# Andrei AgrachevScuola Internazionale Superiore di Studi Avanzati di Trieste | SISSA · Mathematics

Andrei Agrachev

Professor

## About

220

Publications

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6,271

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Citations since 2017

## Publications

Publications (220)

We consider a mechanical system of three ants on the floor, in two situations. In the first situation ants move according to Rule A, which forces the velocity of any given ant to always point at a neighboring ant; in the second situation ants move according to Rule B, which forces the velocity of every ant to be parallel to the line defined by the...

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. Several applications su...

In this paper we discuss a general framework based on symplectic geometry for the study of second order conditions in constrained variational problems on curves. Using the notion of L-derivatives we construct Jacobi curves, which represent a generalisation of Jacobi fields from the classical calculus of variations, but which also works for non-smoo...

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 th...

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...

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 configura...

Given a rank-two sub-Riemannian structure $(M,\Delta)$ and a point $x_0\in M$, a singular curve is a critical point of the endpoint map $F:\gamma\mapsto\gamma(1)$ defined on the space of horizontal curves starting at $x_0$. The typical least degenerate singular curves of these structures are called \emph{regular singular curves}; they are \emph{nic...

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 discont...

We use a control-theoretic setting to model the process of training (deep learning) of Artificial Neural Networks (ANN), which are aimed at solving classification problems. A successful classifier is the network whose input-output map approximates well the classifying map defined on a finite or an infinite training set. A fruitful idea is substitut...

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.

Sub-Riemannian geometry is the geometry of a world with nonholonomic constraints. In such a world, one can move, send and receive information only in certain admissible directions but eventually can reach every position from any other. In the last two decades sub-Riemannian geometry has emerged as an independent research domain impacting on several...

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 semigrou...

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 when n = 3 and the controllable vector fields form a contact distribution, and when the Lie algebra of the contro...

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.

We study the variation of a smooth volume form along extremals of a variational problem with nonholonomic constraints and an action-like Lagrangian. We introduce a new invariant describing the interaction of the volume with the dynamics and we study its basic properties. We then show how this invariant, together with curvature-like invariants of th...

Вторая вариация для регулярной экстремали гладкой задачи оптимального управления есть симметричный фредгольмов оператор. Исследована асимптотика спектра этого оператора, и получено явное выражение его определителя через решения уравнения Якоби. В случае принципа наименьшего действия для гармонического осциллятора это дает классическое тождество Эйл...

Second variation of a smooth optimal control problem at a regular extremal is a symmetric Fredholm operator. We study the asymptotics of the spectrum of this operator and give an explicit expression for its determinant in terms of solutions of the Jacobi equation. In the case of the least action principle for the harmonic oscillator, we obtain the...

In this second paper of the series we specify general theory developed in the first paper. Here we study the structure of Jacobi fields in the case of an analytic system and piece-wise analytic control. Moreover, we consider only 1-dimensional control variations. Jacobi fields are piece-wise analytic in this case but may have much more singularitie...

Given a rank-two sub-Riemannian structure $(M,\Delta)$ and a point $x\in M$, a singular curve is a critical point of the endpoint map $F:\gamma\mapsto\gamma(1)$ defined on the space of horizontal curves starting at $x$. The typical least degenerate singular curves of these structures are often called nice singular curves; another name is "regular a...

In this paper we discuss a general framework based on symplectic geometry for the study of second order conditions in optimal control problems. Using the notion of $L$-derivatives we construct Jacobi curves, which represent a generalization of Jacobi fields from the classical calculus of variations. This construction includes in particular the prev...

The curvature discussed in this paper is a rather far going generalization of
the Riemannian sectional curvature. We define it for a wide class of optimal
control problems: a unified framework including geometric structures such as
Riemannian, sub-Riemannian, Finsler and sub-Finsler structures; a special
attention is paid to the sub-Riemannian (or...

Second variation of a smooth optimal control problem at a regular extremal is a symmetric Fredholm operator. We study asymptotics of the spectrum of this operator and give an explicit expression for its determinant in terms of solutions of the Jacobi equation.

