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Seminars 1999-2000

ODEs Systems of the Jacobi-Sturm type

The Amann Theorem and some application to periodic systems of differential equations

Higher order elliptic operators in conformal geometry

On Gauss Bonnet formula for 4 manifolds

The p-system with frictional damping

The low order dissipative approximations for hyperbolic conservation laws attracted much interest during recent years. We consider the p-system with frictional damping, in Lagrange coordinates. Some results obtained recently will be presented.

On the Leray-Schauder Principle

Delta-Wave and Pressureless Gases

As a new hyperbolic phenomena, delta-wave was discovered recently. Due to its singularity, standard theory of hyperbolic conservation laws fails here. Studying it may give a good understanding of the theory of hyperbolic systems. Pressureless gases is such an important model involving the new phenomena. Its well posedness has been attracted increasing attention.

Stability of switched systems on the plane

We study the stability of the origin of the dynamical system: (1) \dot x(t)=u(t)Ax(t)+(1-u(t))Bx(t), where A and B are two 2 x 2 real matrices with eigenvalues having strictly negative real part, x \in R^2 and u(.) : [0,\infty[ \to [0,1] is an arbitrary measurable control function. In particular we find a necessary and sufficient condition on A and B under which the origin of the system (1) is asymptotically stable for each function u(.). The result is obtained studying the locus in which the two vector fields Ax and Bx are collinear. There are only three relevant parameter: the first depends on the eigenvalues of A, the second on the eigenvalues of B, the third contains the interrelation between the two systems and it is the cross ratio of the four points in the projective line CP^1 that correspond to the four eigenvectors of A and B. In the space of this parameters we study the shape and the convexity of the stability domain. This bidimensional problem assumes particular interest since systems of higher dimension can be reduced to our situation.

Parabolic Problems with Nonlocal Sources

Several parabolic with multiplicative nonlocal source terms will be informally derived motivated by desire to obtain physically realistic models of shearing. We then will give a limited survey of known results for nonlocal parabolic problems.

Stime a priori per soluzioni di problemi semilineari ellittici.

Sunto: Si presenteranno varie tecniche per ottenere stime a priori di soluzioni di problem semilineari ellittici. Queste stime sono necessarie per provare risultati di esistenza quando la struttura variazionale del problema e' assente.

Euler angles and time-optimal factorizations in SO(3)

We consider the following problem: given two generators X, Y of the Lie algebra so(3) and a rotation g \in SO(3) we are looking for a factorization g=exp(t_1 A_1) ... exp(t_k A_k), with k \in N, t_i \in R, A_i \in {X,Y}, such that |t_1| + ... + |t_k| is minimal. If X,Y are perpendicular (i.e., the corresponding axes are perpendicular), then it is well known that every g \in SO(3) can be written as a product of at most three factors, the parameters t_1, t_2, t_3 being nothing but Euler angles. This raises the following questions: (1) Assume that X and Y are perpendicular and normalized. Do Euler angles provide optimal factorizations? (2) If not, which factorizations are optimal and how many factors are needed? (3) What is the situation like for SU(2)? (4) What happens if one considers generators which are not perpendicular to each other? Considering the factorization problem as a control problem on a Lie group we will obtain detailed answers to (1)--(3). In case of (4) we obtain results suggesting a quite natural conjecture.

Nonlocal problems modelling shear banding

The formation of shear bands in metals has important implications to a variety of technological processes. These bands are observed in very thin zones and are generally regarded as a precuror to material failure. Shear band formation is caused by the heat generated in regions with highest strain rate. With insufficient time for diffusion of this heat, a localized thermal softening of the metal occurs which enhances plastic flow in a thin zone. This adiabatic strain localization can be modelled as nonlinear thermally activated reaction-diffusion equations. This leads to a class of nonlocal parabolic problems and their associated steady states. An analysis of these problems will be presented.

Quantized Systems

The aim of this talk is to introduce a new class of control systems called quantized systems. These are precisely discrete systems with a finite number of controls. Explicit application examples motivates the interest in these systems. We will provide some basic definitions and some results for the linear case and for some other particular cases. Moreover, open research problems will be discussed.

Controllability of solvable Lie algebras

Let G be a real connected Lie group and L its Lie algebra (the set of all right-invariant vector fields on G). For right-invariant control systems on G (1) dx/dt = A(x) + u B(x), x \in G, u \in R, where A, B \in L, we are interested in the (global) cotrollability property: when any pair of points in G can be connected by a trajectory of system (1)? This is equivalent to the following property: the subsemigroup of G generated by the set exp(A + R B) coincides with the Lie group G. It turns out that for solvable simply connected Lie groups G the property of controllability of systems (1) depends primarily not on vector fields A, B, but on the Lie group G, i.e., on the Lie algebra L. We call a Lie algebra L controllable if it contains elements A, B such that the corresponding system (1) is controllable. The talk will be devoted to the following questions: Description of controllable solvable Lie algebras (a recent work of D.Mittenhuber), A complete list of low-dimensional controllable Lie algebras (up to dim 6).

