**List of Lecturers**

**List of seminars**

Lecturer |
Title of seminars |
Date |

C. Altafini | Control of Quantum Systems (III and IV) | November 12-14 |

A. Bacciotti | Generalized solitions of discontinious differential equations | September 11 |

D. Bambusi | Some results on Birkhoff Normal Form in Hamiltonian PDEs | September 17 |

M. Berti | Birkhoff-Lewis periodic orbits in some Hamiltonian PDE | November 5 |

P. Bettion | On ergodic problems for Hamilton-Jacobi-Isaacs equations with state constraints | November 18 |

L Biasco | A variational methods for fast Arnold diffusion | November 26 |

A. Bicchi | Quantized Control Systems and Optimal Steering | November 3 |

P. Bolle | Periodic solutions for the completely resonant wave equation with a general non-linearity. | November 11. |

B. Bonnard | Geometric Methods in Orbital Transfers | October 30 |

U. Boscain | Control of Quantum Systems (I and II) | November 7-10 |

G. Capitanio | Minimax solutions to Hamilton-Jacobi equations | November 14 |

M. Chaperon | Old and new results on invariant manifolds and conjugacies | September 16 |

L. Chierchia | Perpetual stability of the Sun-Jupiter-Victoria problem | November 7 |

J.-M. Coron | Exact boundary controllability of a nonlinear KdV equation with critical lengths | November 6 |

A. Davydov | 1. Optimization of cyclic motions
2. Structural stability of controllability |
October 6-15 |

A. Delshams | A geometric method for instability in Hamiltonian systems | October 29 |

B. Dubrovin | Invariant tori for water waves and theta-functions | October 2 and 7 |

D. Ferrario | On the Existence of Collisionless Equivariant Minimizers for the Classical
n-body Problem |
October 22 |

J.-P. Gauthier | 1. Identification of Non Linear Systems
2. Complexity and Motion Planning |
October 29-30 |

A. Grigoriev | 'Subgeneric' sets | November 20 |

C. Liverani | Statistical properties of geodesic flows in negative curvature | September 30 |

A. Marigo | Quantized Control Systems: Group Theoretic Characterization and Encoding | November 3 |

S. Marmi | On the cohomological equation for interval exchange maps | November 13 |

V. Mastropietro | Lindstedt series for periodic solutions on nonlinear wave equations | November 25 |

A. Neishtadt | 1. On stability loss delay for dynamical bifurcations
2. Averaging, passages through resonances and capture into resonance |
November 27 and 28 |

S. Pohozaev | Blow up for nonlinear differential equations (introduction) | September 9 |

M. Procesi | Periodic solutions for the non-linear wave equation | November 6 |

D. Rastovich | Transport theory and systems theory | October 15 |

H. Riahi | On parabolic and quasi-periodic solutions for some singular Hamiltonian systems | October 23 |

Y. Sachkov | The nilpotent (2,3,5) sub-Riemannian problem | November 27 |

M. Sigalotti | Local regularity properties of optimal trajectories | October 1st |

G. Stefani | Graded approximations of vector fields and application to controllability properties | September 18 |

E. Zehnder | Pseudoholomorphic Maps and Dynamics in 3 Dimensions | November 20 |

I. Zelenko | Dynamical Approach to Problem of Equivalence of Rank 2 Vector Distributions, I and II | November 11 and 13 |

A. Zuyev | Localization of the limit set and the stabilization problem in infinite dimensions | November 27 |

**Abstracts**

__Lectures:__

__Andrei Agrachev (SISSA)__**"Differential geometry of optimal control problems and Hamiltonian
systems"**
*whole trimester.*

1. Hamiltonian systems and extremals of optimal control problems.

2. Jacobi curves associated with trajectories of Hamiltonian systems and with various types of extremals.

3. Elements of symplectic geometry: Lagrange Grassmannians, Maslov index, Cross-ratio.

4. Canonical connections and curvatures of Hamiltonian systems and smooth control systems.

5. Geometry of Jacobi curves: their ranks, weights, derivative curves, curvatures, and fundamental forms. Flat curves.

6. "Comparizon theorems" for conjugate points.