In this paper we analyze local regularity of time-optimal controls and trajectories for an n-dimensional affine control system with a control parameter, taking values in a k-dimensional closed ball. In the case of k = n − 1, we give sufficient conditions in terms of Lie bracket relations for all optimal controls to be smooth or to have only isolate...

Given a smooth manifold $M$ and a totally nonholonomic distribution $\Delta\subset TM$ of rank $d$, we study the effect of singular curves on the topology of the space of horizontal paths joining two points on $M$. Singular curves are critical points of the endpoint map $F:\gamma\mapsto\gamma(1)$ defined on the space $\Omega$ of horizontal paths st...

We study local structure of time-optimal controls and trajectories for a 3-dimensional control-affine system with a 2-dimensional control parameter with values in the disk. In particular, we give sufficient conditions, in terms of Lie bracket relations, for optimal controls to be smooth or to have only isolated jump discontinuities.

These are the notes of rather informal lectures given by the first co-author in UPMC, Paris, in January 2017. Practical goal is to explain how to compute or estimate the Morse index of the second variation. Symplectic geometry allows to effectively do it even for very degenerate problems with complicated constraints. Main geometric and analytic too...

Sub-Riemannian geometry is the geometry of spaces with nonholonomic constraints. This paper presents an informal survey of some topics in this area, starting with the construction of geodesic curves and ending with a recent definition of curvature. © 2016 Russian Academy of Sciences (DoM), London Mathematical Society, Turpion Ltd.

https://www.ems-ph.org/books/show_abstract.php?proj_nr=206&vol=1&rank=1

In this paper we analyse local regularity of time-optimal controls and trajectories for an n-dimensional affine control system with a control parameter, taking values in a k-dimensional closed ball. In the case of k equal to n-1, we give sufficient conditions in terms of Lie bracket relations for all optimal controls to be smooth or to have only is...

Субриманова геометрия есть геометрия пространств с неголономными связями. В статье дан неформальный обзор некоторых тем этого предмета, начиная c построения геодезических линий и кончая недавним определением кривизны.

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 $\mathbb{R}^3...

We relate some basic constructions of stochastic analysis to differential
geometry, via random walk approximations. We consider walks on both Riemannian
and sub-Riemannian manifolds in which the steps consist of travel along either
geodesics or integral curves associated to orthonormal frames, and we give
particular attention to walks where the cho...

We study properties of the space of horizontal paths joining the origin with a vertical point on a generic two-step Carnot group. The energy is a Morse-Bott functional on paths and its critical points (sub-Riemannian geodesics) appear in families (compact critical manifolds) with controlled topology. We study the asymptotic of the number of critica...

We compare different notions of curvature on contact sub-Riemannian
manifolds. In particular we introduce canonical curvatures as the coefficients
of the sub-Riemannian Jacobi equation. The main result is that all these
coefficients are encoded in the asymptotic expansion of the horizontal
derivatives of the sub-Riemannian distance. We explicitly c...

We examine the existence of tangent hyperplanes to subriemannian balls.
Strictly abnormal shortest paths are allowed

We study the topology of admissible-loop spaces on a step-two Carnot group G.
We use a Morse-Bott theory argument to study the structure and the number of
geodesics on G connecting the origin with a 'vertical' point (geodesics are
critical points of the 'Energy' functional, defined on the loop space). These
geodesics typically appear in families (c...

Motivated by the study of linear quadratic optimal control problems, we
consider a dynamical system with a constant, quadratic Hamiltonian, and we
characterize the number of conjugate times in terms of the spectrum of the
Hamiltonian vector field $\vec{H}$. We prove the following dichotomy: the
number of conjugate times is identically zero or grows...

We discuss some challenging open problems in the geometric control theory and
sub-Riemannian geometry.

We study homological invariants of smooth families of real quadratic forms as
a step towards a "Lagrange multipliers rule in the large" that intends to
describe topology of smooth vector functions in terms of scalar Lagrange
functions.

