CAGE's activities take place in the field of mathematical control theory, with applications in three main directions: geometric models for vision, control of quantum mechanical systems, and control of systems with uncertain dynamics.

The relations between control theory and geometry of vision rely on the notion of sub-Riemannian structure, a geometric framework which is used to measure distances in nonholonomic contexts and which has a natural and powerful control theoretical interpretation. We recall that nonholonomicity refers to the property of a velocity constraint that cannot be recast as a state constraint. In the language of differential geometry, a sub-Riemannian structure is a (possibly rank-varying) Lie bracket generating distribution endowed with a smoothly varying norm.

Sub-Riemannian geometry, and in particular the theory of associated (hypoelliptic) diffusive processes, plays a crucial role in the neurogeometrical model of the primary visual cortex due to Petitot, Citti and Sarti, based on the functional architecture first described by Hubel and Wiesel. Such a model can be used as a powerful paradigm for bio-inspired image processing, as already illustrated in the recent literature (including by members of our team). Our contributions to this field are based not only on this approach, but also on another geometric and sub-Riemannian framework for vision, based on pattern matching in the group of diffeomorphisms. In this case admissible diffeomorphisms correspond to deformations which are generated by vector fields satisfying a set of nonholonomic constraints. A sub-Riemannian metric on the infinite-dimensional group of diffeomorphisms is induced by a length on the tangent distribution of admissible velocities. Nonholonomic constraints can be especially useful to describe distortions of sets of interconnected objects (e.g., motions of organs in medical imaging).

Control theory is one of the components of the forthcoming quantum revolution

Simultaneous control of a continuum of systems with slightly different dynamics is a typical problem in quantum mechanics and also a special case of the third applicative axis to which CAGE is contributing: control of systems with uncertain dynamics. The slightly different dynamics can indeed be seen as uncertainties in the system to be controlled, and simultaneous control rephrased in terms of a robustness task. Robustification, i.e., offsetting uncertainties by suitably designing the control strategy, is a widespread task in automatic control theory, showing up in many applicative domains such as electric circuits or aerospace motion planning. If dynamics are not only subject to static uncertainty, but may also change as time goes, the problem of controlling the system can be recast within the theory of switched and hybrid systems, both in a deterministic and in a probabilistic setting. Our contributions to this research field concern both stabilization (either asymptotic or in finite time) and optimal control, where redundancies and probabilistic tools can be introduced to offset uncertainties.

The activities of CAGE are part of the research in the wide area of control theory. This nowadays mature discipline is still the subject of intensive research because of its crucial role in a vast array of applications.

More specifically, our contributions are in the area of **mathematical control theory**, which is to say that we are interested in the analytical and geometrical aspects of control applications.
In this approach, a control system is modeled by a system of equations (of many possible types: ordinary differential equations, partial differential equations, stochastic differential equations, difference equations,...), possibly not explicitly known in all its components, which are studied in order to establish qualitative and quantitative properties concerning the actuation of the system through the control.

**Motion planning** is, in this respect, a cornerstone property: it denotes the design and validation of algorithms for identifying a control law steering the system from a given initial state to (or close to) a target one.
Initial and target positions can be replaced by sets of admissible initial and final states as, for instance, in the motion planning task towards a desired periodic solution.
Many specifications can be added to the pure motion planning task, such as robustness to external or endogenous disturbances, obstacle avoidance or penalization criteria.
A more abstract notion is that of **controllability**, which
denotes the property of a system for which any two states can be connected by a trajectory corresponding to an admissible control law.
In mathematical terms, this translates into the surjectivity of the so-called **end-point map**, which associates with a control and an initial state the final point of the
corresponding trajectory. The analytical and topological properties of endpoint maps are therefore crucial in analyzing the properties of control systems.

One of the most important additional objective which can be associated with a motion planning task is **optimal control**, which corresponds to the minimization of a cost (or, equivalently, the maximization of a gain) .
Optimal control theory is clearly deeply interconnected with calculus of variations, even if the non-interchangeable nature of the time-variable results in some important specific features, such as the occurrence of **abnormal extremals** . Research in optimal control encompasses different aspects, from numerical methods to dynamic programming and non-smooth analysis, from regularity of minimizers to high order optimality conditions and curvature-like invariants.

Another domain of control theory with countless applications is **stabilization**. The goal in this case is to make the system
converge towards an equilibrium or some more general safety region. The main difference with respect to motion planning is that here the control law is constructed in feedback form. One of the most important properties in this context is that of **robustness**, i.e., the performance of the stabilization protocol in presence of disturbances or modeling uncertainties.
A powerful framework which has been developed to take into account uncertainties and exogenous non-autonomous disturbances is that of hybrid and switched systems , , . The central tool in the stability analysis of control systems is that of **control Lyapunov function**. Other relevant techniques are based on algebraic criteria or dynamical systems. One of the most important stability property which
is studied in the context of control system is **input-to-state stability** , which measures how sensitive the system is to an external excitation.

