2023Activity reportProject-TeamCAGE

3.1 Research domain

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) 105. 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 extremals81. 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 92, 82, 98. 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 stability96, 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 90 and models for neural activity 74. Therapy analysis from the point of view of optimal control has also attracted a great attention 94.

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 102.

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 75.

3.2 Scientific foundations

At the core of the scientific activity of the team is the geometric control approach. 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 65. 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 in 66 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 100, 93. 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 86, 83, those based on the differentiation of nonlinear flows such as the return method70, 71, and those exploiting the differential flatness of the system 73.

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 52 or shape optimization 56. Examples of the second type are inactivation principles in human motricity 58 or neurogeometrical models for image representation of the primary visual cortex in mammals 63.

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 85, geometric measure theory 54 and hypoelliptic operator theory 59.

4.1 First axis: Quantum control

Quantum control is one of the bricks of quantum engineering, since manipulation of quantum mechanical systems is ubiquitous in applications such as quantum computation, quantum cryptography, and quantum sensing (in particular, imaging by nuclear magnetic resonance).

Quantum control 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. The efficiency of the control action has a dramatic impact on the quality of the coherence and the robustness of the required manipulation. Minimal time constraints and interaction of time scales are important factors for characterizing the efficiency of a quantum control strategy. CAGE works for the improvement of quantum control paradigms, especially for what concerns quantum systems evolving in infinite-dimensional Hilbert spaces. The controllability of quantum system is a well-established topic when the state space is finite-dimensional 72, thanks to general controllability methods for left-invariant control systems on compact Lie groups 64, 78. When the state space is infinite-dimensional, it is known that in general the bilinear Schrödinger equation is not exactly controllable 103. The Lie–Galerkin technique 66 combines finite-dimensional geometric control techniques and the distributed parameter framework in order to provide the most powerful available tests for the approximate controllability of quantum systems defined on infinite-dimensional Hilbert spaces. Another important technique to the development of which we contribute is adiabatic quantum control. Adiabatic approximation theory and, in particular, adiabatic evolution 87, 99, 106 is well-known to improve the robustness of the control strategy and is strongly related to time scales analysis. The advantage of the adiabatic control is that it is constructive and produces control laws which are both smooth and robust to parameter uncertainty 107, 80, 62.

4.2 Second axis: Stability and stabilization

A control application with a long history and still very challenging open problems is stabilization. For infinite-dimensional systems, in particular nonlinear ones, the richness of the possible functional analytical frameworks makes feedback stabilization a challenging and active domain of research. Of particular interest are the different types of stabilization that may be obtained: exponential, polynomial, finite-time, ... Another important aspect of stabilization concerns control of systems with uncertain dynamics, i.e., with dynamics including possibly non-autonomous parameters whose value and dependence on time cannot be anticipated. 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. Switched and hybrid systems constitute a broad framework for the description of the heterogeneous 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 95, energy management 88 and congestion control 84. Even if both controllability 97 and observability 79 of switched and hybrid systems raise several important research issues, the central role in their study is played by uniform stability and stabilizabilization 82, 98. Uniformity is considered with respect to all signals in a given class, and it is well-known that stability of switched systems depends not only on the dynamics of each subsystem but also on the properties of the considered class of switching signals. 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. 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. 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 57, 68, 104.

4.3 Third axis: Motion planning and optimal control

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. Geometric optimal control methods are in particular important to handle subtle situations such as rigid optimal paths and, more generally, optimal syntheses exhibiting abnormal minimizers.

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 100, 93. Applications of optimal control theory considered by CAGE concern, in particular, motion planning problems for aerospace (atmospheric re-entry, orbit transfer, low cost interplanetary space missions, ...) 60, 101.

4.4 Fourth axis: Geometric models for vision and sub-Riemannian geometry

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. In particular, we use control theory to investigate the properties of sub-Riemannian structures, both for the sake of mathematical understanding and as a modeling tool for image and sound perception and processing . We recall that sub-Riemannian geometry is a geometric framework which is used to measure distances in nonholonomic contexts and which has a natural and powerful optimal control interpretation in terms control-linear systems with quadratic cost. Sub-Riemannian geometry, and in particular the theory of their 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 76, 89, 67, 91. Such a model can be used as a powerful paradigm for bio-inspired image processing, as already illustrated in the literature 63, 61. Our contributions to geometry of vision 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 55. Nonholonomic constraints can be especially useful to describe distortions of sets of interconnected objects (e.g., motions of organs in medical imaging).

5.1 Awards

Rémi Robin was awarded with the Prix solennel de thèse de la chancellerie de Paris, the Prix de thèse PGMO 2023, and the Prix de thèse SMAI-GAMNI for his PhD thesis defended in 2022 and obtained while being member of the team CAGE.

Daniele Cannarsa and Mario Sigalotti have been awarded the Brockett-Willems Outstanding Paper Award 2023 for the best paper published in Systems & Control Letters during the two-year period from January 2021 through December 2022 (paper written while Daniele Cannarsa was member of the team CAGE).