7. Application to special classes of systems.

1.Viscosity solutions of Hamilton-Jacobi equations

2. Viscosity solutions of hyperbolic systems of conversation laws

3. Uniqueness theorems

4. The Hamilton-Jacobi-Bellman equation of dynamic programming

5. Applications to n-persons, non cooperative differential games: Existence of Nash equilibrium solutions

__Walter Craig (McMaster Univ., Canada)__**"Invariant tori for Hamiltonian PDE"**
*9/10--16/10.*

The course will address questions on the existence of KAM tori in Hamiltonian PDE, with the point of view being to address problems of exact resonance, as well as classical small divisors.1. Hamiltonian partial differential equations and Hamiltonian systems with infinitely many degrees of freedom.

2. The Lyapunov center theorem and the resonant theorems of A. Weinstein and J. Moser.

3. The Nash Moser method in this setting.

4. The linearized problem: resolvant estimates of J. Froehlich. T. Spencer and J. Bourgain.

5. Construction of Cantor varieties of invariant tori.

The purpose of the lectures is to introduce to the variational approach to the search of periodic solutions for the N-body problems. Such an approach has recently lead to the discovery of some new periodic solutions (see, for example, the paper A. Chenciner and R. Montgomery, A remarkable periodic solution of the three-body problem in the case of equal masses, Ann. of Math. (2), 152 (2000), 881--901).1. Introduction, central configurations.

2. The variational approach and the minimizing properties of the solutions of the 2-body problemand of the central configurations.

3. The lack of compactness of the variational formulation for the 2 and the N-body problem. Symmetry and topological constrains.

4. Collisions, strong and weak forces.

5. Existence of periodic non-collision solutions viavariational methods for perturbed N-body problems.

6. Existence of periodic solutions for the N-body problem: symmetry and variational constrains.

__Bronislaw Jakubczyk (Polish Academy of Sciences)__**"Geometry of control systems and distributions, feedback invariants"**
*10/9--6/10.*

1. Vector fields, Lie bracket, distributions,involutive distributions.

2. State and feedback equivalence. Feedback linearizable systems.

3. Classification of singularities of control systems in the plane. Bifurcations.

4. Local equivalence of families of vector fields and invariants. Cartan's method of graph.

5. Contact distributions and their singularities.

6. Singularities of general distributions. Hamiltonian formalism and characteristic curves (abnormal curves)

7. Hamiltonian description of general control systems. Singularities of the Hamiltonian. Critical Hamiltonians.

8. Microlocal and local feedback invariants via critical Hamiltonians and symbols.

__Frederic Jean (ENSTA, Paris, France)__**"Sub-Riemannian Geometry"**
*13/10--3/11*

Part I. Nonholonomic control systems and SR manifolds:

- Sub-Riemannian metrics

- Chow and Nagano/Sussmann theorems

- Nilpotent approximations

Part II. SR manifolds as metric spaces:

- Algebraic structure of the tangent space

- Ball-box theorem

- Metric tangent space

Part III. Geodesics:

- Optimal control problem

- Abnormal trajectories

- Regularity of the distance

__Sergei Kuksin (Heriot-Watt Univ., Edinburgh, United
Kingdom)__**"Mathematical hydrodynamics"**

PREREQUISITES: weak and strong solutions of nonlinear PDE, Sobolev spaces and the embedding theorems. The knowledge of these topics is sufficient (say) if it allows to read first sections of Chapter 1 of the book J-L Lions "Quelques methodes de resolution des problems aux limites non lineares".1 & 2. 2D and 3D Navier-Stokes (NS) equations Leray theory: weak and strong solutions for 3D NS, weak-strong uniqueness, local in time theory, 2D NS

3. Euler equation in 2D. Strong solutions and the Wolibner theorem, weak solutions and the Yudovich theorem

4. Euler equation in 3D and the inviscous limit

5 & 6. Around the Onsager conjecture.The conjecture, the work of Duchon-Robert,weak solutions of Euler equation of non-constantenergy.

7. Appendix (if time permits): the great open problems of mathematical` hydrodynamics.

__Benedetto Piccoli (ICA, Rome, Italy)__**"Optimal Synthesis Theory"**
*20/10--5/11.*

1. A brief introduction to Optimal Control Theory.

2. Pontryagin Maximum Principle.

3. Comparison between different kinds of solutions for Optimal Control Problems:

- open loop controls

- feedbacks

- syntheses

- value function

4. Boltianskii-Brunovsky Regular Syntheses. Sufficient conditions for Optimality.

5. Minimum Time for Bidimensional Single-input Systems. The Classification Program.

__Yuri Sachkov (Russian Academy of Science, Russia)__**"Controllability of invariant systems on Lie groups and homogeneous
systems"**
*1/11--27/11,*

Part I: Introduction to right-invariant systems on Lie groups:

- Lie groups and right-invariant systems.