One of the areas where control applications have nowadays the most impressive developments is in the field of **biomedicine and neurosciences**.
Improvements both in modeling and in the capability of finely actuating biological systems
have concurred in increasing the popularity of these subjects.
Notable advances concern, in particular, identification and control for biochemical networks and models for neural activity .
Therapy analysis from the point of view of optimal control has also attracted a great attention .

Biological models are not the only one in which stochastic processes play an important role. Stock-markets and energy grids are two major examples where optimal control techniques are applied in the non-deterministic setting. Sophisticated mathematical tools have been developed since several decades to
allow for such extensions. Many theoretical advances have also been required for dealing with complex systems whose description is based on **distributed parameters** representation and **partial differential equations**. Functional analysis, in particular, is a crucial tool to tackle the control of such systems .

Let us conclude this section by mentioning another challenging application domain for control theory: the decision by the European Union to fund a flagship devoted to the development of quantum technologies is a symptom of the role that quantum applications are going to play in tomorrow's society. **Quantum control** is one of the bricks of quantum engineering, and presents many peculiarities with respect to standard control theory, as a consequence of the specific properties of the systems described by the laws of quantum physics. Particularly important for technological applications is the
capability of inducing and reproducing coherent state superpositions and entanglement in a fast, reliable, and efficient way .

At the core of the scientific activity of the team
is the **geometric control** approach, that is,
a distinctive viewpoint issued in particular from
(elementary) differential geometry, to tackle
questions of
controllability, observability,
optimal control... , .
The emphasis of such a geometric approach to control theory is put on
intrinsic properties of the systems and it is particularly well adapted to study
nonlinear and nonholonomic phenomena.

One of the features of the geometric control approach is its capability of exploiting **symmetries and intrinsic structures** of control systems.
Symmetries and intrinsic structures can be used to characterize minimizing trajectories, prove regularity properties and describe invariants.
An egregious example
is given by mechanical systems, which inherently exhibit Lagrangian/Hamiltonian structures which are naturally expressed using the language of symplectic geometry .
The geometric theory of quantum control, in particular, exploits the rich geometric structure encoded in the Schrödinger equation to
engineer adapted control schemes and to characterize their qualitative properties.
The Lie–Galerkin technique that we proposed starting from 2009
builds on this premises in order to provide powerful tests for the controllability of quantum systems defined on infinite-dimensional Hilbert spaces.

Although the focus of geometric control theory is on qualitative properties, its impact can also be disruptive when it is used in combination with quantitative analytical tools, in which case it can dramatically improve the computational efficiency. This is the case in particular in optimal control. Classical optimal control techniques (in particular, Pontryagin Maximum Principle, conjugate point theory, associated numerical methods) can be significantly improved by combining them with powerful modern techniques of geometric optimal control, of the theory of numerical continuation, or of dynamical system theory , . Geometric optimal control allows the development of general techniques, applying to wide classes of nonlinear optimal control problems, that can be used to characterize the behavior of optimal trajectories and in particular to establish regularity properties for them and for the cost function. Hence, geometric optimal control can be used to obtain powerful optimal syntheses results and to provide deep geometric insights into many applied problems. Numerical optimal control methods with geometric insight are in particular important to handle subtle situations such as rigid optimal paths and, more generally, optimal syntheses exhibiting abnormal minimizers.

Optimal control is not the only area where the geometric approach has a great impact.
Let us mention, for instance, motion planning, where different geometric approaches have been developed:
those based on the **Lie algebra** associated with the control system
, , those based on the differentiation of nonlinear flows
such as the **return method** , , and
those exploiting the **differential flatness** of the system .

Geometric control theory is not only a powerful framework to investigate control systems,
but also a useful tool to model and study phenomena that are not *a priori* control-related.
Two occurrences of this property play an important role in the activities of CAGE:

geometric control theory as a tool to investigate properties of mathematical structures;

geometric control theory as a modeling tool for neurophysical phenomena and for synthesizing biomimetic algorithms based on such models.

Examples of the first type, concern, for instance, hypoelliptic heat kernels or shape optimization . Examples of the second type are inactivation principles in human motricity or neurogeometrical models for image representation of the primary visual cortex in mammals .

