We study controlability issues for the group of volume-preserving diffeomorphisms of the torus 𝕋^d for system ẋ=f(x)+u(t), where f is a fixed divergence free vector field on 𝕋^d and u(t) are constant vector fields which generate translations of the torus. Main results concern d equals two or three.

We study $L^1$-optimal stabilization of linear systems with finite and infinite horizons. Main results concern the existence, uniqueness and structure of optimal solutions, and the robustness of optimal cost.

We study the optimal control problem for a control-affine system, where we want to minimize the L^1 norm of the control. First, we show how Pontryagin Maximum Principle (PMP) applies to this problem and we divide the extremal trajectories into two categories: regular and singular extremals. Then, we obtain a strong generalized Legendre-Clebsch condition for singular extremals and we show that this condition together with the absence of conjugate points is sufficient to ensure local strong optimality. We provide also some geometric examples where we apply our results. Finally, we prove that generalized Legendre-Clebsch condition is necessary for optimality.

Through the use of sub-Riemannian metrics we provide quantitative estimates for the maximal tight neighbourhood of a Reeb orbit on a three-dimensional contact manifold. Under appropriate geometric conditions we show how to construct closed curves which are boundaries of overtwisted disks. We introduce the concept of contact Jacobi curve, and prove lower bounds of the so-called tightness radius (from a Reeb orbit) in terms of Schwarzian derivative bounds. We compare these results with the corresponding ones from Etnyre et al (2012 Invent. Math. 188 621-57; 2016 Trans. Am. Math. Soc. 368 7845-81), and we show that our estimates are sharp for classical model structures. We also prove similar, but non-sharp, estimates in terms of sub-Riemannian canonical curvature bounds. We apply our results to K-contact sub-Riemannian manifolds. In this setting, we prove a contact analogue of the celebrated Cartan-Hadamard theorem.

We explicitly compute the maximal Lyapunov exponent for a switched system on SL_2(ℝ) and the corresponding switching function which realizes the maximal exponent. This computation is reduced to the characterization of optimal trajectories for an optimal control problem on the Lie group.

There exist many examples of systems which have some symmetries, and which one may monitor with symmetry preserving controls. Since symmetries are preserved along the evolution, full controllability is not possible, and controllability has to be considered inside sets of states with same symmetries. We prove that generic systems with symmetries are controllable in this sense. This result has several applications, for instance: (i) generic controllability of particle systems when the kernel of interaction between particles plays the role of a mean-field control; (ii) generic controllability for families of vector fields on manifolds with boundary; (iii) universal interpolation for neural networks architectures with "generic" self attention-type layers - a type of layers ubiquitous in recent neural networks architectures, e.g., in the Transformers architecture. The tools we develop could help address various other questions of control of equivariant systems.

We study the small-time controllability problem on the Lie groups $SL_2(\mathbb{R})$ and $SL_2(\mathbb{R})\ltimes H_{d}(\mathbb{R})$ with Lie bracket methods (here $H_{d}(\mathbb{R})$ denotes the $(2d+1)$-dimensional real Heisenberg group). Then, using unitary representations of $SL_2(\mathbb{R})\ltimes H_{d}(\mathbb{R})$ on $L^2(\mathbb{R}^d,\mathbb{C})$ and $L^p(T^*\mathbb{R}^d,\mathbb{R}), p\in[1,\infty)$, we recover small-time approximate reachability properties of the Schr\"odinger PDE for the quantum harmonic oscillator, and find new small-time approximate reachability properties of the Liouville PDE for the classical harmonic oscillator.

We consider a smooth system of the form q̇=f_0(q)+∑ _i=1^ku_if_i(q) , q∈ M, u_i∈ℝ, and study controllability issues on the group DiffM . It is well-known that the system can arbitrarily well approximate the movement in the direction of any Lie bracket polynomial of f_1,… ,f_k . Any Lie bracket polynomial of f_1,… ,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.

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 $\Omega$, allowing the heat to flow outside the domain. In this work, 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 Poincar\'e inequality, including in particular sub-Riemannian structures on compact manifolds and Carnot groups.

In this paper 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. Some applications such as the iteration formulas and lower bounds for the index are proved.

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.