6.1 Quantum control: new results

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

• Achiral molecules can be made temporarily chiral by excitation with electric fields, in the sense that an average over molecular orientations displays a net chiral signal. In 46, we go beyond the assumption of molecular orientations to remain fixed during the excitation process. Treating both rotations and vibrations quantum mechanically, we identify conditions for the creation of chiral vibrational wavepackets – with net chiral signals – in ensembles of achiral molecules which are initially randomly oriented. Based on the analysis of symmetry and controllability, we derive excitation schemes for the creation of chiral wavepackets using a combination of (a) microwave and IR pulses and (b) a static field and a sequence of IR pulses. These protocols leverage quantum rotational dynamics for pump-probe spectroscopy of chiral vibrational dynamics, extending the latter to regions of the electromagnetic spectrum other than the UV.
• In 48, we explore the controllability of a closed multi-input control-affine quantum system. Previous studies have demonstrated that a spectrum connected by conical intersections which do not pile up yields exact controllability in finite dimension and approximate controllability in infinite dimension. Actually, the property that intersections between eigenvalues are conical and that they do not pile up is generic. However, in physical situations, due to symmetry of the system, the spectrum can exhibit intersections that are not conical and possibly pile up. We extend the controllability result to cover this type of situations under the hypothesis that the intersections have at least one conical direction and the piled-up intersections have "rationally unrelated germs". Finally, we provide a testable first-order sufficient condition for controllability. Physically relevant examples are provided.
• In 45, we obtain observability estimates for Schrödinger equations in the plane. More precisely, considering a periodic bounded potential, we prove that the evolution Schrödinger equation is observable from any periodic measurable set, in any small time. We then extend Taüffer's recent result in the two-dimensional case to less regular observable sets and general bounded periodic potentials. The methodology of the proof is based on the use of the Floquet-Bloch transform, Strichartz estimates and semiclassical defect measures for the obtention of observability inequalities for a family of Schrödinger equations posed on the torus.
• In 20, we study, in the semiclassical sense, the global approximate controllability in small time of the quantum density and quantum momentum of the 1-D semiclassical cubic Schrödinger equation with two controls between two states with positive quantum densities. We first control the asymptotic expansions of the zeroth and first order of the physical observables via the Agrachev–Sarychev method. Then we conclude the proof through techniques of semiclassical approximation of the nonlinear Schrödinger equation.
• In 26, we establish some properties of quantum limits on a product manifold, proving for instance that, under appropriate assumptions, the quantum limits on the product of manifolds are absolutely continuous if the quantum limits on each manifolds are absolutely continuous. On a product of Riemannian manifolds satisfying the minimal multiplicity property, we prove that a periodic geodesic can never be charged by a quantum limit.

6.2 Stability and stabilization: new results

Let us list here our new results about stability and stabilization of control and hybrid systems.

• In 23, we study some spectral properties of the scalar dynamical system defined by a linear delay-differential equation with two positive delays. More precisely, the existing links between the delays and the maximal multiplicity of the characteristic roots are explored, as well as the dominancy of such roots compared with the spectrum localization. As a by-product of the analysis, the pole placement issue is revisited with more emphasis on the role of the delays as control parameters in defining a partial pole placement guaranteeing the closed-loop stability with an appropriate decay rate of the corresponding dynamical system.
• In 18, we consider the problem of determining the stability properties, and in particular assessing the exponential stability, of a singularly perturbed linear switching system. One of the challenges of this problem arises from the intricate interplay between the small parameter of singular perturbation and the rate of switching as both tend to zero. Our approach consists in characterizing suitable auxiliary linear systems that provide lower and upper bounds for the asymptotics of the maximal Lyapunov exponent of the linear switching system as the parameter of the singular perturbation tends to zero.
• In 29, we discuss the notion of universality for classes of candidate common Lyapunov functions of linear switched systems. On the one hand, we prove that a family of absolutely homogeneous functions is universal as soon as it approximates arbitrarily well every convex absolutely homogeneous function for the C0 topology of the unit sphere. On the other hand, we prove several obstructions for a class to be universal, showing, in particular, that families of piecewise-polynomial continuous functions whose construction involves at most l polynomials of degree at most m (for given positive integers l,m) cannot be universal.
• One of the central questions in control theory is achieving stability through feedback control. The paper 31 introduces a novel approach that combines Reinforcement Learning (RL) with mathematical analysis to address this challenge, with a specific focus on the Sterile Insect Technique (SIT) system. The objective is to find a feedback control that stabilizes the mosquito population model. Despite the mathematical complexities and the absence of known solutions for this specific problem, our RL approach identifies a candidate solution for an explicit stabilizing control. This study underscores the synergy between AI and mathematics, opening new avenues for tackling intricate mathematical problems.
• Consider a non-autonomous continuous-time linear system in which the time-dependent matrix determining the dynamics is piecewise constant and takes finitely many values $A_1,...,A_N$. The paper 19 studies the equality cases between the maximal Lyapunov exponent associated with the set of matrices $\{A_1,...,A_N\}$, on the one hand, and the corresponding ones for piecewise deterministic Markov processes with modes $A_1,\cdots ,A_N$, on the other hand. A fundamental step in this study consists in establishing a result of independent interest, namely, that any sequence of Markov processes associated with the matrices $A_1,...,A_N$ converges, up to extracting a subsequence, to a Markov process associated with a suitable convex combination of those matrices.
• The article 39 deals with the stability of linear periodic difference delay systems, where the value at time $t$ of a solution is a linear combination with periodic coefficients of its values at finitely many delayed instants $t-{\tau }_1,...,t-{\tau }_N$. We establish a necessary and sufficient condition for exponential stability of such systems when the coefficients have Hölder-continuous derivative, that generalizes the one obtained for difference delay systems with constant coefficients by Henry and Hale in the 1970s. This condition may be construed as analyticity, in a half plane, of the (operator valued) harmonic transfer function of an associated linear control system.
• In 35, we explicitly compute the maximal Lyapunov exponent for a switched system on $S{L}_2\left(ℝ\right)$. This computation is reduced to the characterization of optimal trajectories for an optimal control problem on the Lie group.
• In 38, we prove the (uniform) global exponential stabilization of the cubic defocusing Schrödinger equation on the torus d-dimensional torus, for d=1, 2 or 3, with a linear damping localized in a subset of the torus satisfying some geometrical assumptions. In particular, this answers an open question of Dehman, Gérard, Lebeau from 2006. Our approach is based on three ingredients. First, we prove the well-posedness of the closed-loop system in Bourgain spaces. Secondly, we derive new Carleman estimates for the nonlinear equation by directly including the cubic term in the conjugated operator. Thirdly, by conjugating with energy estimates and Morawetz multipliers method, we then deduce quantitative observability estimates leading to the uniform exponential decay of the total energy of the system. As a corollary of the global stabilization result, we obtain an upper bound of the minimal time of the global null-controllability of the nonlinear equation by using a stabilization procedure and a local null-controllability result.
• The Sterile Insect Technique or SIT is presently one of the most ecological methods for controlling insect pests responsible for disease transmission or crop destruction worldwide. This technique consists of releasing sterile males into the insect pest population. This approach aims at reducing fertility in the population and, consequently, reduce significantly the native insect population after a few generations. In the work 32, we study the global stabilization of a pest population at extinction equilibrium by the SIT method. We construct explicit feedback laws that stabilize the model and do numerical simulations to show the efficiency of our feedback laws. The different feedback laws are also compared taking into account their possible implementation in field interventions.
• In the article 24, we study the problem of stabilizing the traffic flow on a ring road to a uniform steady-state using autonomous vehicles (AV). Traffic is represented at a microscopic level via a Bando-Follow-the-Leader model capable of reproducing phantom jams. For the single-lane case, a single AV can stabilize an arbitrary large ring road with an arbitrary large number of cars. Moreover, this stabilization is exponentially quick with a decay rate independent of the number of cars and a control gain also independent of the number of cars. On the other side, the stabilization domain and stabilization time depend on the number of cars. Two types of controller algorithms are proposed: a proportional control and a proportional-integral control. In both cases, the measurements used by the controller only depend on the local data around the AV, enabling an easy implementation. After numerical tests of the single-lane case, a multilane model is described using a safety-incentive mechanism for lane change. Numerical simulations for the multilane ring road suggest that the control strategy is also very efficient in such a setting, even with a single AV.