- Basic properties of attainable sets of invariant systems.

- Homogeneous spaces and bilinear systems.

- Symmetric invariant systems.

Part II: Invariant systems on particular classes of Lie groups:

- Compact Lie groups and semidirect products.

- Semisimple Lie groups.

- Nilpotent Lie groups.

- Solvable Lie groups.

__Andrei Sarychev (Univ. of Florence, Italy)__**"Lie extensions and controllability results for nonlinear
systems"**
*6/10--19/10.*

1. Brief introduction into stability theory. Basic definitions. Linearization principle. Beyond the linearization principle: differential-geometric methods. Symmetric and nonsymmetric systems; orbit theorem

2. Extension of nonlinear control systems.

3. Convexification and introduction into theory of relaxed (sliding mode) controls.

4. Extensions beyond convexification: Lie saturation.

5. Nonlinear systems with generalized controls.

6. Extended (high-order) averaging and stability.

7. Lie extensions and controllability for some classes of PDE

__Armen Shirikyan (Université Paris-Sud, Orsay, France)__**"Stochastic hydrodynamics: the dynamical system approach"**
*14/9--27/9.*

1. Random dynamical systems generated by 2D Navier-Stokes equations

2. Maximal coupling of measures in finite-dimensional spaces

3. Coupling operators associated with a random dynamical system

4. Uniqueness of a stationary measure and exponential mixing

5. Ergodic theorems

1. Symplectic and contact geometry of control systems. Lagrangian and Legendre varieties. Caustics and wavefronts.

2. Generating families of lagrangian and Legendre varieties. Local and global settings.

3. Arnold theory of simple singularities of wavefronts and caustics.

4. Generic one-parameter bifurcations of wavefronts.

5. Nye-Chekanov theorem of admissible bifurcations in pseudooptical case. Maslov class.

6. Boundary singularities and control theory.

7. Avoiding an obstacle problem and first examples of stable singular lagrangian varieties.

Applications to control systems:

8. Singularites of relative minima function.

9. Caustic of exponential mapping in contact 3-dimensional subriemannian problem.

10. Singularities of attainability domains of systems determined by translation invariant finsler metric.

__Mikhail Zhitomirskii (Technion, Israel)__**"Singularities of vector distributions and foliations"**
*15/9--15/10.*

1. Equations A(x,y)dx+B(x,y)dy=0 as a starting point for several modern math topics.

2. Vector fields and differential forms on the plane. Phase portraits. Integrability. Center-focus problem and other open problems.

3. Foliations. Frobenius theorem. Malgrange theorem (Frobenius through singularities) and Kupka phenomenon. Other results and open problems on singular foliations.

4. Generic vector distributions. Non-holonomic dynamical systems. Darboux theorem. Engel theorem.

5. Singularities of distributions. Contact geometry. Darboux-Givental' theorem. Geometric point of view on PDE's. Recent works by V. Arnol'd on local contact algebra.

6. Cartan's prolongation. Cartan-Goursat flags and curves in contact space.

7. Control systems - invariant point of view. Darboux and Martinet theorems in terms of control theory.

8. Basic methods and techniques.

__Seminars:__

__C. Altafini (SISSA)__**"Control of Quantum Systems (III and IV)"**
*November 12 and 14*

The focus of these two lectures will be controllability of "closed" and "open" quantum systems. In particular, unitary controllability of wavefunctions for finite dimensional Schrodinger equations will be discussed using the structure of the associated semisimple Lie algebras. For density operators, it will be shown how to reformulate the same control problem in terms of Liouville equations and how to treat the case of quantum systems dissipating into the environment in the so-called Markovian approximation, still under unitary control.

Abstract: In this lecture we survey some results and counterexamples, with the aim of giving a complete representation of relationships among various possible definitions of solution for ordinary differential equations with a discontinuous right hand side. The interest in this topic is motivated by some problems in stability theory and feedback stabilization of nonlinear control systems.