We study the Monge's optimal transportation prob-lem where the cost is given by optimal control cost. We prove the existence and uniqueness of optimal map under certain regularity conditions on the Lagrangian, absolute continuity of the measures and most importantly the absent of sharp abnormal minimizers. In particular, this result is applicable i...

We present a spectral sequence which efficiently computes Betti numbers of a
closed semi-algebraic subset of RP^n defined by a system of quadratic
inequalities and the image of the homology homomorphism induced by the
inclusion of this subset in RP^n. We do not restrict ourselves to the term E_2
of the spectral sequence and give a simple explicit f...

The prominent Russian mathematician Vladimir Mikhailovich Zakalyukin died suddenly in Moscow on 30 December 2011.

For a regular sub-Riemannian manifold we study the Radon-Nikodym derivative
of the spherical Hausdorff measure with respect to a smooth volume. We prove
that this is the volume of the unit ball in the nilpotent approximation and it
is always a continuous function. We then prove that up to dimension 4 it is
smooth, while starting from dimension 5, i...

This paper presents new sufficient conditions for exponential stability of switched linear systems under arbitrary switching, which involve the commutators (Lie brackets) among the given matrices generating the switched system. The main novel feature of these stability criteria is that, unlike their earlier counterparts, they are robust with respec...

We give a complete classification of left-invariant sub-Riemannian structures
on three dimensional Lie groups in terms of the basic differential invariants.
As a corollary we explicitly find a sub-Riemannian isometry between the
nonisomorphic Lie groups SL(2) and $A^+(\R)\times S^1$, where $A^+(\R)$ denotes
the group of orientation preserving affin...

We study homological structure of the filtrations of the spaces of
self-adjoint operators by the multiplicity of the ground state. We consider
only operators acting in a finite dimensional complex or real Hilbert space but
infinite dimensional generalizations are easily guessed.

An invariant formulation of the maximum principle in optimal control is presented, and some second-order invariants are discussed.

We prove a Bishop volume comparison theorem and a Laplacian comparison
theorem for three dimensional contact subriemannian manifolds with symmetry.

We study solutions of modified Hamilton-Jacobi equations H(du/dq,q) + cu(q) = 0, q \in M, on a compact manifold M .

We study homological structure of the filtration of the space of self-adjoint operators by the multiplicity of the ground state. We consider only operators acting in a finite dimensional complex or real Hilbert space but infinite dimensional generalizations are easily guessed.

This paper presents new sufficient conditions for exponential stability of switched linear systems under arbitrary switching, which involve the commutators (Lie brackets) among the given matrices generating the switched system. The main novel feature of these stability criteria is that, unlike their earlier counterparts, they are robust with respec...

We prove continuity of certain cost functions arising from optimal control of affine control systems. We give sharp sufficient
conditions for this continuity. As an application, we prove a version of weak KAM theorem and consider the Aubry–Mather problems
corresponding to these systems.
Mathematics Subject Classification (2000)35F21-49L25

We consider a smooth bracket-generating control-affine system in
\mathbbRd {\mathbb{R}^d} and show that any orientation-preserving diffeomorphism of
\mathbbRd {\mathbb{R}^d} can be approximated, in a very strong sense, by a diffeomorphism included in the flow generated by a time-varying feedback
control which is polynomial with respect to the s...

Given a compact manifold M, we prove that any bracket generating family of vector fields on M, which is invariant under multiplication by smooth functions, generates the connected component of the identity of the group of diffeomorphisms of M.RésuméSoit M une variété compacte, nous montrons que toute famille de champs de vecteurs satisfaisant la co...

Two-dimensional almost-Riemannian structures are generalized Riemannian structures on surfaces for which a local orthonormal frame is given by a Lie bracket generating pair of vector ?elds that can become collinear. We study the relation between the topological invariants of an almost-Riemannian structure on a compact oriented surface and the rank-...