6.3 Motion planning and optimal control: new results

Let us list here our new results on controllability and motion planning algorithms, including optimal control, optimization beyond the quantum control framework.

• In 27, we address the problem of catching all speed 1 geodesics of a Riemannian manifold with a moving ball: given a compact Riemannian manifold $(M,g)$ and small parameters $ϵ>0$ and $v>0$, is it possible to find $T>0$ and an absolutely continuous map $x:[0,T]\to M$,$t\mapsto x\left(t\right)$ satisfying $\parallel \dot{x}{\parallel }_{\infty }\le v$ and such that any geodesic of $(M,g)$ traveled at speed 1 meets the open ball $B_g(x\left(t\right),ϵ)\subset M$ within time $T$? Our main motivation comes from the control of the wave equation: our results show that the controllability of the wave equation can sometimes be improved by allowing the domain of control to move adequately, even very slowly. We first prove that, in any Riemannian manifold $(M,g)$ satisfying a geodesic recurrence condition (GRC), our problem has a positive answer for any $ϵ>0$ and $v>0$, and we give examples of Riemannian manifolds $(M,g)$ for which (GRC) is satisfied.
• In the lecture notes 51 we introduce controllability, tabilization and optimal control for systems in finite and infinite dimension.
• In 25, we prove the small-time global null-controllability of forward (resp. backward) semilinear stochastic parabolic equations with globally Lipschitz nonlinearities in the drift and diffusion terms (resp. in the drift term). In particular, we solve the open question posed by S. Tang and X. Zhang, in 2009. We propose a new twist on a classical strategy for controlling linear stochastic systems. By employing a new refined Carleman estimate, we obtain a controllability result in a weighted space for a linear system with source terms. The main novelty here is that the Carleman parameters are made explicit and are then used in a Banach fixed point method. This allows to circumvent the well-known problem of the lack of compactness embeddings for the solutions spaces arising in the study of controllability problems for stochastic PDEs.
• The article 42 deals with the controllability of linear one-dimensional hyperbolic systems. Reformulating the problem in terms of linear difference equations and making use of infinite-dimensional realization theory, we obtain both necessary and sufficient conditions for approximate and exact controllability, expressed in the frequency domain. The results are applied to flows in networks.
• In 36, we consider a linear quadratic (LQ) optimal control problem in both finite and infinite dimensions. We derive an asymptotic expansion of the value function as the fixed time horizon $T$ tends to infinity. The leading term in this expansion, proportional to $T$, corresponds to the optimal value attained through the classical turnpike theory in the associated static problem. The remaining terms are associated with optimal stabilization problems towards the turnpike.
• In 16, we deal with the global exact controllability to the trajectories of the Boussinesq system posed in 2D or 3D smooth bounded domains. The velocity field of the fluid must satisfy a Navier slip-with-friction boundary condition and a Robin boundary condition is im- posed to the temperature. We assume that one can act on the velocity and the temperature on a small part of the boundary. For the proof, we first transform the boundary control problem into a distributed control problem. Then, we prove a global approximate controllability result by adapting the strategy of Coron et al [J. Eur. Math. Soc., 22 (2020), pp. 1625–1673]; this relies on the controllability properties of the inviscid Boussinesq system and the analysis of appropriate asymptotic boundary layer expansions. Finally, we conclude with a local control- lability result; as in many other cases, this can be established as a consequence of the null controllability of a linearized system through a fixed-point argument. Our contribution can be viewed as an extension of the results in [J. Eur. Math. Soc., 22 (2020), pp. 1625–1673], where thermal effects were not considered. Thus, we prove that the ideas behind the controllability properties of the Euler system and the well-prepared dissipation technique can be adapted to the present situation. Furthermore, we cover all the classical boundary conditions for the temperature, that is, those of the Robin, Neumann and Dirichlet kinds.
• The article 17 deals with the controllability of finite-dimensional linear difference delay equations, i.e., dynamics for which the state at a given time $t$ is obtained as a linear combination of the control evaluated at time t and of the state evaluated at finitely many previous instants of time $t-{\Lambda }_1,...,t-{\Lambda }_N$. Based on the realization theory developed by Y.Yamamoto for general infinite-dimensional dynamical systems, we obtain necessary and sufficient conditions, expressed in the frequency domain, for the approximate controllability in finite time in $L^q$ spaces, $q\in [1,+\infty )$. We also provide a necessary condition for $L^1$ exact controllability, which can be seen as the closure of the $L^1$ approximate controllability criterion. Furthermore, we provide an explicit upper bound on the minimal times of approximate and exact controllability, given by $dmax\{{\Lambda }_1,...,{\Lambda }_N\}$, where $d$ is the dimension of the state space.
• In 50, we consider the internal control of linear parabolic equations through on-off shape controls, i.e., controls of the form $M\left(t\right){\chi }_{\omega \left(t\right)}$ with $M\left(t\right)\ge 0$ and $\omega \left(t\right)$ with a prescribed maximal measure. We establish small-time approximate controllability towards all possible final states allowed by the comparison principle with nonnegative controls. We manage to build controls with constant amplitude $M\left(t\right)\equiv \overline{M}$. In contrast, if the moving control set $\omega \left(t\right)$ is confined to evolve in some region of the whole domain, we prove that approximate controllability fails to hold for small times. The method of proof is constructive. Using Fenchel-Rockafellar duality and the bathtub principle, the on-off shape control is obtained as the bang-bang solution of an optimal control problem, which we design by relaxing the constraints. Our optimal control approach is outlined in a rather general form for linear constrained control problems, paving the way for generalisations and applications to other PDEs and constraints.
• The goal of 21 is to obtain observability estimates for non-homogeneous elliptic equations in the presence of a potential, posed on a smooth bounded domain $\Omega$ in 2d and observed from a non-empty open subset $\omega \subset \Omega$. More precisely, for every real-valued bounded potential $V$, our main result shows that, when $\Omega$ has a finite number of holes, the observability constant of the elliptic operator $-\Delta +V$, with domain $H^2\cap H_0^1\left(\Omega \right)$, is of the form ${C\exp \left(C\right|V|}^{1/2}{log}^{1/2}\left(\right|V\left|\right))$ where $C$ is a positive constant depending only on $\Omega$ and $\omega$. Our methodology of proof is crucially based on the one recently developed by Logunov, Malinnikova, Nadirashvili, and Nazarov, in the context of the Landis conjecture on exponential decay of solutions to homogeneous elliptic equations in the plane. The main difference and additional difficulty is that the zero set of the solutions to elliptic equations with source term can be very intricate and should be dealt with carefully. As a consequence of these new observability estimates, we obtain new results concerning control of semi-linear elliptic equations in the spirit of Fernández-Cara, Zuazua's open problem concerning small-time global null-controllability of slightly super-linear heat equations.