__D. Bambusi (Università degli
studi di Milano, Italy)__**"Some results on Birkhoff Normal Form in Hamiltonian
PDEs"**
*September 17.*

Consider a Hamiltonian system with a linearly stable equilibrium point. Assume that the frequencies of small oscillation are

nonresonant then there exist coordinates $(p,q)$ in which the Hamiltonian is the sum of a part depending only the actions $(p_j^2

+ q_j^2) / 2$ and a small remainder. From this one can deduce that all small amplitude solutions remain close to a torus up to very

long times. I will show how this result can be extended to a quite general class of Hamiltonian semilinear PDEs including the

Nonlinear Wave Equation in one space dimensions and the Nonlinear Schr\"odinger Equation in arbitrary space dimensions. An

extension to quasilinear equations will also be discussed.

__M. Berti (SISSA)__**"Birkhoff-Lewis periodic orbits in some Hamiltonian
PDE"**
*November 5.*

__P. Bettiol (SISSA)__**"On ergodic problems for Hamilton-Jacobi-Isaacs
equations with state constraints"**
*November 18.*

We study the asymptotic behavior as $\lambda\rightarrow 0^+$ of $\lambda v_\lambda$, where $v_\lambda$ is the viscosity solution of the following Hamilton-Jacobi-Isaacs (HJI) equation (infinite horizon case) $$\lambda v_\lambda + H(x,Dv_\lambda)=0, $$ with $$ H(x,p):=\min_{b\in B}\max_{a \in A} \{-f(x,a,b)\cdot p -l(x,a,b)\}. $$ We discuss the case in which the state of the system is required to stay in the closure of a bounded domain $\Omega\subset{\Bbb R}^n$ with sufficiently smooth boundary (for example $\partial \Omega\in {\cal C}^2$). Under the uniform approximate controllability assumption by the first player, we extend to Differential Games the convergence result of the term $\lambda v_\lambda (x)$ to a constant as $\lambda\rightarrow 0^+$. We also show how to contruct nonanticipative strategies which satisfies some "good" estimates in order to obtain H\"older regularity of the value function.

__L. Biasco (Univ. of Roma 3, Italy)__**"'A variational methods for fast Arnold diffusion"**
*November 26.*

__A. Bicchi (University of Pisa, Italy)__**"Quantized Control Systems and Optimal Steering"**
*November 3.*

In this talk we consider the problem of optimal control (specifically,minimum-time steering) for systems with quantized inputs. In particular, we propose a new approach to the solution of the optimal control problem for an important class of nonlinear systems, i.e. chained--form systems. By exploiting results on the structure of the reachability set of these systems under quantized control, the optimal solution is determined solving an integer linear programming problem. Our algorithm represents an improvement with respect to classical approaches in terms of exactness, as it does not resort to any a priori state-space discretization. Although the computational complexity of the problem in our formulation is still exponential, it lends itself to application of Branch and Bound techniques, which substantially cuts down computations in many cases, as it has been experimentally observed.

__P. Bolle (Université d'Avignon,
France)__**" Periodic solutions for the completely resonant
wave equation with a general non-linearity"**
*November 11.*

We consider a non-linear wave equation for which all the solutions of the linearized problem at 0 are $2\pi$-periodic in time. Existence and multiplicity results for periodic solutions of fixed period are presented, under quite general assumptions on the non-linearity. These results are obtained by means of a variational principle.

__B. Bonnard (Université de Bourgogne,
Dijon, France)__**"Geometric Methods in Orbital Transfers"**
*October 30.*

An important problem in aeorospace mechanic is to transfer a satellite from a given elliptic orbit to a final orbit, e.g. circular. Also it is connected to recent projects to transfer the satellite with a low thrust engine.The objective of this talk is to present the tools from geometric control theory to handle the problem by both stabilization, motion planning and optimal control methods, The numerical aspects of the problem will be discussed at the end of the talk.

__G. Capitanio (Univ. of Paris 7, France)__**"Minimax solutions to Hamilton-Jacobi equations"**
*November 14*

Minimax solutions are weak solutions of Cauchy problems for Hamilton-Jacobi equations, constructed from generating families (quadratic at infinity) of the geometric solutions. We describe a new construction of the minimax in terms of Morse theory, and we show its stability by small perturbations of the generating family. We consider the wave front corresponding to the geometric solution as the graph of a multi-valued solution of the Cauchy problem, and we give a geometric criterion to find the graph of the minimax.