A particularly relevant class of
control systems, both from the point of view of
theory and applications,
is characterized by the linearity of the controlled vector field with respect to the
control parameters.
When the controls are unconstrained in norm, this means that the admissible velocities form a distribution in the tangent bundle to the state manifold. If the distribution is equipped with a point-dependent quadratic form (encoding the cost of the control), the resulting geometrical structure is said to be
**sub-Riemannian**.
Sub-Riemannian geometry appears as the underlying geometry of nonlinear control systems: in a similar way
as the linearization of a control system provides local informations
which are
readable using the Euclidean metric scale, sub-Riemannian geometry provides an adapted non-isotropic class of lenses which are often much more informative.
As such, its study is fundamental for control design. The importance of sub-Riemannian geometry goes beyond control theory and it is an active field of research both in differential geometry ,
geometric measure theory
and hypoelliptic operator theory .

The geometric control approach has historically been related to the development of finite-dimensional control theory. However, its impact in the analysis of distributed parameter control systems and in particular systems of controlled partial differential equations has been growing in the last decades, complementing analytical and numerical approaches, providing dynamical, qualitative and intrinsic insight . CAGE's ambition is to be at the core of this development in the years to come.

A suggestive application of sub-Riemannian geometry and in particular of hypoelliptic diffusion comes from a model of geometry of vision describing the functional architecture of the primary visual cortex V1.
In 1958, Hubel and Wiesel (Nobel in 1981)
observed that the visual cortex V1 is endowed with the so-called **pinwheel structure**, characterized by neurons grouped into orientation columns, that are sensible both to positions and directions .
The mathematical rephrasing of this discovery is that the visual cortex lifts an image from

A simplified version of the model can be described as follows: neurons of V1 are grouped into orientation columns, each of them
being sensitive to visual stimuli at a given point of the retina and for a given direction
on it. The retina is
modeled by the real plane, i.e., each point is represented by a pair

Orientation columns are connected between them in two different ways. The first kind of connections are the vertical (inhibitory) ones, which connect orientation columns belonging to the same hypercolumn and sensible to similar directions. The second kind of connections are the horizontal (excitatory) connections, which connect neurons belonging to different (but not too far) hypercolumns and sensible to the same directions. The resulting metric structure is sub-Riemannian and the model obtained in this way provides a convincing explanation in terms of sub-Riemannian geodesics of gestalt phenomena such as Kanizsa illusory contours.

The sub-Riemannian model for image representation of V1
has a great potential of yielding powerful
bio-inspired
image processing
algorithms , .
Image inpainting, for instance, can be implemented by reconstructing an incomplete image by activating orientation columns in the missing regions
in accordance with sub-Riemannian non-isotropic constraints.
The process intrinsically defines an hypoelliptic heat equation on

We have been working on the model and its software implementation since 2012. This work has been supported by several project, as the ERC starting grant GeCoMethods and the ERC Proof of Concept ARTIV1 of U. Boscain, and the ANR GCM.

A parallel approach that we will pursue and combine with this first one is based on **pattern matching in the group of diffeomorphisms**. We want to extend this approach, already explored in the Riemannian setting , , to the general sub-Riemannian framework.
The paradigm of the approach is the following:
consider a distortable object, more or less rigid, discretized into a certain number of points. One may track its distortion by considering the paths drawn by these points. One would however like to know how the object itself (and not its discretized version) has been distorted. The study in , shed light on the importance of Riemannian geometry in this kind of problem. In particular, they study the Riemannian submersion obtained by making the group of diffeomorphisms act transitively on the manifold formed by the points of the discretization, minimizing a certain energy so as to take into account the whole object.
Settled as such, the problem is Riemannian, but if one considers objects involving connections, or submitted to nonholonomic constraints,
like in medical imaging where one tracks the motions of organs, then one comes up with a sub-Riemannian problem. The transitive group is then far bigger, and the aim is to lift curves submitted to these nonholonomic constraints into curves in the set of diffeomorphisms satisfying the corresponding constraints, in a unique way and minimizing an energy (giving rise to a sub-Riemannian structure).

The goal of quantum control is to design efficient protocols for tuning the occupation probabilities of the energy levels of a system. This task is crucial in atomic and molecular physics, with applications ranging from photochemistry to nuclear magnetic resonance and quantum computing. A quantum system may be controlled by exciting it with one or several external fields, such as magnetic or electric fields. The goal of quantum control theory is to adapt the tools originally developed by control theory and to develop new specific strategies that tackle and exploit the features of quantum dynamics (probabilistic nature of wavefunctions and density operators, measure and wavefunction collapse, decoherence, ...). A rich variety of relevant models for controlled quantum dynamics exist, encompassing low-dimensional models (e.g., single-spin systems) and PDEs alike, with deterministic and stochastic components, making it a rich and exciting area of research in control theory.