• In 33, we consider a smooth system of the form $q^{\text{'}}=f_0\left(q\right)+{\sum }_{i=1}^k{u}_i{f}_i\left(q\right)$, $q\in M$, $u_i\in ℝ$, and study controllability issues on the group of diffeomorphisms of $M$. It is well-known that the system can arbitrarily well approximate the movement in the direction of any Lie bracket polynomial of $f_1,\cdots ,f_k$. Any Lie bracket polynomial of $f_1,\cdots ,f_k$ is good in this sense. Moreover, some combinations of Lie brackets which involve the drift term $f_0$ are also good but surely not all of them. In this paper we try to characterize good ones and, in particular, all universal good combinations, which are good for any nilpotent truncation of any system.
• In 12, we consider a mechanical system of three ants on the floor, which move according to two independt rules: Rule A - forces the velocity of any given ant to always point at a neighboring ant, and Rule B - forces the velocity of every ant to be parallel to the line defined by the two other ants. We observe that Rule A equips the 6-dimensional configuration space of the ants with a structure of a homogeneous (3,6) distribution, and that Rule B foliates this 6-dimensional configuration space onto 5-dimensional leaves, each of which is equiped with a homogeneous (2,3,5) distribution. The symmetry properties and Bryant-Cartan local invariants of these distributions are determined. In the case of Rule B we study and determine the singular trajectories (abnormal extremals) of the corresponding distributions. We show that these satisfy an interesting system of two ODEs of Fuchsian type.
• In the work 43, we present a general framework which guarantees the existence of optimal domains for isoperimetric problems within the class of $C^{1,1}$-regular domains satisfying a uniform ball condition as long as the desired objective function satisfies certain properties. We then verify that the helicity isoperimetric problem studied by Cantarella, DeTurck, Gluck and Teytel in 2002 satisfies the conditions of our framework and hence establish the existence of optimal domains within the given class of domains. We additionally use the same framework to prove the existence of optimal domains among uniform $C^{1,1}$-domains for a first curl eigenvalue problem which has been studied recently for other classes of domains.
• In 44, we investigate properties of the image and kernel of the Biot-Savart operator in the context of stellarator designs for plasma fusion. We first show that for any given coil winding surface (CWS) the image of the Biot-Savart operator is $L^2$-dense in the space of square-integrable harmonic fields defined on a plasma domain surrounded by the CWS. Then we show that harmonic fields which are harmonic in a proper neighbourhood of the underlying plasma domain can in fact be approximated in any $C^k$-norm by elements of the image of the Biot-Savart operator. In the second part of this work we establish an explicit isomorphism between the space of harmonic Neumann fields and the kernel of the Biot-Savart operator which in particular implies that the dimension of the kernel of the Biot-Savart operator coincides with the genus of the coil winding surface and hence turns out to be a homotopy invariant among regular domains in 3-space. Lastly, we provide an iterative scheme which we show converges weakly in $W^{-\frac{1}{2},2}$-topology to elements of the kernel of the Biot-Savart operator.
• In 37, considering a general nonlinear dissipative finite dimensional optimal control problem in fixed time horizon $T$, we establish a two-term asymptotic expansion of the value function as $T\to +\infty$. The dominating term is T times the optimal value obtained from the optimal static problem within the classical turnpike theory. The second term, of order unity, is interpreted as the sum of two values associated with optimal stabilization problems related to the turnpike.
• The turnpike phenomenon stipulates that the solution of an optimal control problem in large time, remains essentially close to a steady-state of the dynamics, itself being the optimal solution of an associated static optimal control problem. Under general assumptions, it is known that not only the optimal state and the optimal control, but also the adjoint state coming from the application of the Pontryagin maximum principle, are exponentially close to a steady-state, except at the beginning and at the end of the time frame. In such results, the turnpike set is a singleton, which is a steady-state. In the paper 30, we establish a turnpike result for finite-dimensional optimal control problems in which some of the coordinates evolve in a monotone way, and some others are partial steady-states of the dynamics. We prove that the discrepancy between the optimal trajectory and the turnpike set is then linear, but not exponential: we thus speak of a linear turnpike theorem.
• Consider, on the one part, a general nonlinear finite-dimensional optimal control problem and assume that it has a unique solution whose state is denoted by $x^{*}$. On the other part, consider the sampled-data control version of it. Under appropriate assumptions, in 41, we prove that the optimal state of the sampled-data problem converges uniformly to $x^{*}$ as the norm of the corresponding partition tends to zero. Moreover, applying the Pontryagin maximum principle to both problems, we prove that, if $x^{*}$ has a unique weak extremal lift with a costate $p$ that is normal, then the costate of the sampled-data problem converges uniformly to $p$. In other words, under a nondegeneracy assumption, control sampling commutes, at the limit of small partitions, with the application of the Pontryagin maximum principle.
• In this paper 34, we prove Morse index theorems for a big class of constrained variational problems on graphs. Such theorems are useful in various physical and geometric applications. Our formulas compute the difference of Morse indices of two Hessians related to two different graphs or two different sets of boundary conditions. Several applications such as the iteration formulas or lower bounds for the index are proved.
• The work 49 tackles the open pit planning problem in an optimal control framework. We study the optimality conditions for the so-called continuous formulation using Pontryagin’s Maximum Principle, and introduce a new, semi-continuous formulation that can handle the optimization of a two-dimensional mine profile. Numerical simulations are provided for several test cases, including global optimization for the one-dimensional final open pit, and first results for the two-dimensional sequential open pit. Theses indicate a good consistency between the different approaches, and with the theoretical optimality conditions.
• In the paper 14, we consider a measure-theoretical formulation of the training of NeurODEs in the form of a mean-field optimal control with $L^2$-regularization of the control. We derive first order optimality conditions for the NeurODE training problem in the form of a mean-field maximum principle, and show that it admits a unique control solution, which is Lipschitz continuous in time. As a consequence of this uniqueness property, the mean-field maximum principle also provides a strong quantitative generalization error for finite sample approximations. Our derivation of the mean-field maximum principle is much simpler than the ones currently available in the literature for mean-field optimal control problems, and is based on a generalized Lagrange multiplier theorem on convex sets of spaces of measures. The latter is also new, and can be considered as a result of independent interest.