Singular Lagrangian manifolds in Sub-Riemannian geometry

The construction of the ball of small radius in Sub-Riemannian geometry shows the importance of Lagrangian stratifications when there exists abnormal geodesics. We discuss this stratification using the example of Martinet SR geometry. We show that such a stratification is also related to the computations of limit cycles in 16-th Hilbert problem.

Car towing several trailers and Goursat distributions

In the first half of the 90's, through the work of Laumond, Sordalen, Risler, Jean and others it had become clear that the kinematic model of a car drawing a given number l of attached trailers represents a rank--2 distribution D (on the configuration space \Sigma = R^2 \times (S^1)^{l+1}) having the property that the Lie square [D, D] of D is a rank--3 distribution, the Lie square of [D, D] is a rank--4 distribution, and so on. Distributions satisfying this condition are called Goursat. Every Goursat distribution around a generic point behaves (locally) in a unique way best visualised by chained model known in control theory. For trailer systems this behaviour happens around positions where NONE of the angles between: the car and 1st trailer, the 1st and 2nd trailer, ... , the (l-2)th and (l-1)th, is right. All other positions in \Sigma are singular. I will speak about: 1) a fundamental stratification of \Sigma by Jean '96 separating different singular behaviours of the system and drawing attention to specific values of the angles. This includes the computation of the growth vector at a point, hence -- also the nonholonomy degree of any given position of the system. 2) how Jean's classification prompts a basic geometric classification (if only rough; the local classification problem for Goursat objects is a lot more hard) of singularities of Goursat objects. 3) whether the car with l trailers is a universal model for local behaviours of Goursat distributions on manifolds of dimension l+3 (i.e., whether any Goursat germ is equivalent to the kinematic model at certain point of \Sigma). For l \le 5 the affirmative answer was known to me in '97. Pasillas--Lepine informed me in '98 that a general answer was also yes. In '99 this result was elegantly shown by Montgomery & Zhitomirskii in their big work on Goursat. It is a byproduct of performing -- over Goursat distributions -- so-called Cartan's prolongations as described by Bryant & Hsu in their '93 paper. I will recall that construction and explain the resulting universality of the trailer systems. 4) [time permitting; briefly and only giving references] which evolutions of the trailer systems are locally C^1--rigid.

Optimal Existence Theorems for Nonhomogeneous Differential Inclusions

In this paper we address the question of solvability of the differential inclusions Du(.) \in K(.,u(.)), u is Lischitz, u = f in the boundary, where Df(.) \in F(.,f(.)) a.e. in the domain and F is a relevant multi-valued function. Our approach to these problems is based on the idea to construct a sequence of approximate solutions which converges strongly and makes use of Gromov's idea (following earlier work of Nash and Kuiper) to control convergence of gradients by appropriate selection of the elements of the sequence. In this paper we identify an optimal setting of this method. We discuss some applications of the result: Hamilton-Jacobi systems of equations, convex multi-valued mappings F (here we identify the minimal sets K to solve all inclusions), and a nonhomogeneous case of phase transitions K(.) = SO(2)A(.) U SO(2)B(.).

Euler-Lagrange and Hamiltonian formalisms in variational problems involving differential inclusions

Analysis of variational problems involving differential inclusions dx/dt \in F(t, x) is inevitably connected with nonsmoothness unless the set-valued mapping F admits a smooth parametrization. Still it appears possible to develop a theory of necessary conditions for such problems containing the corresponding classical results and, moreover, extending the Hamiltonian formalism of calculus of variations to such problems. In the talk we mainly discuss the following questions: (a) what is the analogue of the Euler-Lagrange equation for variational problems with differential inclusions? (b) what is the analogue of the Hamiltonian necessary condition for such problems? (c) how Euler-Lagrange and Hamiltonian conditions are connected? We shall also discuss (if time permits) extensions for differential inclusions involving partial differential operators and/or some problems which remain open.

Stabilization of Control Systems by Discontinuous Feedback Laws

We are concerned with the problem of constructing discontinuous state feedbacks that asymptotically stabilize a nonlinear control system. We will first review the basic theory developed in the past few years. Next, we shall introduce a family of discontinuous, piecewise smooth vector fields and a class of ``patchy feedbacks'' which are obtained by patching together a locally finite family of smooth controls. Relying on the main properties of this class of feedback laws we will show that, if a system is asymptotically controllable at the origin, then it can be stabilized by a piecewise constant patchy feedback control. Robustness properties of such feedback with respect to external and internal perturbations will be also investigated.