The controllability of quantum system
is
a well-established topic when the state space
is finite-dimensional , thanks to general controllability methods for left-invariant control systems on compact Lie groups , .
When the state space
is
infinite-dimensional, it is known that in general
the bilinear Schrödinger equation is not exactly controllable . Nevertheless, weaker
controllability properties, such as approximate controllability or controllability between eigenstates of the internal Hamiltonian
(which are the most relevant physical states), may hold.
In certain cases, when the state space
is a function space on a 1D manifold, some rather precise description of the set of reachable states has
been provided . A similar description for higher-dimensional manifolds seems intractable and at the moment only approximate controllability results
are available , , .
The most widely applicable tests for controllability of quantum systems in infinite-dimensional Hilbert spaces are based on the **Lie–Galerkin technique** , , . They allow, in particular, to show that the controllability property is generic among this class of systems .

A family of algorithms
which are specific to quantum systems are those based on adiabatic evolution , , .
The basic principle of adiabatic control is
that the flow of a slowly varying Hamiltonian can be approximated (up to a phase factor) by a quasi-static evolution, with a precision proportional to the velocity of variation of the Hamiltonian.
The advantage of the **adiabatic approach** is that it is constructive and produces control laws which are both smooth and
robust to parameter uncertainty. The paradigm is based on
the adiabatic perturbation theory developed in mathematical physics , , ,
where it plays an important role for understanding molecular dynamics.
Approximation theory by adiabatic perturbation can be used to
describe the evolution of the occupation probabilities of
the energy levels of a slowly varying
Hamiltonian.
Results from the last 15 years, including those by members of our team , , have highlighted the effectiveness of control techniques based on adiabatic path following.

Switched and hybrid systems constitute a broad framework for the description of the heterogeneous aspects of systems in which continuous dynamics (typically pertaining to physical quantities) interact with discrete/logical components. The development of the switched and hybrid paradigm has been motivated by a broad range of applications, including automotive and transportation industry , energy management and congestion control .

Even if both controllability and observability of switched and hybrid systems have attracted much research efforts, the central role in their study is played by the problem of stability and stabilizability. The goal is to determine whether a dynamical or a control system whose evolution is influenced by a time-dependent signal is uniformly stable or can be uniformly stabilized , . Uniformity is considered with respect to all signals in a given class. Stability of switched systems lead to several interesting phenomena. For example, even when all the subsystems corresponding to a constant switching law are exponentially stable, the switched systems may have divergent trajectories for certain switching signals . This fact illustrates the fact that stability of switched systems depends not only on the dynamics of each subsystem but also on the properties of the class of switching signals which is considered.

The most common class of switching signals which has been considered in the literature is made of all piecewise constant signals. In this case uniform stability of the system is equivalent to the existence of a common quadratic Lyapunov function . Moreover, provided that the system has finitely many modes, the Lyapunov function can be taken polyhedral or polynomial , , . A special role in the switched control literature has been played by common quadratic Lyapunov functions, since their existence can be tested rather efficiently (see the surveys , and the references therein). It is known, however, that the existence of a common quadratic Lyapunov function is not necessary for the global uniform exponential stability of a linear switched system with finitely many modes. Moreover, there exists no uniform upper bound on the minimal degree of a common polynomial Lyapunov function . More refined tools rely on multiple and non-monotone Lyapunov functions . Let us also mention linear switched systems technics based on the analysis of the Lie algebra generated by the matrices corresponding to the modes of the system .

For systems evolving in the plane, more geometrical tests apply, and yield a complete characterization of the stability , . Such a geometric approach also yields sufficient conditions for uniform stability in the linear planar case .

In many situations,
it is interesting for modeling purposes to
specify the features
of the switched system by introducing
**constrained switching rules**. A typical constraint is that each mode is activated for at least a fixed minimal amount of time, called the dwell-time.
Switching rules can also be imposed, for instance, by a timed automata.
When constraints apply, the common Lyapunov function approach becomes conservative and new tools have to be developed to give more detailed characterizations of stable and unstable systems.

Our approach to constrained switching is based on the idea of relating the analytical properties of the classes of constrained switching laws (shift-invariance, compactness, closure under concatenation, ...) to the stability behavior of the corresponding switched systems.
One can introduce
**probabilistic uncertainties** by endowing the classes of admissible signals with suitable probability measures.
One then looks at the corresponding Lyapunov exponents, whose existence is established by the multiplicative ergodic theorem.
The interest of this approach is that probabilistic stability analysis filters out highly `exceptional' worst-case trajectories.
Although less explicitly characterized from a dynamical viewpoint than its deterministic counterpart, the probabilistic notion of uniform exponential stability can be studied
using several
reformulations of Lyapunov exponents proposed in the literature , , .

The theoretical questions raised by the different applicative area will be pooled in a research axis on the transversal aspects of geometric control theory and sub-Riemannian structures.