6.4 Geometric models for vision and sub-Riemannian geometry: new results

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

• In the article 28, we study the observability (or, equivalently, the controllability) of some subelliptic evolution equations depending on their step. This sheds light on the speed of propagation of these equations, notably in the “degenerated directions” of the subelliptic structure. First, for any $\gamma \ge 1$, we establish a resolvent estimate for the Baouendi-Grushin-type operator ${\Delta }_{\gamma }={\partial }_x^2+{\left|x\right|}^{2\gamma }{\partial }_y^2$, which has step $\gamma +1$. We then derive consequences for the observability of the Schrodinger type equation $i{\partial }_{t}u-{(-{\Delta }_{\gamma })}^{s}u=0$ where $s\in \mathbb{N}$. We identify three different cases: depending on the value of the ratio $(\gamma +1)/s$, observability may hold in arbitrarily small time, or only for sufficiently large times, or even fail for any time. As a corollary of our resolvent estimate, we also obtain observability for heat-type equations ${\partial }_{t}u+{(-{\Delta }_{\gamma })}^{s}u=0$ and establish a decay rate for the damped wave equation associated with ${\Delta }_{\gamma }$.
• In 40, we consider the evolution of a free quantum particle on the Grushin cylinder, under different type of quantizations. In particular we are interested to understand if the particle can cross the singular set, i.e., the set where the structure is not Riemannian. We consider intrinsic and extrinsic quantizations, where the latter are obtained by embedding the Grushin structure isometrically in ${ℝ}^3$ (with singularities). As a byproduct we provide formulas to embed the Grushin cylinder in ${ℝ}^3$ that could be useful for other purposes. Such formulas are not global, but permit to study the embedding arbitrarily close to the singular set. We extend these results to the case of $\alpha$-Grushin cylinders.
• In 47, we establish two results concerning the Quantum Limits (QLs) of some sub-Laplacians. First, under a commutativity assumption on the vector fields involved in the definition of the sub- Laplacian, we prove that it is possible to split any QL into several pieces which can be studied separately, and which come from well-characterized parts of the associated sequence of eigenfunctions. Secondly, building upon this result, we study in detail the QLs of a particular family of sub-Laplacians defined on products of compact quotients of Heisenberg groups. We express the QLs through a disintegration of measure result which follows from a natural spectral decomposition of the sub-Laplacian in which harmonic oscillators appear. Both results are based on the construction of an adequate elliptic operator commuting with the sub-Laplacian, and on the associated joint spectral calculus. They illustrate the fact that, because of the possible high degeneracies in the spectrum, the spectral theory of sub-Laplacians is very rich.
• In 15, we consider surfaces embedded in a 3D contact sub-Riemannian manifold and the problem of the finiteness of the induced distance (i.e., the infimum of the length of horizontal curves that belong to the surface). Recently it has been proved that for a surface having the topology of a sphere embedded in a tight co-orientable structure, the distance is always finite. In this paper we study closed surfaces of genus larger than 1, proving that such surfaces can be embedded in such a way that the induced distance is finite or infinite. We then study the structural stability of the finiteness/not-finiteness of the distance.
• The relative heat content associated with a subset $\Omega \subset M$ of a sub-Riemannian manifold, is defined as the total amount of heat contained in $\Omega$ at time $t$, with uniform initial condition on Ω, allowing the heat to flow outside the domain. In the work 13, we obtain a fourth-order asymptotic expansion in square root of t of the relative heat content associated with relatively compact non-characteristic domains. Compared to the classical heat content that we studied in [Rizzi, Rossi - J. Math. Pur. Appl., 2021], several difficulties emerge due to the absence of Dirichlet conditions at the boundary of the domain. To overcome this lack of information, we combine a rough asymptotic for the temperature function at the boundary, coupled with stochastic completeness of the heat semi-group. Our technique applies to any (possibly rank-varying) sub-Riemannian manifold that is globally doubling and satisfies a global weak Poincaré inequality, including in particular sub-Riemannian structures on compact manifolds and Carnot groups.
• In 22, we study the isoperimetric problem for anisotropic left-invariant perimeter measures on ${ℝ}^3$, endowed with the Heisenberg group structure. The perimeter is associated with a left-invariant norm $\varphi$ on the horizontal distribution. We first prove a representation formula for the $\varphi$-perimeter of regular sets and, assuming some regularity on $\varphi$ and on its dual norm ${\varphi }_{*}$, we deduce a foliation property by sub-Finsler geodesics of $C^2$-smooth surfaces with constant $\varphi$-curvature. We then prove that the characteristic set of $C^2$-smooth surfaces that are locally extremal for the isoperimetric problem is made of isolated points and horizontal curves satisfying a suitable differential equation. Based on such a characterization, we characterize $C^2$-smooth $\varphi$-isoperimetric sets as the sub-Finsler analogue of Pansu's bubbles. We also show, under suitable regularity properties on $\varphi$, that such sub-Finsler candidate isoperimetric sets are indeed $C^2$-smooth. By an approximation procedure, we finally prove a conditional minimality property for the candidate solutions in the general case (including the case where $\varphi$ is crystalline).