Smooth Geometric Evolutions of Curves

In these lectures we consider curves evolving in the plane by the gradient flow of some geometric functionals, with particular attention to the problem of regularity. In a first part we discuss briefly the motion of a regular planar curve by mean curvature and we will analyze the cases where smoothness is preserved during the flow. Then we will study the works of Polden about evolution of curves by the gradient of functionals involving the curvature: we will discuss regularity, convergence and classification of the limit curves. If there will be time, we will present some extensions of these results, applications, open problems and research directions.

Observability, Observers and Dynamic Output Stabilization

Given a nonlinear system, we have already studied (previous lecture) the observable cases where the state->output map is regular (they are of two types, according to the fact that the number of observations is larger or smaller than the number of controls). Now, we study the case where this state->output map is degenerate. As in the theory of differentiable functions, it looks that the right notion of a degenerate mapping is related to the notion of a "finite mapping". In all situations (the degenerate and the non degenerate one), we can construct nonlinear asymptotic observers, with prescribed arbitrary exponential convergence, and we can use them to stabilise asymptotically the system via dynamic output feedback.

Complex Hamiltonians, Tops and Elastic Curves

The class of curves in the plane that L. Euler called elasticae in 1744 admit natural generalizations to arbitrary Riemannian manifolds M as the projections of solution curves of certain variational problems on the ortonormal frame bundle of M. For manifolds in which the frame bundle coincides with the isometry group, the elastic curves show remarkable connections with the movements of multi-dimensional mechanical tops. These, somewhat mysterious connections between tops, elastic curves and problems of optimal control constitute a major theme of the talk.

Pressurized shape memory thin films

While in bulk shape memory materials interfaces separate austenite and a very fine mixture of variants of martensite, in thin films the formation of a wide variety of exact austenite-martensite interfaces is possible. These are used to construct some deformations, tents and tunnels, which are energy minimizing and could be used as the basis of simple actuators. In the proposed model, a deformable film subject to hydrostatic pressure on its lower surface is considered. Following an approach first introduced by Bhattacharya and James, a direct derivation of a theory of a very thin film from three dimensional nonlinear elasticity is given. The theory is then specialized to a film in the form of a long strip made of a material which can undergo a martensitic phase transformation. It is found that, if the pressure is less than a temperature-dependent critical value, the film bulges up in a tunnel-like configuration and the resulting microstructure is a fine mixture of austenite and martensite. If the pressure is increased above the critical value, only martensite is present. In both cases, the cross-section of the tunnel takes the form of a circular arch.

Optimization techniques applied to route tracking and multiagent coordinated planning

We consider two problems in planning and controlling a class of vehicles along planar routes. The vehicle is supposed to move forward only with a given velocity profile, and to have bounds on its turning radius. This model, sometimes referred to as the ``Dubins' vehicle'', is relevant to the kinematics of road vehicles as well as aircraft cruising at constant altitude, or sea vessels. We consider first the optimal control problem consisting in minimizing the length travelled by the vehicle starting from a generic configuration to connect to the specified route. A feedback law is proposed, such that straight routes can be approached optimally, while system is asymptotically stabilized. A generalization to curved paths is also described. In the second part, we will consider the problem of optimal coordinated conflict management scheme for a simplified model of traffic, where several Dubins' vehicles share their operational space with possibly conflicting requirements on minimization of path length. A technique to minimize total fuel consumption while guaranteeing safety against collisions is described, in the hypothesis of full cooperation among agents. A decentralized implementation of such a scheme is then introduced, and its features as a hybrid control scheme are described. Finally, we discuss the tradeoff between performance and fault tolerance that goes with decentralization, and assess it by extensive simulation trials. Links with work at U.C. Berkeley on optimization under a non--cooperative assumption will be considered.