We recall that sub-Riemannian geometry is a generalization of Riemannian geometry, whose birth dates back to Carathéodory's seminal paper on the foundations of Carnot thermodynamics , followed by E. Cartan's address at the International Congress of Mathematicians in Bologna . In the last twenty years, sub-Riemannian geometry has emerged as an independent research domain, with a variety of motivations and ramifications in several parts of pure and applied mathematics. Let us mention geometric analysis, geometric measure theory, stochastic calculus and evolution equations together with applications in mechanics and optimal control (motion planning, robotics, nonholonomic mechanics, quantum control) , .

One of the main open problems in sub-Riemannian geometry concerns the regularity of length-minimizers , .
Length-minimizers are solutions to a variational problem with constraints and satisfy a first-order necessary condition resulting from the Pontryagin Maximum Principle (PMP).
Solutions of the PMP are either *normal* or *abnormal*.
Normal length-minimizer are well-known to be smooth, i.e.,

An interesting set of recent results in sub-Riemannian geometry concerns the extension to such a setting of the Riemannian notion of sectional curvature. The curvature operator can be introduced in terms of the symplectic invariants of the Jacobi curve , , , a curve in the Lagrange Grassmannian related to the linearization of the Hamiltonian flow. Alternative approaches to curvatures in metric spaces are based either on the associated heat equation and the generalization of the curvature-dimension inequality , or on optimal transport and the generalization of Ricci curvature , , , .

**Emmanuel Trélat** has been invited speaker
at the International Congress of Mathematicians (ICM2018) in Rio, Brazil, in the session “Control theory and optimization".

The poster
“Adaptive Stimulation
Strategy for Selective Brain Oscillations Disruption in a
Neuronal Population Model with Delays" by **Jakub Orlowski**, Antoine Chaillet, **Mario Sigalotti**, and Alain Destexhe, has received the
CPHS 2018 Best Poster Prize at the 2nd IFAC Conference on Cyber-Physical & Human Systems.

Let us list here our new results in the geometry of vision axis and, more generally, on hypoelliptic diffusion and sub-Riemannian geometry.

In we present a new image inpainting algorithm, the Averaging and Hypoelliptic Evolution (AHE) algorithm, inspired by the one presented in and based upon a (semi-discrete) variation of the Citti–Petitot–Sarti model of the primary visual cortex V1. In particular, we focus on reconstructing highly corrupted images (i.e. where more than the 80% of the image is missing).

In
we address the double bubble problem for the anisotropic Grushin perimeter

We would also like to mention the defense of the PhD thesis of Ludovic Saccchelli on the subject.

Let us list here our new results in quantum control theory.

In we consider a quantum particle in a potential

Let us list here our new results about stability and stabilization of control systems, on the properties of systems with uncertain dynamics.

Based on the notion of generalized homogeneity, we develop in a new algorithm of feedback control design for a plant modeled by a linear evolution equation in a Hilbert space with a possibly unbounded operator. The designed control law steers any solution of the closed-loop system to zero in a finite time. Method of homogeneous extension is presented in order to make the developed control design principles to be applicable for evolution systems with non-homogeneous operators. The design scheme is demonstrated for heat equation with the control input distributed on the segment

Motivated by improved ways to disrupt brain oscillations linked to Parkinson's disease, we propose in an adaptive output feedback strategy for the stabilization of nonlinear time-delay systems evolving on a bounded set. To that aim, using the formalism of input-to-output stability (IOS), we first show that, for such systems, internal stability guarantees robustness to exogenous disturbances. We then use this feature to establish a general result on scalar adaptive output feedback of time-delay systems inspired by the “

In we consider open channels represented by Saint-Venant equations that are monitored and controlled at the downstream boundary and subject to unmeasured flow disturbances at the upstream boundary. We address the issue of feedback stabilization and disturbance rejection under Proportional-Integral (PI) boundary control. For channels with uniform steady states, the analysis has been carried out previously in the literature with spectral methods as well as with Lyapunov functions in Riemann coordinates. In , our main contribution is to show how the analysis can be extended to channels with non-uniform steady states with a Lyapunov function in physical coordinates.

Given a discrete-time linear switched system

The exponential stability problem of the nonlinear Saint-Venant equations is addressed in . We consider the general case where an arbitrary friction and space-varying slope are both included in the system, which lead to non-uniform steady-states. An explicit quadratic Lyapunov function as a weighted function of a small perturbation of the steady-states is constructed. Then we show that by a suitable choice of boundary feedback controls, that we give explicitly, the local exponential stability of the nonlinear Saint-Venant equations for the

Given a linear control system in a Hilbert space with a bounded control operator, we establish in a characterization of exponential stabilizability in terms of an observability inequality. Such dual characterizations are well known for exact (null) controllability. Our approach exploits classical Fenchel duality arguments and, in turn, leads to characterizations in terms of observability inequalities of approximately null controllability and of

Let us also mention the lecture notes on stabilization of semilinear PDE's, which have been published this year.

Let us list here our new results in optimal control theory beyond the sub-Riemannian framework.