We give an effective sufficient condition for a variational problem with infinite horizon on a compact Riemannian manifold
M to admit a smooth optimal synthesis, i.e., a smooth dynamical system on M whose positive semi-trajectories are solutions to the problem. To realize the synthesis, we construct an invariant Lagrangian
submanifold (well-project...

We present an invariant definition of the hypoelliptic Laplacian on sub-Riemannian structures with constant growth vector using the Popp's volume form introduced by Montgomery. This definition generalizes the one of the Laplace–Beltrami operator in Riemannian geometry. In the case of left-invariant problems on unimodular Lie groups we prove that it...

Measure contraction property is one of the possible generalizations of Ricci
curvature bound to more general metric measure spaces. In this paper, we
discover sufficient conditions for a three dimensional contact subriemannian
manifold to satisfy this property.

We prove the result stated in the title that is equivalent to the existence of a regular point of the sub-Riemannian exponential
mapping. In the case of a complete real-analytic sub-Riemannian manifold, we prove that the metric is analytic on an open
everywhere dense subset.

We study Hamiltonian systems which generate extremal flows of regular
variational problems on smooth manifolds and demonstrate that
negativity of the generalized curvature of such a system implies
the existence of a global smooth optimal synthesis for the infinite
horizon problem.
We also show that in the Euclidean case negativity of the generalize...

These notes are based on the mini-course given in June 2004 in Cetraro, Italy, in the frame of a C.I.M.E. school. Of course, they contain much more material that I could present in the 6 h course. The idea was to explain a general variational and dynamical nature of nice and powerful concepts and results mainly known in the narrow framework of Riem...

Given a compact manifold M, we prove that any bracket generating and invariant under multiplication on smooth functions family of vector fields on M generates the connected component of unit of the group Diff(M).

Solutions of any optimal control problem are described by trajectories of a Hamiltonian system. The system is intrinsically
associated to the problem by a procedure that is a geometric elaboration of the Lagrange multipliers rule. The intimate relation
of the optimal control and Hamiltonian dynamics is fruitful for both domains; among other things,...

We study the existence and the structure of smooth optimal synthesis for regular variational problems with infinite horizon. To do that we investigate the asymptotic behavior of the flows generated by the extremals (of finite horizon problems) using curvature-type invariants of the flows and some methods of hyperbolic dynamics.

Jacobi curves are far going generalizations of the spaces of “Jacobi fields” along Riemannian geodesics. Actually, Jacobi
curves are curves in the Lagrange Grassmannians. Differential geometry of these curves provides basic feedback or gauge invariants
for a wide class of smooth control systems and geometric structures. In the present paper we main...

We study Monge's optimal transportation problem, where the cost is given by optimal control cost. We prove the existence and uniqueness of an optimal map under certain regularity conditions on the Lagrangian, absolute continuity of the measures with respect to Lebesgue, and most importantly the absence of sharp abnormal minimizers. In particular, t...

A method is presented for reducing a smooth system linear in the control on an n-dimensional manifold M to a nonlinear system on an
(n – 1)-dimensional manifold. This reduction is used to obtain sufficient conditions for a high order of local controllability of the system, and the problem of a time-optimal control of the angular momentum of a rota...

With the help of a symplectic technique the concept of a field of extremals in the classical calculus of variations is generalized to optimal control problems. This enables us to get new optimality conditions that are equally suitable for regular, bang-bang, and singular extremals. Special attention is given to systems of the form with a scalar con...

In this paper, a general necessary optimality condition of second order is proved for a time-optimal problem. Necessary optimality conditions of second order are applied, in general, in studying singular optimal regimes which can not be found with the aid of the first-order necessary condition for optimality, i.e. with the aid of the Pontrjagin max...

The level set M of a smooth mapping F in a neighborhood of an anormal point is investigated. Concepts of 2-regularity are introduced for it. It is proved that if the mapping is 2-regular at the point under consideration, then in a neighborhood of it the set M is locally diffeomorphic to the set of zeros of the second differential of F.