7.1 Bilateral contracts with industry

Contract with MBDA (Palaiseau), 2021–2023. Subject: “Contrôle optimal pour la planification de trajectoires et l’estimation des ensembles accessibles". Pariticpants: V. Askovic (MBDA & CAGE), E. Trélat (coordinator).

7.2 Bilateral grants with industry

Grant by AFOSR (Air Force Office of Scientific Research), 2020–2023. Participants : Mohab Safey El Din (LIP6), E. Trélat.

8.1 International research visitors

8.2 National initiatives

8.3 Regional initiatives

The Bourse Emergence(s) de la Ville de Paris “Morphométrie sous contrainte pour l’analyse de données biologiques : un nouvel outil pour la communauté scientifique”, whose principal investigator is Barbara Gris, runs from 2022 to 2025.

9.1 Promoting scientific activities

9.2 Teaching - Supervision - Juries

10.1 Major publications

• 1 articleD.Davide Barilari, Y.Yacine Chitour, F.Frédéric Jean, D.Dario Prandi and M.Mario Sigalotti. On the regularity of abnormal minimizers for rank 2 sub-Riemannian structures.Journal de Mathématiques Pures et Appliquées1332020, 118-138HALDOI
• 2 articleR.Riccardo Bonalli, B.Bruno Hérissé and E.Emmanuel Trélat. Optimal Control of Endo-Atmospheric Launch Vehicle Systems: Geometric and Computational Issues.IEEE Transactions on Automatic Control6562020, 2418--2433HALDOI
• 3 miscY.Yves Colin de Verdìère, L.Luc Hillairet and E.Emmanuel Trélat. Spectral asymptotics for sub-Riemannian Laplacians.December 2022HAL
• 4 articleJ.-M.Jean-Michel Coron, A.Amaury Hayat, S.Shengquan Xiang and C.Christophe Zhang. Stabilization of the linearized water tank system.Archive for Rational Mechanics and Analysis24432022, 1019–1097HAL
• 5 articleJ.-M.Jean-Michel Coron, F.Frédéric Marbach and F.Franck Sueur. Small-time global exact controllability of the Navier-Stokes equation with Navier slip-with-friction boundary conditions.Journal of the European Mathematical Society225May 2020, 1625--1673HALDOI
• 6 articleJ.-M.Jean-Michel Coron and H.-M.Hoai-Minh Nguyen. Optimal time for the controllability of linear hyperbolic systems in one dimensional space.SIAM Journal on Control and Optimization572April 2019, 1127-1156HALDOI
• 7 articleS.Sylvain Ervedoza, K.Kévin Le Balc'H and M.Marius Tucsnak. Reachability results for perturbed heat equations.Journal of Functional Analysis28310November 2022HAL
• 8 articleM.Monika Leibscher, E.Eugenio Pozzoli, C.Cristobal Pérez, M.Melanie Schnell, M.Mario Sigalotti, U.Ugo Boscain and C. P.Christiane P. Koch. Full quantum control of enantiomer-selective state transfer in chiral molecules despite degeneracy.Communications PhysicsMay 2022HALDOI
• 9 articleO.Ozan Öktem, B.Barbara Gris and C.Chong Chen. Image reconstruction through metamorphosis.Inverse Problems362020HALDOI
• 10 articleY.Yannick Privat, R.Rémi Robin and M.Mario Sigalotti. Optimal shape of stellarators for magnetic confinement fusion.Journal de Mathématiques Pures et Appliquées2022HALDOI
• 11 articleR.Rémi Robin, N.Nicolas Augier, U.Ugo Boscain and M.Mario Sigalotti. Ensemble qubit controllability with a single control via adiabatic and rotating wave approximations.Journal of Differential Equations318May 2022HALDOI