We study possibilities to control an ensemble (a parameterized family) of nonlinear control systems by a single parameter-independent control. Proceeding by Lie algebraic methods we establish genericity of exact controllability property for finite ensembles, prove sufficient approximate controllability condition for a model problem in R3, and provide a variant of RashevskyChow theorem for approximate controllability of control-linear ensembles.

Deep learning of the artificial neural networks (ANN) can be treated as a particular class of interpolation problems. The goal is to find a neural network whose input-output map approximates well the desired map on a finite or an infinite training set. Our idea consists of taking as an approximant the input-output map, which arises from a nonlinear continuous-time control system. In the limit such control system can be seen as a network with a continuum of layers, each one labelled by the time variable. The values of the controls at each instant of time are the parameters of the layer.

Given a rank-two sub-Riemannian structure ( M , Δ) and a point x 0 ∈ M , a singular curve is a critical point of the endpoint map F : γ ↦ γ (1) defined on the space of horizontal curves starting at x 0 . The typical least degenerate singular curves of these structures are called regular singular curves ; they are nice if their endpoint is not conjugate along γ . The main goal of this paper is to show that locally around a nice singular curve γ , once we choose a suitable topology on the control space we can find a normal form for the endpoint map, in which F writes essentially as a sum of a linear map and a quadratic form. This is a preparation for a forthcoming generalization of the Morse theory to rank-two sub-Riemannian structures.

In this paper which is closely related to the previous paper [3] we specify general theory developed there. We study the structure of Jacobi fields in the case of an analytic system and piecewise analytic control. Moreover, we consider only 1-dimensional control variations. Jacobi fields are piecewise analytic in this case but may have jump discontinuities. We derive ODEs that these fields satisfy on the intervals of regularity and study behavior of the fields in a neighbourhood of a singularity where the ODE becomes singular and the Jacobi fields may have jumps.

We study the controlled dynamics of the {\it ensembles of points} of a Riemannian manifold $M$. Parameterized ensemble of points of $M$ is the image of a continuous map $\gamma:\Theta \to M$, where $\Theta$ is a compact set of parameters. The dynamics of ensembles is defined by the action $\gamma(\theta) \mapsto P_t(\gamma(\theta))$ of the semigroup of diffeomorphisms $P_t:M \to M, \ t \in \mathbb{R}$, generated by the controlled equation $\dot{x}=f(x,u(t))$ on $M$. Therefore any control system on $M$ defines a control system on (generally infinite-dimensional) space $\mathcal{E}_\Theta(M)$ of the ensembles of points. We wish to establish criteria of controllability for such control systems. As in our previous work ([1]) we seek to adapt the Lie-algebraic approach of geometric control theory to the infinite-dimensional setting. We study the case of finite ensembles and prove genericity of exact controllability property for them. We also find sufficient approximate controllability criterion for continual ensembles and prove a result on motion planning in the space of flows on $M$. We discuss the relation of the obtained controllability criteria to various versions of Rashevsky-Chow theorem for finite- and infinite-dimensional manifolds.

Floer homology is a good example of homological invariants living in the infinite dimension. We suggest a way to construct this kind of invariants using only soft essentially finite-dimensional tools; no hard analysis or PDE is involved. This work is partially inspired by the M. Gromov's survey ``Soft and hard symplectic geometry'' at ICM-86.

In this paper which is closely related to the previous paper [3] we specify general theory developed there. We study the structure of Jacobi fields in the case of an analytic system and piecewise analytic control. Moreover, we consider only 1-dimensional control variations. Jacobi fields are piecewise analytic in this case but may have jump discontinuities. We derive ODEs that these fields satisfy on the intervals of regularity and study behavior of the fields in a neighbourhood of a singularity where the ODE becomes singular and the Jacobi fields may have jumps.

We establish a new extremal property of the classical Chebyshev polynomials in the context of best rank-one approximation of tensors. We also give some necessary conditions for a tensor to be a minimizer of the ratio of spectral and Frobenius norms.

In this paper, we analyse the optimality of broken Pontryagin extremal for an n-dimensional affine control system with a control parameter, taking values in a k-dimensional closed ball. We prove the optimality of broken normal extremals in many cases that include (but not are exhausted by) the cases of the involutive driftless part of the system for any n > k and of the contact driftless part for n = 3 and k = 2.