In this paper a second-order optimality condition is obtained for the general nonlinear problem with arbitrary boundary conditions and an integral-type minimized functional. Moreover, a second-order sufficient condition for controllability is derived.
Bibliography: 3 titles.

In this semi-expository paper we disclose hidden symmetries of a classical nonholonomic kinematic model and try to explain
the geometric meaning of the basic invariants of vector distributions.

The paper is devoted to studying the image of probability measures on a Hilbert space under finite-dimensional analytic maps. We establish sufficient conditions under which the image of a measure has a density with respect to the Lebesgue measure and continuously depends on the map. The results obtained are applied to the 2D Navier–Stokes equations...

Curvature-type invariants of Hamiltonian systems generalize sectional curvatures of Riemannian manifolds: the negativity of
the curvature is an indicator of the hyperbolic behavior of the Hamiltonian flow. In this paper, we give a self-contained
description of the related constructions and facts; they lead to a natural extension of the classical re...

We survey results of recent activity towards studying controllability and accessibility issues for equations of dynamics of incompressible fluids controlled by low-dimensional or, degenerate, forcing. New results concerning controllability of Navier-Stokes/Euler systems on two-dimensional sphere and on a generic two-dimensional domain are represent...

We study a randomization of the standard finite dimensional optimal control problem: we just assume that boundary values of the trajectory are not fixed but have some probability distributions and try to minimize the expected cost. This is actually a control version of the “optimal mass transportation”. We are busy with the existence, uniqueness an...

In: Subelliptic PDE's and applications to geometry and finance, vol. 6 of Lect. Notes Semin. Interdiscip. Mat., Semin. Interdiscip. Mat.(S.I.M.), Potenza

We consider a generalization of Riemannian geometry that naturally arises in the framework of control theory. Let $X$ and $Y$ be two smooth vector fields on a two-dimensional manifold $M$. If $X$ and $Y$ are everywhere linearly independent, then they define a classical Riemannian metric on $M$ (the metric for which they are orthonormal) and they gi...

We study controllability issues for the 2D Euler and Navier-Stokes (NS) systems under periodic boundary conditions. These systems describe the motion of the homogeneous ideal or viscous incompressible fluid on a two-dimensional torus
\mathbbT2\mathbb{T}^2. We assume the system to be controlled by a degenerate forcing applied to a fixed number of m...

The function of ATP-activated P2X3 receptors involved in pain sensation is modulated by desensitization, a phenomenon poorly understood. The present study used patch-clamp recording from cultured rat or mouse sensory neurons and kinetic modeling to clarify the properties of P2X3 receptor desensitization. Two types of desensitization were observed,...

Geometric control theory and Riemannian techniques are used to describe
the reachable set at time t of left invariant single-input control systems
on semi-simple compact Lie groups and to
estimate the minimal time needed to reach any point from identity.
This method provides an effective way to give an upper and a lower bound
for the minimal time n...

As was shown recently by P. Nurowski, to any rank 2 maximally nonholonomic vector distribution on a 5-dimensional manifold M one can assign the canonical conformal structure of signature (3,2). His construction is based on the properties of the special 12-dimensional coframe bundle over M, which was distinguished by E. Cartan during his famous cons...

In this paper, we study basic differential invariants of the pair (vector field, foliation). As a result, we establish a dynamic
interpretation and a generalization of the Levi-Civita connection and Riemannian curvature treated as invariants of the geodesic
flow on the tangent bundle.

As was shown recently by P. Nurowski, to any rank 2 maximally
nonholonomic vector distribution on a 5-dimensional manifold M one can
assign the canonical conformal structure of signature (3,2). His
construction is based on the properties of the special 12-dimensional
coframe bundle over M, which was distinguished by E. Cartan during his
famous cons...

The paper is devoted to studying the image of probability measures on a
Hilbert space under finite-dimensional analytic maps. We establish sufficient
conditions under which the image of a measure has a density with respect to the
Lebesgue measure and continuously depends on the map. The results obtained are
applied to the 2D Navier--Stokes equation...