Omega _{F}}m_{c}\left \vert \sigma (\phi,\,\psi )n\right \vert ^{2}dx, }$$
(12)

for all solution (ϕ, ψ) of Eq. (11). The observability inequality Eq. (12) may be proved by establishing weighted energy estimates called “Carleman-type estimates”; see Fernandez-Cara et al. (2004) and Fursikov and Imanuvilov (1996).

Additional Controllability Results for Other Fluid Flow Models

The null controllability of the 2D incompressible Euler equation has been obtained by J.-M. Coron with the so-called Return Method (Coron 1996). See also Coron (2007) for additional references (in particular, the 3D case has been treated by O. Glass).

Some null controllability results for the one-dimensional compressible Navier-Stokes equations have been obtained in Ervedoza et al. (2012).

Fluid-Structure Models

Fluid-structure models are obtained by coupling an equation describing the evolution of the fluid flow with an equation describing the evolution of the structure. The coupling comes from the balance of momentum and by writing that at the fluid-structure interface, the fluid velocity is equal to the displacement velocity of the structure.

The most important difficulty in studying those models comes from the fact that the domain occupied by the fluid at time t evolves and depends on the displacement of the structure. In addition, when the structure is deformable, its evolution is usually written in Lagrangian coordinates while fluid flows are usually described in Eulerian coordinates.

The structure may be a rigid or a deformable body immersed into the fluid. It may also be a deformable structure located at the boundary of the domain occupied by the fluid.

A Rigid Body Immersed in a Three-Dimensional Incompressible Viscous Fluid

In the case of a 3D rigid body \(\Omega _{S}(t)\) immersed in a fluid flow occupying the domain \(\Omega _{F}(t)\), the motion of the rigid body may be described by the position \(h(t) \in \mathbb{R}^{3}\) of its center of mass and by a matrix of rotation \(Q(t) \in \mathbb{R}^{3\ \times \ 3}\). The domain \(\Omega _{S}(t)\) and the flow X S associated with the motion of the structure obey

$$\displaystyle{ \begin{array}{l} X_{S}(y,\,t) = h(t) + Q(t)Q_{0}^{-1}(y - h(0)), \\ \mbox{ for}\;\,y \in \Omega _{S}(0) = \Omega _{S}, \\ \Omega _{S}(t) = X_{S}(\Omega _{S}(0),\,t),\\ \end{array} }$$
(13)

and the matrix Q(t) is related to the angular velocity \(\omega : (0,T)\ \mapsto \ \mathbb{R}^{3}\), by the differential equation

$$\displaystyle{ Q\prime(t) =\omega (t) \times Q(t),\quad Q(0) = Q_{0}. }$$
(14)

We consider the case when the fluid flow satisfies the incompressible Navier-Stokes system Eq. (3) in the domain \(\Omega _{F}(t)\) corresponding to Fig. 1. Denoting by \(J(t) \in \mathbb{R}^{3\times 3}\) the tensor of inertia at time t, and by m the mass of the rigid body, the equations of the structure are obtained by writing the balance of linear and angular momenta

$$\displaystyle{ \begin{array}{l} mh\prime\prime =\int _{\partial \Omega _{S}(t)}\sigma (u,\,p)ndx,\\ \\ \\ J\omega \prime = J\omega \times \omega +\int _{\partial \Omega _{S}(t)}(x - h) \times \sigma (u,\,p)ndx,\\ \\ \\ h(0) = h_{0},\,h\prime(0) = h_{1},\,\,\omega (0) =\omega _{0},\\ \end{array} }$$
(15)

where n is the normal to \(\partial \Omega _{S}(t)\) outward \(\Omega _{F}(t)\). The system Eqs. (3) and (13)–(15) has to be completed with boundary conditions. At the fluid-structure interface, the fluid velocity is equal to the displacement velocity of the rigid solid:

$$\displaystyle{ u(x,\,t) = h\prime(t) +\omega (t) \times (x - h(t)), }$$
(16)

for all \(x \in \partial \Omega _{S}(t)\), t > 0. The exterior boundary of the fluid domain is assumed to be fixed \(\Gamma _{e} = \partial \Omega _{F}(t)\setminus \partial \Omega _{S}(t)\). The boundary condition on \(\Gamma _{e} \times (0,T)\) may be of the form

$$\displaystyle{ u = m_{c}\,f\quad \mbox{ on}\,\;\,\Gamma _{e} \times (0,\,\infty ), }$$
(17)

with \(\int _{\Gamma _{e}}m_{c}\,f \cdot n = 0\), f is a control, and m c a localization function.