In control theory the term chattering is used to refer to fast oscillations of controls, such as an infinite number of switchings over a finite time interval. In we focus on three typical instances of chattering: the Fuller phenomenon, referring to situations where an optimal control features an accumulation of switchings in finite time; the Robbins phenomenon, concerning optimal control problems with state constraints, where the optimal trajectory touches the boundary of the constraint set an infinite number of times over a finite time interval; and the Zeno phenomenon, for hybrid systems, referring to a trajectory that depicts an infinite number of location switchings in finite time. From the practical point of view, when trying to compute an optimal trajectory, for instance, by means of a shooting method, chattering may be a serious obstacle to convergence. In we propose a general regularization procedure, by adding an appropriate penalization of the total variation. This produces a family of quasi-optimal controls whose associated cost converge to the optimal cost of the initial problem as the penalization tends to zero. Under additional assumptions, we also quantify quasi-optimality by determining a speed of convergence of the costs.

The Allee threshold of an ecological system distinguishes the sign of population growth either towards extinction or to carrying capacity. In practice human interventions can tune the Allee threshold for instance thanks to the sterile male technique and the mating disruption. In we address various control objectives for a system described by a diffusion-reaction equation regulating the Allee threshold, viewed as a real parameter determining the unstable equilibrium of the bistable nonlinear reaction term. We prove that this system is the mean field limit of an interacting system of particles in which individual behaviours are driven by stochastic laws. Numerical simulations of the stochastic process show that population propagations are governed by wave-like solutions corresponding to traveling solutions of the macroscopic reaction-diffusion system. An optimal control problem for the macroscopic model is then introduced with the objective of steering the system to a target traveling wave. The relevance of this problem is motivated by the fact that traveling wave solutions model the fact that bounded space domains reach asymptotically an equilibrium configuration. Using well known analytical results and stability properties of traveling waves, we show that well-chosen piecewise constant controls allow to reach the target approximately in sufficiently long time. We then develop a direct computational method and show its efficiency for computing such controls in various numerical simulations. Finally we show the efficiency of the obtained macroscopic optimal controls in the microscopic system of interacting particles and we discuss their advantage when addressing situations that are out of reach for the analytical methods. We conclude the article with some open problems and directions for future research.

Consider a general nonlinear optimal control problem in finite dimension, with constant state and/or control delays. By the Pontryagin Maximum Principle, any optimal trajectory is the projection of a Pontryagin extremal. In we establish that, under appropriate assumptions, Pontryagin extremals depend continuously on the parameter delays, for adequate topologies. The proof of the continuity of the trajectory and of the control is quite easy, however, for the adjoint vector, the proof requires a much finer analysis. The continuity property of the adjoint with respect to the parameter delay opens a new perspective for the numerical implementation of indirect methods, such as the shooting method. We also discuss the sharpness of our assumptions.

In we consider a state-constrained optimal control problem of a system of two non-local partial-differential equations, which is an extension of the one introduced in a previous work in mathematical oncology. The aim is to minimize the tumor size through chemotherapy while avoiding the emergence of resistance to the drugs. The numerical approach to solve the problem was the combination of direct methods and continuation on discretization parameters, which happen to be insufficient for the more complicated model, where diffusion is added to account for mutations. In , we propose an approach relying on changing the problem so that it can theoretically be solved thanks to a Pontryagin Maximum Principle in infinite dimension. This provides an excellent starting point for a much more reliable and efficient algorithm combining direct methods and continuations. The global idea is new and can be thought of as an alternative to other numerical optimal control techniques.

We would also like to mention the defense of the PhD theses of Riccardo Bonalli and Antoine Olivier on the subject.

A bilateral contract with CNES funded the PhD thesis of Antoine Olivier, who defended in October 2018.

ANR SRGI, for *Sub-Riemannian Geometry and Interactions*, coordinated by **Emmanuel Trélat**, started in 2015 and runs until 2020. Other partners: Toulon University and Grenoble University. SRGI deals with sub-Riemannian geometry, hypoelliptic diffusion and geometric control.

ANR Finite4SoS, for *Commande et estimation en temps fini pour les Systèmes de Systèmes*, coordinated by Wilfrid Perruquetti, started in 2015 and runs until 2019. Other partners: Inria Lille, CAOR - ARMINES. Finite4SoS aims at developing a new promising framework to address control and estimation issues of Systems of Systems subject to model diversity, while achieving robustness as well as severe time response constraints.

ANR QUACO, for *QUAntum COntrol: PDE systems and MRI applications*, coordinated by Thomas Chambrion, started in 2017 and runs until 2021. Other partners: Lorraine University. QUACO aims at contributing to quantum control theory in two directions: improving the comprehension of the dynamical properties of controlled quantum systems in infinite-dimensional state spaces, and improve the efficiency of control algorithms for MRI.