10.2 Publications of the year

10.3 Cited publications

• 52 articleA.Andrei Agrachev, U.Ugo Boscain, J.-P.Jean-Paul Gauthier and F.Francesco Rossi. The intrinsic hypoelliptic Laplacian and its heat kernel on unimodular Lie groups.J. Funct. Anal.25682009, 2621--2655URL: https://doi.org/10.1016/j.jfa.2009.01.006DOIback to text
• 53 bookA. A.Andrei A. Agrachev and Y. L.Yuri L. Sachkov. Control theory from the geometric viewpoint.87Encyclopaedia of Mathematical SciencesControl Theory and Optimization, IISpringer-Verlag, Berlin2004, xiv+412URL: https://doi.org/10.1007/978-3-662-06404-7DOIback to text
• 54 bookL.Luigi Ambrosio and P.Paolo Tilli. Topics on analysis in metric spaces.25Oxford Lecture Series in Mathematics and its ApplicationsOxford University Press, Oxford2004, viii+133back to text
• 55 articleS.Sylvain Arguillère, E.Emmanuel Trélat, A.Alain Trouvé and L.Laurent Younes. Shape deformation analysis from the optimal control viewpoint.J. Math. Pures Appl. (9)10412015, 139--178URL: https://doi.org/10.1016/j.matpur.2015.02.004DOIback to text
• 56 articleT.Térence Bayen. Analytical parameterization of rotors and proof of a Goldberg conjecture by optimal control theory.SIAM J. Control Optim.4762008, 3007--3036DOIback to text
• 57 articleM.Michel Benaim, S.Stéphane Le Borgne, F.Florent Malrieu and P.-A.Pierre-André Zitt. Qualitative properties of certain piecewise deterministic Markov processes.Ann. Inst. Henri Poincaré Probab. Stat.5132015, 1040--1075URL: https://doi.org/10.1214/14-AIHP619DOIback to text
• 58 articleB.Bastien Berret, C.Christian Darlot, F.Frédéric Jean, T.Thierry Pozzo, C.Charalambos Papaxanthis and J. P.Jean Paul Gauthier. The inactivation principle: mathematical solutions minimizing the absolute work and biological implications for the planning of arm movements.PLoS Comput. Biol.4102008, e1000194, 25URL: https://doi.org/10.1371/journal.pcbi.1000194DOIback to text
• 59 bookA.A. Bonfiglioli, E.E. Lanconelli and F.F. Uguzzoni. Stratified Lie groups and potential theory for their sub-Laplacians.Springer Monographs in MathematicsSpringer, Berlin2007, xxvi+800back to text
• 60 bookB.Bernard Bonnard, L.Ludovic Faubourg and E.Emmanuel Trélat. Mécanique céleste et contrôle des véhicules spatiaux.51Mathématiques & Applications (Berlin) [Mathematics & Applications]Springer-Verlag, Berlin2006, xiv+276back to text
• 61 articleU.U. Boscain, R. A.R. A. Chertovskih, J. P.J. P. Gauthier and A. O.A. O. Remizov. Hypoelliptic diffusion and human vision: a semidiscrete new twist.SIAM J. Imaging Sci.722014, 669--695DOIback to text
• 62 articleU.Ugo Boscain, F.Francesca Chittaro, P.Paolo Mason and M.Mario Sigalotti. Adiabatic control of the Schroedinger equation via conical intersections of the eigenvalues.IEEE Trans. Automat. Control5782012, 1970--1983back to text
• 63 articleU.Ugo Boscain, J.Jean Duplaix, J.-P.Jean-Paul Gauthier and F.Francesco Rossi. Anthropomorphic image reconstruction via hypoelliptic diffusion.SIAM J. Control Optim.5032012, 1309--1336DOIback to textback to text
• 64 articleR. W.R. W. Brockett. System theory on group manifolds and coset spaces.SIAM J. Control101972, 265--284back to text
• 65 bookF.Francesco Bullo and A. D.Andrew D. Lewis. Geometric control of mechanical systems.49Texts in Applied MathematicsModeling, analysis, and design for simple mechanical control systemsSpringer-Verlag, New York2005, xxiv+726DOIback to text
• 66 articleT.Thomas Chambrion, P.Paolo Mason, M.Mario Sigalotti and U.Ugo Boscain. Controllability of the discrete-spectrum Schrödinger equation driven by an external field.Ann. Inst. H. Poincaré Anal. Non Linéaire2612009, 329--349URL: https://doi.org/10.1016/j.anihpc.2008.05.001DOIback to textback to text
• 67 articleG.G. Citti and A.A. Sarti. A cortical based model of perceptual completion in the roto-translation space.J. Math. Imaging Vision2432006, 307--326URL: http://dx.doi.org/10.1007/s10851-005-3630-2DOIback to text
• 68 articleF.Fritz Colonius and G.Guilherme Mazanti. Decay rates for stabilization of linear continuous-time systems with random switching.Math. Control Relat. Fields2019back to text
• 69 bookJ.-M.Jean-Michel Coron. Control and nonlinearity.136Mathematical Surveys and MonographsAmerican Mathematical Society, Providence, RI2007, xiv+426back to text
• 70 articleJ.-M.Jean-Michel Coron. Global asymptotic stabilization for controllable systems without drift.Math. Control Signals Systems531992, 295--312URL: https://doi.org/10.1007/BF01211563DOIback to text
• 71 inproceedingsJ.-M.Jean-Michel Coron. On the controllability of nonlinear partial differential equations.Proceedings of the International Congress of Mathematicians. Volume IHindustan Book Agency, New Delhi2010, 238--264back to text
• 72 bookD.Domenico D'Alessandro. Introduction to quantum control and dynamics.Chapman & Hall/CRC Applied Mathematics and Nonlinear Science SeriesChapman & Hall/CRC, Boca Raton, FL2008, xiv+343back to text
• 73 articleM.Michel Fliess, J.Jean Lévine, P.Philippe Martin and P.Pierre Rouchon. Flatness and defect of non-linear systems: introductory theory and examples.Internat. J. Control6161995, 1327--1361URL: https://doi.org/10.1080/00207179508921959DOIback to text
• 74 articleA.Alessio Franci and R.Rodolphe Sepulchre. A three-scale model of spatio-temporal bursting.SIAM J. Appl. Dyn. Syst.1542016, 2143--2175DOIback to text
• 75 articleS. ..S .J. Glaser, U.U. Boscain, T.T. Calarco, C. ..C .P. Koch, W.W. Köckenberger, R.R. Kosloff, I.I. Kuprov, B.B. Luy, S.S. Schirmer, T.T. Schulte-Herbrüggen, D.D. Sugny and F. ..F .K. Wilhelm. Training Schrödinger's cat: quantum optimal control. Strategic report on current status, visions and goals for research in Europe.European Physical Journal D692015, 279DOIback to text
• 76 bookD.D.H. Hubel and T.T.N. Wiesel. Brain and Visual Perception: The Story of a 25-Year Collaboration.OxfordOxford University Press2004back to text
• 77 bookV.Velimir Jurdjevic. Geometric control theory.52Cambridge Studies in Advanced MathematicsCambridge University Press, Cambridge1997, xviii+492back to text
• 78 articleV.Velimir Jurdjevic and H. J.Héctor J. Sussmann. Control systems on Lie groups.J. Differential Equations121972, 313--329URL: https://doi.org/10.1016/0022-0396(72)90035-6DOIback to text