Fig. 1
figure 325figure 325

Control of Fluids and Fluid-Structure Interactions

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An Elastic Beam Located at the Boundary of a Two-Dimensional Domain Filled by an Incompressible Viscous Fluid

When the structure is described by an infinite-dimensional model (a partial differential equation or a system of p.d.e.), there are a few existence results for such systems and mainly existence of weak solutions (Chambolle et al. 2005). But for stabilization and control problems of nonlinear systems, we are usually interested in strong solutions. Let us describe a two-dimensional model in which a one-dimensional structure is located on a flat part \(\Gamma _{S} = (0,L)\times\) {y0} of the boundary of the reference configuration of the fluid domain \(\Omega _{F}\). We assume that the structure is a Euler-Bernoulli beam with or without damping. The displacement η of the structure in the direction normal to the boundary \(\Gamma _{S}\) is described by the partial differential equation

$$\displaystyle{ \begin{array}{l} \eta _{tt} - b\eta _{xx} - c\eta _{txx} + a\eta _{xxxx} = F,\mbox{ in}\,\Gamma _{S} \times (0,\,\infty ), \\ \eta = 0\quad \mbox{ and}\quad \eta _{x} = 0\,\,\,\mbox{ on}\,\,\,\partial \Gamma _{S} \times (0,\,\infty ), \\ \eta (0) =\eta _{ 1}^{0}\quad \mbox{ and}\quad \eta _{t}(0) =\eta _{ 2}^{0}\,\,\mbox{ in}\,\;\Gamma _{S},\\ \end{array} }$$
(18)

where η x , η xx , and η xxxx stand for the first, the second, and the fourth derivative of η with respect to \(x\ \in \Gamma _{S}\). The other derivatives are defined in a similar way. The coefficients b and c are nonnegative, and a > 0. The term cη txx is a structural damping term. At time t, the structure occupies the position \(\Gamma _{S}(t) = \left \{(x,\,y)\left \vert x \in (0,\,L),\,\,y = y_{0} +\eta (x,\,t)\right.\right \}\). When \(\Omega _{F}\) is a two-dimensional model, \(\Gamma _{S}\) is of dimension one, and \(\partial \Gamma _{S}\) is reduced to the two extremities of \(\Gamma _{S}\). The momentum balance is obtained by writing that F in Eq. (18) is given by \(F = -\sqrt{1 +\eta _{ x }^{2}}\ \sigma (u,\,p)\tilde{n} \cdot n\), where \(\tilde{n}(x,\,y)\) is the unit normal at \((x,y) \in \Gamma _{S}(t)\) to \(\Gamma _{S}(t)\) outward \(\Omega _{F}(t)\), and n is the unit normal to \(\Gamma _{S}\) outward \(\Omega _{F}(0)\ =\ \Omega _{F}\). If in addition, a control f acts as a distributed control in the beam equation, we shall have

$$\displaystyle{ F = -\sqrt{1 +\eta _{ x }^{2}}\ \sigma (u,\,p)\tilde{n} \cdot n + f }$$
(19)

The equality of velocities on \(\Gamma _{S}(t)\) reads as

$$\displaystyle\begin{array}{rcl} & & u(x,\,y_{0} +\eta (x,\,t)) = (0,\,\eta _{t}(x,\,t)), \\ & & x \in (0,\,L),\,t> 0. {}\end{array}$$
(20)

Control of Fluid-Structure Models

To control or to stabilize fluid-structure models, the control may act either in the fluid equation or in the structure equation or in both equations. There are a very few controllability and stabilization results for systems coupling the incompressible Navier-Stokes system with a structure equation. We state below two of those results. Some other results are obtained for simplified one-dimensional models coupling the viscous Burgers equation coupled with the motion of a mass; see Badra and Takahashi (2013) and the references therein.

We also have to mention here recent papers on control problems for systems coupling quasi-stationary Stokes equations with the motion of deformable bodies, modeling microorganism swimmers at low Reynolds number; see Alouges et al. (2008).

Null Controllability of the Navier-Stokes System Coupled with the Motion of a Rigid Body

The system coupling the incompressible Navier-Stokes system Eq. (3) in the domain drawn in Fig. 1, with the motion of a rigid body described by Eqs. (13)–(16), with the boundary control Eq. (17) is null controllable locally in a neighborhood of 0. Before linearizing the system in a neighborhood of 0, the fluid equations have to be rewritten in Lagrangian coordinates, that is, in the cylindrical domain \(\Omega _{F} \times (0,\infty )\). The linearized system is the Stokes system coupled with a system of ordinary differential equations. The proof of this null controllability result relies on a Carleman estimate for the adjoint system; see, e.g., Boulakia and Guerrero (2013).

Feedback Stabilization of the Navier-Stokes System Coupled with a Beam Equation

The system coupling the incompressible Navier-Stokes system Eq. (3) in the domain drawn in Fig. 2, with beam Eqs. (18)–(20), can be locally stabilized with any prescribed exponential decay rate −α < 0, by a feedback control f acting in Eq. (18) via Eq. (19); see Raymond (2010). The proof consists in showing that the infinitesimal generator of the linearized model is an analytic semigroup (when c > 0), that its resolvent is compact, and that the Hautus criterion is satisfied.

Fig. 2
figure 326figure 326

Control of Fluids and Fluid-Structure Interactions

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When the control acts in the fluid equation, the system coupling Eq. (3) in the domain drawn in Fig. 2, with the beam Eqs. (18)–(20), can be stabilized when c > 0. To the best of our knowledge, there is no null controllability result for such systems, even with controls acting both in the structure and fluid equations. The case where the beam equation is approximated by a finite-dimensional model is studied in Lequeurre (2013).

Cross-References