Program: ERC Proof of Concept

Project acronym: ARTIV1

Project title: An artificial visual cortex for image processing

Duration: From April 2017 to September 2018.

Coordinator: Ugo Boscain

Abstract: The ERC starting grant GECOMETHODS, on which this POC is based, tackled problems of diffusion equations via geometric control methods. One of the most striking achievements of the project has been the development of an algorithm of image reconstruction based mainly on non-isotropic diffusion. This algorithm is bio-mimetic in the sense that it replicates the way in which the primary visual cortex V1 of mammals processes the signals arriving from the eyes. It has performances that are at the state of the art in image processing. These results together with others obtained in the ERC project show that image processing algorithms based on the functional architecture of V1 can go very far. However, the exceptional performances of the primary visual cortex V1 rely not only on the particular algorithm used, but also on the fact that such algorithm 'runs' on a dedicated hardware having the following features: 1. an exceptional level of parallelism; 2. connections that are well adapted to transmit information in a non-isotropic way as it is required by the algorithms of image reconstruction and recognition. The idea of this POC is to create a dedicated hardware (called ARTIV1) emulating the functional architecture of V1 and hence having on one hand a huge degree of parallelism and on the other hand connections among the CPUs that reflect the non-isotropic structure of the visual cortex V1.

Jean-Michel Coron was at EPFL (Switzerland) from January to June 2018.

Ugo Boscain and Mario Sigalotti were Members of the Organizing Committee of the Workshop “Sub-Riemannian Geometry and Topolò(gy)", Topolò/Topolove, Italy, June 2018

Emmanuel Trélat was Member of the Program Committee of the 18th French-German-Italian Conference on Optimization (FGI'2018).

Emmanuel Trélat was Member of the Scientific Committee of the 23rd International Symposium on Mathematical Programming (ISMP 2018).

Ugo Boscain is Associate editor of SIAM Journal of Control and Optimization

Ugo Boscain is Managing editor of Journal of Dynamical and Control Systems

Jean-Michel Coron is Editor-in-chief of Comptes Rendus Mathématique

Jean-Michel Coron is Member of the editorial board of Journal of Evolution Equations

Jean-Michel Coron is Member of the editorial board of Asymptotic Analysis

Jean-Michel Coron is Member of the editorial board of ESAIM : Control, Optimisation and Calculus of Variations

Jean-Michel Coron is Member of the editorial board of Applied Mathematics Research Express

Jean-Michel Coron is Member of the editorial board of Advances in Differential Equations

Jean-Michel Coron is Member of the editorial board of Math. Control Signals Systems

Jean-Michel Coron is Member of the editorial board of Annales de l'IHP, Analyse non linéaire

Mario Sigalotti is Associate editor of ESAIM : Control, Optimisation and Calculus of Variations

Mario Sigalotti is Associate editor of Journal on Dynamical and Control Systems

Emmanuel Trélat is Editor-in-chief of ESAIM : Control, Optimisation and Calculus of Variations

Emmanuel Trélat is Associate editor of Syst. Cont. Letters

Emmanuel Trélat is Associate editor of J. Dynam. Cont. Syst.

Emmanuel Trélat is Associate editor of Bollettino dell'Unione Matematica Italiana

Emmanuel Trélat is Associate editor of ESAIM Math. Modelling Num. Analysis

Emmanuel Trélat is Editor of BCAM Springer Briefs

Emmanuel Trélat is Associate editor of J. Optim. Theory Appl.

Emmanuel Trélat is Associate editor of Math. Control Related fields

Ugo Boscain was invited speaker at the International Conference “Optimal Control and Differential Games”, dedicated to the 110th anniversary of L.S. Pontryagin, Dec. 2018.

Ugo Boscain was invited speaker at the conference “Dynamics, Control, and Geometry", Banach Center, Warsaw, Sept. 2018.

Ugo Boscain was invited speaker at Linkopyng University, Department of Electrical Engineering, Nov. 2018.

Ugo Boscain was invited speaker at the conference “Analysis, Control and Inverse Problems for PDEs”, Napoli (Italy), Nov. 2018.

Mario Sigalotti was invited speaker at the Workshop Quantum control and feedback: foundations and applications, Paris, Jun. 2018.

Emmanuel Trélat was invited speaker at ICM 2018, Rio, section “Control Theory and Optimization", Aug. 2018.

Emmanuel Trélat was invited speaker at Analysis, Control and Inverse Problems for PDEs, Naples, Nov. 2018.

Emmanuel Trélat was invited speaker at Dynamics Control and Geometry, Varsovie, Sept. 2018.

Emmanuel Trélat was invited speaker at 14th Viennese Conference on Optimal Control and Dynamic Games, Vienna, July 2018.