__M. Chaperon (Université Paris7,
France)__**"Old and new results on invariant manifolds
and conjugacies"**
*September 16.*

In the early nineties, Alexander Shoshitashvili made the speaker observe that $h:X\rightarrow Y$ is a semi-conjugacy between

$f:X\rightarrow X$ and $g:Y\rightarrow Y$ (i.e. satisfies $g\circ h=h\circ f$) if and only if the graph of $h$ is invariant by $f\times

g:(x,y)\mapsto\big(f(x),g(y)\big)$\/. Since 1993, the speaker has been building up a theory of invariant manifolds which includes many (semi-)conjugacy theorems---and other, related results---as corollaries. The talk is a survey of this theory.

KAM theory is a powerful tool apt to prove perpetual stability in Hamiltonian systems, which are a perturbation of integrable ones; the smallness requirements for its applicability are well known to be extremely stringent. A long standing problem, in this context, is the application of KAM theory to "physical systems" for "observable" values of the perturbation parameters. Here, we consider the Restricted, Planar, Circular, Three-Body Problem (RPCTBP), i.e., the problem of studying the planar motions of a minor body subject to the gravitational attraction of two primary bodies revolving on a circular Keplerian orbit (which is assumed not to be influenced by the minor body). When the mass ratio of the two primary bodies is small the RPCTBP is described by a nearly-integrable Hamiltonian system with two degrees of freedom; in region of phase space corresponding to nearly elliptical motions with non small eccentricities, the system is well described by Delaunay variables. The Sun-Jupiter observed motion is nearly circular and an asteroid of the Asteroidal belt may be certainly assumed not to influence the Sun-Jupiter motion. The Jupiter-Sun mass ratio is slightly less than 1/1000. We consider the motion of the asteroid 12 Victoria taking into account only the Sun-Jupiter gravitational attraction regarding such a system as a prototype of a RCPTBP. For values of mass ratios up to 1/1000, we prove the existence of two-dimensional KAM tori on a fixed three-dimensional energy level corresponding to the observed energy of the Sun-Jupiter-Victoria system; such tori trap the evolution of phase points "close" to the observed physical data of the Sun-Jupiter-Victoria system. As a consequence, in the RCPTBP description, the motion of Victoria is proven to be forever close to an elliptical motion. The proof is based on: 1) a new iso-energetic KAM theorem; 2) an algorithm for computing iso-energetic, approximate Lindstedt series; 3) a computer-aided application of 1)+2) to the Sun-Jupiter-Victoria system.

We study the boundary controllability of a nonlinear Korteweg-de Vries equation with the Dirichlet boundary condition on an interval with a critical length for which it has been shown by Rosier that the linearized control system around the origin is not controllable. We prove that the nonlinear term gives the local controllability around the origin.

1.

For dynamic inequalities with locally bounded derivatives on the circle we optimize the time averaged profit along admissible

motions which are defined for all nonnegative time. We describeReferences

- the nature of optimal strategies;
- generic singularities of maximal and minimal velocities used by ones;
- generic switchings between the strategies when the problem depends additionally from a one dimensional parameter and also

- V.I.Arnol'd, Averaged optimization and phase transition in control dynamical systems, Funct. Anal. and its Appl. {\bf 36} (2002), 1-11.
- V.I.Arnol'd, On a Variational Problem Connected with Phase Transitions of Means in Controllable Dynamical Systems, in M.Birman (ed) et al., Nonlinear Problems in Mathematical Physics and Related Topics I, Kluwer/Plenum Publishers, ISBN 0-306-47333-X, July 2002.
- Davydov, A.A.; Jongen H.Th., Normal forms in One-Parametric Optimization, Annals of operations Research 101. 255 - 265, 2001.
- Davydov, A.A.; Zakalyukin, V.M. Coincidence of generic relative minimum singularities in problems with explicit and implicit constraints, J. Math. Sci., New York 103, No.6, 709-724 (2001).
- Davydov, A.A.; Zakalyukin, V.M. Classification of relative minima singularities Janeczko, Stanislaw (ed.) et al., Geometry and topology of caustics - CAUSTICS '98. Proceedings of the Banach Center symposium, Warsaw, Poland, June 15-27, 1998. Warsaw: Polish Academy of Sciences, Institute of Mathematics, Banach Cent.Publ. 50, 75-90 (1999).
- Zhikov V.V., Mathematical problems of the retrieval theory, Proceedings of Vladimir Politekhnical Institute, 1968, 263-270 (in Russian).

For generic control systems we study the stability of domains of points having the same controllability properties. In particular

there are discussedReferences

- the stability of local controllability properties for generic systems and dynamic inequalities on surfaces;
- the stability of nonlocal controllability properties for generic control system on closed surface;
- the classifications of singularities on the boundaries of domains with the same controllability properties.