On high-order averaging and stability of time-varying systems

We study the questions of stability and asymptotic stability for time-varying systems described by ODE's with the right-hand sides of the form $f(\omega t, x)$, where $f(t,x)$ is 1-periodic with respect to $t$. The systems of this type draw much attention for a number of reasons. Since the discovery of stabilizing effect of vibration in the reverse pendulum example, there was much study regarding utilization of time varying feedback laws for stabilization of the systems which are not stabilizable by time-invariant feedback. Systems 'with a deficit of control' or control systems for nonholonomic mechanical objects (e.g. mobile robots) are natural examples of this kind. Many questions related to the issue of stability are not yet answered even for the linear time-varying systems. Two main approaches to the stability issue - the one based on Lyapunov functions and another one based on averaging - both encounter difficulties when dealing with the problem of fast oscillations. The Lyapunov functions must be very complicated in order to decrease along the trajectories generated by fast-oscillating controls while the standard averaging procedures result in systems with critical equilibria. Our approach is kind of high-order averaging procedure which is based and makes use of the tools of chronological calculus developed by A.A.Agrachev and R.V.Gamkrelidze in 70's-80's. The method is invariant - the high-order averagings are computed via Lie algebraic brackets of vector fields (right-hand sides). We apply the high-order averaging to study stability issues for both linear and nonlinear systems. In particular we derive conditions of stability for the second and third order linear differential equations with periodic fast-oscillating coefficients, study output-feedback stabilization of bilinear systems, consider averaging procedures for nonlinear systems under homogeneity assumptions, study the problem of stabilization of nonholonomic (control-linear) systems by means of time-varying feedbacks.

Projections Singularities of Extremals for Planar Systems

For a generic single input planar control system \dot x=F(x)+uG(x) (x belonging to R^2, F,G \in {\cal C}^3( R^2,R^2), F(0)=0, |u| <= 1) we are concerning with the problem of stabilizing the origin in minimum time. We compute the extremal synthesis in the cotangent bundle and we classifies the singularities of its projection on R^2. Moreover we show that the extremal locus is a stratified set in R^4 and we give a condition under which it is in fact a piecewise--{\cal C}^1 manifold with boundary. The most delicate trajectories are the Abnormal Extremals (i.e. the trajectories such that the corresponding PMP--Hamiltonian vanishes). We show that these trajectories can be classified through "words" belonging to a language recognizable by an Automaton.

Symplectic invariants for problems with nonholonomic constrains

In this talk I want to report about the work which I do now with Prof A. Agrachev. Using Hamiltonian approach, one can reduce the problem of finding feedback invariants of smooth control systems to the problem of finding symplectic invariants of curve in Lagrangian Grassmannian (the set of all Lagrangian subspaces of some symplectic space). For the problems with nonholonomic constrains, a special class of curves in Lagrangian Grassmannian is obtained in this way. We develope the differential geometry of these curves for $4$-dimensional symplectic space ( the first nontrivial case): we introduce canonical moving frame, find its structural equation, and obtain complete system of invariants (so-called, metric and projective curvature). One can apply the theory for developing differential geometry of some concrete problems with nonholonomic constrains. As an example we take contact structure of $3$- dimensional manifold provided with sub-Riemannian metric and find connection and curvature tensor for this problem. We are also able to write the equation for conjugate points along the extremals of this problem and to obtain comparison theorems.

Nonholonomic Motion Planning in Mobile Robotics

Nonholonomic motion planning for mobile robots requires the combination of tools from both computational geometry (collision checking, configuration space searching) and control theory (steering methods). The talk will put emphasis on criticial issues allowing such a combination: -- Controllabilities: We first present the modeling of several types of mobile robots (two-driving wheels mobile robots, car-like robots, robots with trailers). Then we show that the decision part of the motion planning problem (existence of a collision-free admissible path) is solved for small-time controllable systems, while it remains open for general (not small-time) controllable systems. --Steering Methods: Devising effective nonholonomic motion planners requires steering methods which account for small-time controllabilities. We show that, today, the problem is solved only for some classes of systems. --Combinatorial Complexity: We show how the worst case combinatorial complexity of a collision-free admissible path is related to the number of sub-riemannian balls covering the path. Then we will propose another complexity model which opens new direction of research. This talk will be illustrated by real experiments conducted at LAAS-CNRS on the Hilare mobile robots.

Deterministic observation theory and applications (joint work with IAK Kupka).

A complete theory of observability and observation in the deterministic setting, together with applications to dynamic output stabilization, will be presented. Several applications to the concrete world of chemical engineering will be briefly discussed.

Nilpotenization and Nonholonomic Motion Planning

We study local behavior of smooth nonlinear bracket generating control systems $\dot q=f(q,u)$ near an equilibrium $(q_0,0)$, $f(q_0,0)=0$. We are interested in the cases of degenerate linearization of $f$ at $(q_0,0)$, in particular, in linear with respect to control parameters systems $f(q,u)=\sum\limits_i f_i(q) u^i$. Nilpotenization is an intrinsically defined and effectively computed quasi-homogeneous approximation of the original system which preserves the key bracket generating property. One can use trajectories of the rather simple approximating system to construct a kind of ``nonholonomic Euler poligons'' for the original one and thus obtain a natural way for the motion planning.

Last modified: 18 December 2006

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