**ARGOMENTI:** Seminari

SEMINARI DI EQUAZIONI DIFFERENZIALI E APPLICAZIONI

Our SADCO fellow Nguyen Van Luong will defend his PhD thesis on Wednesday, Sept. 24 in the morning.

We have organized also a short workshop with talks by the members of the jury, as scheduled below:

Wednesday, September 24, Room 1BC50

9:15 Helene Frankowska (CNRS and UPMC, Paris): First and second order sensitivity relations in optimal control

Abstract: Sensitivity analysis in optimal control brings on relations between the co-state of the maximum principle and various generalized differentials of the value function along a given optimal trajectory. For Fr?chet superdifferentials these relations help to express the optimal synthesis and lead to sufficient optimality conditions. In this talk I will discuss some very recent results on second order sensitivity relations obtained together with P. Cannarsa and T. Scarinci.

10:05 Carlo Sinestrari (Roma 2): Global propagation of singularities for the time dependent eikonal equation

Abstract: We study the properties of the generalized characteristics associated with the solution of the eikonal equation on a euclidean space or a riemannian manifold. It is well known that such a solution in general has singularities, i.e. points of non differentiability. We show that any singularities propagates forward in time globally along a generalized characteristic. This statement extends to general dimension a classical result by Dafermos for hyperbolic conservation laws in one dimension, and is the time-dependent counterpart of the recent study by Albano-Cannarsa-Nguyen Khai-S. in the stationary case. The proof relies on a sharp estimate of the monotonicity of the superdifferential of the solution jointly in space and time, which is related to the semiconcavity property. The results are in collaboration with P. Cannarsa and M. Mazzola.

10:55 - 11:10 coffee break

11:10 Fabio Ancona (Padova): On quantitative compactness estimates for hyperbolic conservation laws and HJ equations

Abstract: Inspired by a question posed by Lax, we wish to study quantitative compactness estimates for the map S_t, t>0, that associates to every given initial data u_0 the corresponding solution S_t u_0 of a conservation law or of a first order Hamilton-Jacobi equation. Estimates of this type play a central roles in various areas of information theory and statistics as well as of ergodic and learning theory. In the present setting, this concept could provide a measure of the order of "resolution" of a numerical method for the corresponding equation. In this talk we shall first review the results obtained in collaboration with O. Glass and K.T. Nguyen, concerning the compactness estimates for solutions to conservation laws. Next, we shall turn to the more recent analysis of the Hamilton-Jacobi equation pursued in collaboration with P. Cannarsa and K.T. Nguyen. A control-type analysis of such equations turns out to be fundamental to establish some of these properties.

12:00 - 12:50 Nguyen Van Luong (Padova): On regular and singular points of the minimum time function

Abstract: In this thesis, we study the regularity of the minimum time function T for both linear and nonlinear control systems in Euclidean space. We first consider nonlinear problems satisfying Petrov condition. In this case, T is locally Lipschitz and then is differentiable almost everywhere. In general, T fails to be differentiable at points where there are multiple time optimal trajectories and its differentiability at a point does not guarantee continuous differentiability around this point. We show that, under some regularity assumptions, the nonemptiness of proximal subdifferential of the minimum time function at a point x implies its continuous differentiability on a neighborhood of x. The technique consists of deriving sensitivity relations for the proximal subdifferential of the minimum time function and excluding the presence of conjugate points when the proximal subdifferential is nonempty. We then study the regularity the minimum time function to reach the origin under controllability conditions which do not imply the Lipschitz continuity of T. Basing on the analysis of zeros of the switching function, we find out singular sets (e.g., non-Lipschitz set, non-differentiable set) and establish rectifiability properties for them. The results imply further regularity properties of T such as the SBV regularity, the differentiability and the analyticity. The results are mainly for linear control problems.

Rif. Int. G. Colombo, F. Ancona