- V.I.Arnol'd - Catastrophe Theory, Third Revised Edition, Berlin a.o.: Springer-Verlag, 1992;
- Davydov, A.A. - Qualitative Theory of Control Systems, viii + 147. Translations of Mathematical Monographs, 141. American Mathematical Society, Providence, RI, 1994. ISBN: 0-8218-4590-X.
- Davydov, A.A. - Local controllability of typical dynamic inequalities on surfaces, Proceedings of the Steklov Institute of Mathematics, 209 (1995), pp. 73--106.
- Davydov, A.A., Basto-Goncalves, J. - Controllabilty of generic inequalities near singular points, J. of Dyn. and Control Systems , 7 (2001), pp. 77--99

Authors: Amadeu Delshams, Universitat Politècnica de Catalunya, Barcelona

Abstract: We overview a geometrical mechanism, based on combining standard perturbation methods (Averaging, K.A.M. theory, Melnikov method) with new geometric ones (particularly, the computation of the scattering map of a normally hyperbolic invariant manifold along transverse homoclinic orbits), to find landmarks for instability trajectories in Hamiltonian systems.

This geometrical mechanism is applied to two paradigmatic examples. First, to general geodesic flows on manifolds, to show the existence of trajectories with unbounded energy when a generic quasi-periodic potential is added to the system. Second, to a Hamiltonian formed by a pendulum and a rotor, to show the existence of trajectories with arbitrary increasing of energy when a general periodic non-autonomous Hamiltonian perturbation is added to the system. (In the terminology of the Hamiltonian systems, the first case is called ``a priori-chaotic'', whereas the second one is called ``a priori-unstable''.)

This talk is based on joint work with Rafael de la Llave and Tere M. Seara, and related preprints can be found in the mp_arc archive, see:http://www.maia.ub.es/cgi-bin/mps?src=abstracts&key=delshams+and+llave+and+seara

__B. Dubrovin (SISSA-ISAS, Trieste, Italy)____"__Invariant tori for water waves and
theta-functions"*October 2 and 7.*

__D. Ferrario (Univ. of Milano, Italy)__**"On the Existence of Collisionless Equivariant
Minimizers for the Classical n-body Problem" **(joint work with S. Terracini)
*October 22.*

__J.-P. Gauthier (Université de Bourgogne,
Dijon, France)__

1. **"Identification of Non Linear Systems"**
*October 29.*

2. **"Complexity and Motion Planning"**
*October 30.*

__A. Grigoriev (SISSA)__**"'Subgeneric' sets"**
*November 20.*

Regularity properties of subanalytic sets are often used in control theory to establish regularity of various characteristics of real analytic control systems. In some cases, the regularity of the corresponding characteristics of generic smooth control systems, while being of interest, is not known. We suggest an approach to these questions based on older observations due to Yu. Ilyashenko and S. Yakovenko, and S. Yakovenko and the speaker (obtained in a different context). Specifically, we intend to sketch a proof of a partial result in this direction. It is convenient to formulate this result as the logic-theoretic assertion that the extension of the first order theory of the real numbers by (a finite number of) generic smooth functions, restricted to the unit cube, is o-minimal. We will attempt to give a clear and concise explanation of the concepts involved, stressing the usefulness of the logic-theoretic point of view in the present context. In particular, no background in mathematical logic is assumed.

Abstract: In recent years some new ideas have allowed to make substantial progresses in the investigation of the statistical properties of some classes of flows. In particular, the first quantitative informations on the rate of mixing for geodesic flows in negative curvature have been obtained. I will review some of this ideas and present a recent new result.

__A Marigo (IAC, CNR, Rome, Italy)__**"Quantized Control Systems: Group Theoretic
Characterization and Encoding"**
*November 3.*

A quantized control system is a control system where the control input set is a finite set of constant functions of time. Some examples will show that the analysis of the set of reachable points may be very complicated. However precise results can be given under suitable hypotheses. Moreover, if some assumptions are satisfied, it turns out that introducing quantization on purpose can help in designing closed form control algorithms.

