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## Section: New Results

### Control, stabilization and optimization of heterogeneous systems

Participants : Rémi Buffe, Thomas Chambrion, Eloïse Comte, Arnab Roy, Takéo Takahashi, Jean-François Scheid, Julie Valein.

Control and optimization

The use of measures (instead of functions) as controls is usually referred to as “impulsive control”. While the theory is now well understood for finite dimensional dynamics, many questions are still open for the control of PDEs. In [19], Thomas Chambrion and his co-authors discuss the notion of solution for the impulsive control (using measures instead of functions for the control) of the general bilinear Schrödinger equations. The results are adapted in [35] to the case of potentials with high regularity. These techniques have been used to extend the celebrated obstruction to controllability by Ball, Marsden and Slemrod to the case of abstract bilinear equations with bounded potentials [33] and the Klein-Gordon equation [20]. Other obstructions to controllability (preservation of regularity) have been investigated for the Gross-Pitaiewski equation with unbounded potentials [34].

In [15], an optimal control problem for groundwater pollution due to agricultural activities is considered, the objective being the optimization of the trade-off between the fertilizer use and the cleaning costs. The spread of the pollution is modeled by a convection-diffusion-reaction equation. We are interested in the buffer zone around the captation well and we determine its optimal size.

In [44], Eduardo Cerpa, Emmanuelle Crépeau and Julie Valein study the boundary controllability of the Korteweg-de Vries equation on a tree-shaped network, with less controls than equations.

In [27], Jérôme Lohéac and Takéo Takahashi study the locomotion of a ciliated microorganism in a viscous incompressible fluid. They use the Blake ciliated model: the swimmer is a rigid body with tangential displacements at its boundary that allow it to propel in a Stokes fluid. This can be seen as a control problem: using periodical displacements, is it possible to reach a given position and a given orientation? They are interested in the minimal dimension $d$ of the space of controls that allows the microorganism to swim. Their main result states the exact controllability with $d=3$ generically with respect to the shape of the swimmer and with respect to the vector fields generating the tangential displacements. The proof is based on analyticity results and on the study of the particular case of a spheroidal swimmer.

In [31], Arnab Roy and Takéo Takahashi study the controllability of a fluid-structure interaction system. They consider a viscous and incompressible fluid modeled by the Boussinesq system and the structure is a rigid body with arbitrary shape which satisfies Newton’s laws of motion. They assume that the motion of this system is bidimensional in space. They prove the local null controllability for the velocity and temperature of the fluid and for the position and velocity of the rigid body for a control acting only on the temperature equation on a fixed subset of the fluid domain.

Rémi Buffe and Ludovick Gagnon consider $N$ manifolds without boundary that intersect each other. They assume that the speed of propagation on each manifold is different, which implies that the Snell conditions applies at the interface. They give sufficient geometric conditions to ensure the controllability with distributed controls on $N-1$ manifolds.

Stabilization

In [17], Lucie Baudouin, Emmanuelle Crépeau and Julie Valein study the exponential stability of the nonlinear Korteweg-de Vries equation with boundary time-delay feedback. Two different methods are employed: a Lyapunov functional approach (allowing to have an estimation on the decay rate, but with a restrictive assumption on the length of the spatial domain of the KdV equation) and an observability inequality approach, with a contradiction argument (for any non-critical lengths but without estimation on the decay rate).

In [55], Julie Valein shows the semi-global exponential stability of the nonlinear Korteweg-de Vries equation in the presence of a delayed internal feedback, for any lengths, in the case where the weight of the feedback with delay is smaller than the weight of the feedback without delay. In the case where the support of the feedback without delay is not included in the support of the feedback with delay, a local exponential stability result is proved if the weight of the delayed feedback is small enough.

Optimization

J.F. Scheid, V. Calesti (PhD Student) and I. Lucardesi study an optimal shape problem for an elastic structure immersed in a viscous incompressible fluid. They want to establish the existence of an optimal elastic domain associated with an energy-type functional for a Stokes-Elasticity system. We want to find an optimal reference domain (the domain before deformation) for the elasticity problem that minimizes an energy-type functional. This problem is concerned with 2D geometry and is an extension of the work of [113] for a 1D problem. The optimal domain is seeking in a class of admissible open sets defined with a diffeomorphism of a given domain. The main difficulty lies on the coupling between the Stokes problem written in a eulerian frame and the linear elasticity problem written in a lagrangian form. The shape derivative of the energy-type functional is also aimed to be determined in order to numerically obtain an optimal elastic domain. This work is in progress.