Emmanuel Trélat was invited speaker at Portuguese Meeting on Optimal Control 2018, Coimbra (Portugal), June 2018.

Emmanuel Trélat was invited speaker at International Symposium on Mathematical Control Theory, Shanghai, June 2018.

Emmanuel Trélat was invited speaker at GAMM Munich, March 2018.

Emmanuel Trélat is director of the Fondation Sciences Mathématiques de Paris (FSMP).

Ugo Boscain thought “Sub-elliptic diffusion” to PhD students at SISSA, Trieste Italy

Ugo Boscain thought “Automatic Control” (with Mazyar Mirrahimi) at Ecole Polytechnique

Ugo Boscain thought “MODAL of applied mathematics. Contrôle de modèles dynamiques” at Ecole Polytechnique

Emmanuel Trélat thought “Control in finite and infinite dimension” at Master 2, Sorbonne Université

PhD: Riccardo Bonalli, Optimal Control of Aerospace Systems with Control-State Constraints and Delays, Sorbonne Université, July 2018, supervised by Emmanuel Trélat.

PhD: Antoine Olivier, Optimal and robust attitude control of a launcher, Sorbonne Université, October 2018, supervised by Emmanuel Trélat and co-supervised by Thomas Haberkorn, Éric Bourgeois, David-Alexis Handschuh.

PhD: Ludovic Sacchelli, Singularities in sub-Riemannian geometry, Université Paris-Saclay, September 2018, supervised by Ugo Boscain and Mario Sigalotti.

PhD in progress: Nicolas Augier, “Contrôle adiabatique des systèmes quantiques", started in September 2016, supervisors: Ugo Boscain, Mario Sigalotti.

PhD in progress: Amaury Hayat, “ Contrôle et stabilisation en mécanique des fluides", started in October 2016, supervisors: Jean-Michel Coron and Sébastien Boyaval

PhD in progress: Mathieu Kohli, “Volume and curvature in sub-Riemannian geometry", started in September 2016, supervisors: Davide Barilari, Ugo Boscain.

PhD in progress: Gontran Lance, started in September 2018, supervisors: Emmanuel Trélat and Enrique Zuazua.

PhD in progress: Cyril Letrouit, “Équation des ondes sous-riemanniennes", started in September 2019, supervisor Emmanuel Trélat.

PhD in progress: Jakub Orłowski, “Modeling and steering brain oscillations based on in vivo optogenetics data", started in September 2016, supervisors: Antoine Chaillet, Alain Destexhe, and Mario Sigalotti.

PhD in progress: Eugenio Pozzoli, “Adiabatic Control of Open Quantum Systems", started in September 2018, supervisors: Ugo Boscain and Mario Sigalotti.

PhD in progress: Shengquan Xiang, Stabilisation des fluides par feedbacks non-linéaires, September 2016, supervisor: Jean-Michel Coron.

PhD in progress: Christophe Zhang, started in October 2016, supervisor: Jean-Michel Coron

Ugo Boscain was referee and member of the jury of the HDR of Jean-Marie Mirebeau, Université Paris-Sud.

Mario Sigalotti was member of the jury of the PhD thesis of Abdelkrim Bahloul, Univ. Paris-Saclay.

Emmanuel Trélat was co-supervisor and member of the jury of the PhD thesis of Camille Pouchol, Sorbonne Université.

Emmanuel Trélat was member of the jury of the PhD thesis of F. Omnès, Sorbonne Université.

Emmanuel Trélat was referee and member of the jury of the PhD thesis of S. Mitra, Univ. Toulouse.

Emmanuel Trélat was referee and member of the jury of the PhD thesis of T. Weisser, Univ. Toulouse.

Emmanuel Trélat was referee and member of the jury of the PhD thesis of S. Maslovskaya, Univ. Paris-Saclay.

Emmanuel Trélat was referee and member of the jury of the PhD thesis of A. Vieira, Grenoble University.

Emmanuel Trélat was member of the jury of the HDR of F. Chittaro, Univ. Toulon.

Emmanuel Trélat is member of the Comité d'Honneur du Salon des Jeux et Culture Mathématique since November 2018

Nicolas Augier, Ugo Boscain, and Mario Sigalotti are authors of the popularization article explaining how broken adiabatic paths can be used to enhance the control of a quantum systems.

Ugo Boscain and Jean-Michel Coron gave a lecture at journée ENS-UPS, ENS Paris

Emmanuel Trélat gave a lecture at ENS Ulm to first-year students

Emmanuel Trélat gave a lecture at Université Paris-Diderot to first- and second-year students

Ugo Boscain gave a lecture at Alfaclass, Saint-Barthélemy, Aosta, Italy

Emmanuel Trélat gave a lecture at Salon des Jeux et Culture Mathématique