• 79 articleF.Ferdinand Küsters and S.Stephan Trenn. Switch observability for switched linear systems.Automatica J. IFAC872018, 121--127URL: https://doi.org/10.1016/j.automatica.2017.09.024DOIback to text
• 80 articleZ.Z. Leghtas, A.A. Sarlette and P.P. Rouchon. Adiabatic passage and ensemble control of quantum systems.Journal of Physics B44152011back to text
• 81 bookD.Daniel Liberzon. Calculus of variations and optimal control theory.A concise introductionPrinceton University Press, Princeton, NJ2012, xviii+235back to text
• 82 bookD.Daniel Liberzon. Switching in systems and control.Systems & Control: Foundations & ApplicationsBirkhäuser Boston, Inc., Boston, MA2003, xiv+233URL: https://doi.org/10.1007/978-1-4612-0017-8DOIback to textback to text
• 83 articleW.Wensheng Liu. Averaging theorems for highly oscillatory differential equations and iterated Lie brackets.SIAM J. Control Optim.3561997, 1989--2020DOIback to text
• 84 articleL.L. Massoulié. Stability of distributed congestion control with heterogeneous feedback delays.IEEE Trans. Automat. Control476Special issue on systems and control methods for communication networks2002, 895--902URL: https://doi.org/10.1109/TAC.2002.1008356DOIback to text
• 85 bookR.Richard Montgomery. A tour of subriemannian geometries, their geodesics and applications.91Mathematical Surveys and MonographsAmerican Mathematical Society, Providence, RI2002, xx+259back to text
• 86 articleR. M.Richard M. Murray and S. S.S. Shankar Sastry. Nonholonomic motion planning: steering using sinusoids.IEEE Trans. Automat. Control3851993, 700--716URL: https://doi.org/10.1109/9.277235DOIback to text
• 87 articleG.G. Nenciu. On the adiabatic theorem of quantum mechanics.J. Phys. A1321980, L15--L18URL: http://stacks.iop.org/0305-4470/13/L15back to text
• 88 articleD.Diego Patino, M.Mihai Bâja, P.Pierre Riedinger, H.Hervé Cormerais, J.Jean Buisson and C.Claude Iung. Alternative control methods for DC-DC converters: an application to a four-level three-cell DC-DC converter.Internat. J. Robust Nonlinear Control21102011, 1112--1133URL: https://doi.org/10.1002/rnc.1651DOIback to text
• 89 bookJ.Jean Petitot. Neurogéomètrie de la vision. Modèles mathématiques et physiques des architectures fonctionnelles.Les Éditions de l'École Polythechnique2008back to text
• 90 articleJ.Jakob Ruess and J.John Lygeros. Moment-based methods for parameter inference and experiment design for stochastic biochemical reaction networks.ACM Trans. Model. Comput. Simul.2522015, Art. 8, 25URL: https://doi.org/10.1145/2688906DOIback to text
• 91 articleA.Alessandro Sarti, G.Giovanna Citti and J.Jean Petitot. The symplectic structure of the primary visual cortex.Biol. Cybernet.9812008, 33--48URL: http://dx.doi.org/10.1007/s00422-007-0194-9DOIback to text
• 92 bookA.Arjan van der Schaft and H.Hans Schumacher. An introduction to hybrid dynamical systems.251Lecture Notes in Control and Information SciencesSpringer-Verlag London, Ltd., London2000, xiv+174URL: https://doi.org/10.1007/BFb0109998DOIback to text
• 93 bookH.Heinz Schättler and U.Urszula Ledzewicz. Geometric optimal control.38Interdisciplinary Applied MathematicsTheory, methods and examplesSpringer, New York2012, xx+640URL: https://doi.org/10.1007/978-1-4614-3834-2DOIback to textback to text
• 94 bookH.Heinz Schättler and U.Urszula Ledzewicz. Optimal control for mathematical models of cancer therapies.42Interdisciplinary Applied MathematicsAn application of geometric methodsSpringer, New York2015, xix+496URL: https://doi.org/10.1007/978-1-4939-2972-6DOIback to text
• 95 articleS.Selim Solmaz, R.Robert Shorten, K.Kai Wulff and F.Fiacre Ó Cairbre. A design methodology for switched discrete time linear systems with applications to automotive roll dynamics control.Automatica J. IFAC4492008, 2358--2363URL: https://doi.org/10.1016/j.automatica.2008.01.014DOIback to text
• 96 incollectionE. D.E. D. Sontag. Input to state stability: basic concepts and results.Nonlinear and optimal control theory1932Lecture Notes in Math.Springer, Berlin2008, 163--220URL: https://doi.org/10.1007/978-3-540-77653-6_3back to text
• 97 articleZ.Zhendong Sun, S. S.S. S. Ge and T. H.T. H. Lee. Controllability and reachability criteria for switched linear systems.Automatica J. IFAC3852002, 775--786URL: https://doi.org/10.1016/S0005-1098(01)00267-9DOIback to text
• 98 bookZ.Zhendong Sun and S. S.Shuzhi Sam Ge. Stability theory of switched dynamical systems.Communications and Control Engineering SeriesSpringer, London2011, xx+253URL: https://doi.org/10.1007/978-0-85729-256-8DOIback to textback to text
• 99 bookS.S. Teufel. Adiabatic perturbation theory in quantum dynamics.1821Lecture Notes in MathematicsBerlinSpringer-Verlag2003, vi+236back to text
• 100 bookE.Emmanuel Trélat. Contrôle optimal.Mathématiques Concrètes. [Concrete Mathematics]Théorie & applications. [Theory and applications]Vuibert, Paris2005, vi+246back to textback to text
• 101 articleE.E. Trélat. Optimal control and applications to aerospace: some results and challenges.J. Optim. Theory Appl.15432012, 713--758URL: https://doi.org/10.1007/s10957-012-0050-5DOIback to text
• 102 bookM.Marius Tucsnak and G.George Weiss. Observation and control for operator semigroups.Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks]Birkhäuser Verlag, Basel2009, xii+483URL: https://doi.org/10.1007/978-3-7643-8994-9DOIback to text
• 103 inproceedingsG.Gabriel Turinici. On the controllability of bilinear quantum systems.Mathematical models and methods for ab initio Quantum Chemistry74Lecture Notes in ChemistrySpringer2000back to text
• 104 bookM.Marcelo Viana. Lectures on Lyapunov exponents.145Cambridge Studies in Advanced MathematicsCambridge University Press, Cambridge2014, xiv+202URL: https://doi.org/10.1017/CBO9781139976602DOIback to text
• 105 bookR.Richard Vinter. Optimal control.Systems & Control: Foundations & ApplicationsBirkhäuser Boston, Inc., Boston, MA2000, xviii+507back to text
• 106 inproceedingsD.DA Wisniacki, G.GE Murgida and P.PI Tamborenea. Quantum control using diabatic and adiabatic transitions.AIP Conference Proceedings9632AIP2007, 840--842back to text
• 107 articleL.L.P. Yatsenko, S.S. Guérin and H.H.R. Jauslin. Topology of adiabatic passage.Phys. Rev. A652002, 043407, 7back to text