This is a joint work with Alberto Bressan. The talk will begin with a brief introduction to non-cooperative differential game, which is actually an optimal control problem. I will formally derive the Hamilton Jacobian equations and the corresponding system of conservation laws. The hyperbolicity of the system is essential, and neccesary conditions are derived. In the positive case, the weak solution of the system of conservation laws determines an n-tuple of feedback strategies. These yield Nash equilibrium solutions to the non-cooperative differential game. Some discussion about extensions and future work will be given at the end of the talk, where we will also discuss partially cooperative strategies which always yield a hyperbolic system.

Symmetric functions of the critical Hamiltonians, called symbols, play an important role in such problems of nonlinear control as characterization of the symmetries and the feedback invariants. We derive here a stabilizability condition in the class of almost continuous feedback controls based on the symbols. The methodology proposed consists in defining a selector of the multivalued covector field satisfying extra degree condition on a sphere close to the origin. Then the above selector is extended to some neighborhood in order to get an exact differential form. Integration of that form gives us a control Lyapunov function candidate. The condition proposed is applied for analysis of a planar control-affine system with polynomial homogeneous vector fields. Unlike known results in the literature, we consider the case when each vector field vanishes at the origin and the control is bounded. We give stabilizability conditions explicitly in terms of the control system parameters. This is joint work with Prof. Bronislaw Jakubczyk.

We develop the approach to the time-optimal problem based on the Markov moment problem method. The initial point in our analysis is the representation of affine control systems in a neighborhood of the equilibrium as the series of nonlinear power moments. That allows, in particular, to examine analytic changes of variables in the system as the corresponding analytic transformations of its series. We consider the free algebra of nonlinear power moments A an d study the properties of finite-generated right ideals of this algebra. In this way we obtain the following genera lization of the remarkable R. Ree's theorem: Let J be an ideal generated by finite number of Lee elements of A and J^\perp be its orthogonal complement in A. Th en J^\perp admits the orthogonal decomposition J^\perp = L \oplus L^{sh}, where L is the orthogonal projection of the Lee subalgebra of A to J^\perp and L^{sh} = \{l1* l2* . . . *lk, k \geq2, l_j\in L\} , where * means the shuffle product in A. Using this theorem we obtain the result on the reduction (by an analytic transformation) of the system series to th e canonical form and define the (homogeneous) principle part of the series. This, in turn, gives the asymptotic app roximation of the time optimal problems for affine systems by problems for the homogeneous systems whose series hav e the same principle part. Joint work with Svetlana Ignatovich (Kharkov National University).

We study controllability of the 2 and 3-dimensional Navier--Stokes equations $$ \frac{\partial u}{\partial t}+(u\cdot\nabla)u+\nabla p-\nu\Delta u=f, \quad \mbox{div} u=0 \eqno (1) $$ with periodic data. Here $f$ is control which has the shape of a low order (we stay with order two) trigonometric polynomial with respect to the state variables: $$ f(t,x)=\sum\limits_{|k|\le 2}e^{ik\cdot x}a_k(t),\quad a_{-k}=\bar a_k,\quad a_k\cdot k=0, $$ where $k=(k_1,k_2)$ or $k=(k_1,k_2,k_3),\ k_j\in\mathbb Z,\ |k|=\sum\limits_j|k_j|. $$ The goal is to understand how the structure of the equations serves for the energy propagation from low to higher frequencies. e denote by $v^N$ the $N$-th order Fourier polynomial of the function $v(t,x)$ with respect to the state variables. Consider the finite dimensional Galerkin truncation $$ \frac{\partial u^N}{\partial t}+\left((u^N\cdot\nabla)u^N\right)^N+ \nabla p^N-\nu\Delta u^N=f, \quad \mbox{div} u^N=0 \eqno (1^N) $$ of the equation (1). We prove that ($1^N$) is globally controllable for any $N$. In the case of the 2-dimensional Navier--Stokes equation the same class of controls provides the global approximate controllability of the original (not truncated) system (1). This is a joint work with Andrei Sarychev.

Discuteremo una classe di equazioni di tipo Hamilton-Jacobi a coefficienti discontinui. Per tali equazioni e' appropriato il concetto di soluzione di viscosita', che garantisce l' esistenza di soluzioni. In generale pero' problemi al contorno hanno soluzioni multiple. Presentero' alcune condizioni necessarie e sufficienti per l' unicita' ed alcuni espliciti risultati di unicita' sotto ipotesi opportune, che si applicano ad esempio al caso dell' equazione iconale dell' ottica geometrica con indice di rifrazione discontinuo sotto la condizione di Hormander.

When studying flatness of control systems, there appears some (overdetermined) systems of PDEs whose order is not fixed, yet finite for any proper solution. In order to analyse such equations, the main tool introduced here is a valuation, that seems very natural to control system, and allows one to compute formal power series solution although they potentially depend on infinitely many variables. (joint work with David Avanessoff et Laurent Baratchar)

Partial differential operators in $G \times I$, where $G \subset \R^n$ is a bounded set and $I \subset \R$ an interval of time variable, and their generalizations, pseudo-differential operators, are studied here keeping in mind the goal of modelling physical distributed-parameter phenomena. Control actions of such systems usually take place on the boundary $\partial G$. Symbolic calculus applied gives tools to form e.g. compositions, formal adjoints, generalized right and left inverses and so-called compatibility conditions. The operators form an algebra $\mathcal D$ by using of which typical boundary-value control problems can be formulated. Parametrizability, which is a concept closely related to flatness of ordinary controlled differential systems, means that for a given control system $\L u=0$, where $\L$ is the system operator, and the variable $u$ includes the actual control, state and output variables, one can find an operator $\S$ such that $\L u=0$ if and only if $u = \S f$ for some relevant function $f$. It is required that the components $f_i$ of $f$ are $\mathcal D$-linearly independent. The pseudo-differential operators and boundar-value operators are formed by the matrix-like operators: r^+A+B K \L= T Q where the $KQ$-column operates on the functions defined on $\partial G \times I$, and the first column on functions in $\bar G \times I$. The operator $r^+$ restricts the global operator $A$ to act in $\bar G \times I$. $T$ is the trace operator. In our applications the computation rules in $\mathcal D$ give explicitly the parametrization operator $\S$ subject to certain existence assumptions of related pseudo-differential operators or of one-sided inverse operators. This construction is based on methods of homological algebra. Projective freeness of a certain factor module (defined by the system equations) implies parametrizability. Some examples of partial differential control systems including boundary conditions are presented to illustrate the parametrizability concept and construction of the operator $\S$.

Lie group formalism and Riemaniann techniques can be used to get an estimation of the diameter of SU(n) under a single input left invariant control system, that is to say a system with a drift and one unbounded real control. The first step is to apply the technique of variation of the constant to approximate the reachable set at time T of the initial system. The second step consits in the replacement of the initial system by a very close Riemaniann one. Then, using the particular structure of the Lie algebra su(n), it is possible to use well-known Riemaniannian results to get an upper bound for the diameter of the group.

This talk deals with sub-Riemannian metrics in the quasi-contact case. First, in any even dimension, we construct normal coordinates, a normal form and invariants, which are the analogs of normal coordinates, normal form and classical invariants in Riemannian geometry. Second, in dimension 4, and thanks to this "normal form", we study the local singularities of the exponential mapping.

The controllability of the Schrodinger equation and of the Markovian master equation for N-level quantum mechanical systems is discussed using Lie algebraic tools and notions from Lie semigroup theory. The two level case is treated in detail.