__S. Marmi (Scuola Normale Superiore,
Pisa, Italy)__**"On the cohomological equation for interval
exchange maps"**
*November 13.*

Area-preserving vector fields on Riemann surfaces are related via Poincare' section / singular suspension to interval exchange maps. Interval exchange maps also arise naturally in the study of polygonal billiards. Following Rauzy-Veech-Zorich one can introduce a generalized continued fraction algorithm: its acceleration can be used to exhibit a full measure class of ''diophantine Roth type''interval exchange maps for which the (linear) cohomological equation has a bounded solution provided that the datum belongs to a finite codimension subspace of the space of functions with derivative of bounded variation on each interval.has a bounded solution provided that the datum belongs to a finite codimension subspace of the space of functions with derivatives of bounded variation on each interval. The diophantine condition to be used is formulated in terms of the growth rate of the continued fraction and is quite explicit and the loss of differentiability is optimal.

This is joint work with Pierre Moussa and Jean-Christophe Yoccoz.

I describe a work, in collaboration with G.Gentile, in which we have defined a Linsdedt series expansion for the periodic solutions of nonlinear wave equations like the Klein-Gordon or the sine-Gordon equations. After suitable resummation this expansion is convergent for a large set of amplitudes and a large set of masses of full Lebesgue measure. We have recently extended this result, in collaboration also with M.Procesi, to the case of the string equation.

In classical bifurcation theory the behavior of systems, depending on a parameter, is considered for values of the parameter close to some critical, bifurcational one. In theory of dynamical bifurcations the parameter is changing slowly in time and passes through the value, which would be bifurcational in classical static theory. Some phenomena, arising here, are drastically different from predictions, derived by static approach. Let at bifurcational value of the parameter the equilibrium or the limit cycle loses its asymptotic linear stability but remains nondegenerate. It turns out that in analytic systems the stability loss is inevitably delayed: the phase point remains near unstable equilibrium (cycle) for a long time after bifurcation; during this time the parameter changes by a quantity of order 1. Such delay is not in general found in nonanalytic (even infinitely smooth) systems. The maximal delay time is controlled by singularities of the system in the plane of complex time.

**"Averaging, passages through
resonances and capture into resonance"**
*November 28.*

Small perturbations imposed on an integrable system produce an evolution of the motion. For the approximate description of this evolution the classical averaging method prescribes to average the differential equations of the perturbed motion over the phases of the unperturbed motion. In the course of the evolution the frequencies of motion are changing slowly and at some time they become resonant. Due to influence of resonances the actual motion can be considerably different from the one predicted by the averaging method. The two basic phenomena that takes place here in the case of two-frequency systems are quasi-random phenomena of capture into resonances and scattering by the resonances. For Hamiltonian systems with slow and fast motions these phenomena lead to destruction of adiabatic invariance, dynamical chaos and transport in large domains in the phase space. The major topics discussed in the lecture are estimates of accuracy of the averaging method, calculation of probability of capture into resonance, description of motion of captured phase points, calculation of a probabilistic distribution of the scattering amplitude.

__S. Pohozaev (Steklov math. Inst., Moscow,
Russia)__**"Blow up for nonlinear differential equations
(introduction)"**
*September 9.*

__M. Procesi (SISSA)__**"Periodic solutions for the non-linear wave
equation"**
*November 6.*

__D. Rastovich (Zagreb, Croatia)__**"Transport theory and systems theory"**
*October 15.*

The simulation of singular nonlinear transport equation is obtained viacorresponding neutron or photon kinetic equation . The conditions for convergence of the nonstationary transport process toward the purediffusion across the equilibriums are presented . For such purpose the method of transport scattering is exploited . The goal of these results is optimization of fusion fuels via neutron diagnostics .

__H. Riahi (Ecole Nationale des Ingénieurs
de Tunis, Tunisia)__**"On parabolic and quasi-periodic
solutions for some singular Hamiltonian systems"**
*October 23.*

We will describe some methods to get parabolic orbits, also called homoclinics to infinity, as well as quasi-periodic orbits for some singular Hamiltonian systems satisfying the strong force hypothesis. For parabolic orbits, we use some double approximatiom method, applying first classical variational methods to get heteroclinic orbits between two fixed points, then we let the points go to infinity to get parabolic orbits. As for quasi-periodic orbits, we get the solutions by applying variational methods to an associated second order elliptic P.D.E. on a torus using a generalized Poincare inequality, the we show that the obtained weak solutions correspond to regular quasi-periodic solutions of the Hamiltonian system. These are joint works with Pablo Padilla (UNAM, Mexico).

__Y. Sachkov (Russian Academy of Science,
Russia)__**"The nilpotent (2,3,5) sub-Riemannian problem"**
*November 27.*

We consider the nilpotent left-invariant sub-Riemannian structure with the growth vector (2,3,5). The Hamiltonian system for sub-Riemannian geodesics is integrable in elliptic Jacobi functions. With the aim to characterize the length-minimizing geodesics,

__M. Sigalotti (SISSA)__**"Local regularity properties of optimal trajectories"**
*October 1st.*

__G. Stefani (Università degli studi
di Firenze, Italy)__**"Graded approximations of vector fields and
application to controllability properties"**
*September 18.*

Abstract:A graded structure induced by an increasing filtration of a Lie algebra and the associated graded approximation of vector fields will be defined. We use the properties of the approximation to study controllability properties. Namely we investigate whether an equilibrium point of a control system belongs to the interior of the reachable set at some time. Some ideas on the use of the graded approximations to study controllability along a trajectory will be also given.PREREQUISITES: Vector fields, Lie brackets, Lie Algebras, graded structures.The definition of graded structure can be found in:R.W. Goodman, Nilpotent Lie Groups: Structure and Applications to Analysis, Lecture Notes in Mathematics, 562, Springer-Verlag, Berlin, New York,1976.A summary and the properties used in the lecture are summarized in Sec. 2, pp. 910-912, of:R.M. Bianchini and G. Stefani, Graded Approximations and Controllability along a Trajectory, SIAM J. Control and Optimization, 28, 1990, pp.903-924

Dipartimento di Matematica Applicata, Via S. Marta 3 - 50139 Firenze, Italia E-mail address: stefani@dma.unifi.it

__E. Zehnder (Universität Zürich,
Switzerland)__**"Pseudoholomorphic Maps and Dynamics in 3
Dimensions"**
*November 20*

We shall describe tools useful in the study of 3-dimensional flows. The tools are based on partial differential equations of Cauchy-Riemann type. Their solutions allow the construction of global surfaces of sections and, as a generalization, of 2-dimensional foliations transversal to the flow in the complement of finitely many distinguished periodic orbits. Such a global system of transversal sections is established for Reeb-flows on the tight 3-sphere. The applications cover Hamiltonian systems on starlike energy surfaces.

__I. Zelenko (SISSA)__**"Dynamical Approach to Problem of Equivalence
of Rank 2 Vector Distributions, I and II"**
*November 11 and 13.*

Rank $k$ vector distribution on the $n$-dimensional manifold $M$ or $(k,n)$-distribution (where $k<n$) is by definition a $k$-dimensional subbundle of the tangent bundle $TM$. Two distributions are called (locally) equivalent, if one can be transformed to another by (local) diffeomorphism. The case $k=2$, $n=5$ (the first case containing functional parameters) was treated by E. Cartan in 1910 by ingenious use of his "reduction-prolongation" procedure. In particular, he found the covariant fourth-order tensor invariant (Cartan tensor) for such distributions. In our talk we will describe completely different, more geometric approach for construction of invariants of rank 2 vector distributions for arbitrary $n\geq 5$. This approach is based on dynamics of the field of abnormal extremals of rank 2 distribution and on theory of curves in the Lagrange Grassmannian, developed in our previous works with A, Agrachev. In this way we construct the {\it fundamental form} and the {\it projective Ricci curvature} of $(2, n)$-distributions for arbitrary $n\geq 5$. It turns out that in the case $n=5$ our fundamental form coincides with Cartan's tensor. In the case $n=5$ we will present explicit formulas for computation of our invariants. We will apply this formulas to several natural examples, showing simultaneously the efficiency of our invariants in proving that distributions are not equivalent. Finally, for $n=5$ we will construct the canonical frame of distribution with nonzero fundamental form. This gives the way to prove that distributions are equivalent.

In this talk we use Lyapunov s direct method to characterize the limit set of a nonlinear continuous semigroup in a Banach space.A sufficient condition for partial asymptotic stability of the equilibrium with respect to a continuous functional is proved. For dynamical systems on a finite dimensional Euclidean space, this result extends the Risito-Rumyantsev theorem. The technique proposed is applied to stabilize the attitude of a rigid body endowed with an arbitrary number of elastic beams. The dynamics is described by a system of the Euler-Bernoulli PDEs together with an ODE on the angular coordinate. A peculiarity of this nonlinear control system is that the semigroup generator is not accretive, and some perturbation analysis is carried out to verify precompactness of the closed-loop trajectories. In addition, we prove strong (non-asymptotic) stability of the equilibrium in the resonance case.