MCT'07 mini-course lectures description

Piermarco Cannarsa, Università degli Studi di Roma "Tor Vergata"

“Semiconcavity and Control Theory”

Abstract

This series of lectures will focus on the role of semiconcave functions in control theory, and is based in part on the speaker’s recent book [Cannarsa, P., and C. Sinestrari, Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control, Birkhäuser, Boston, 2004].

The topics covered include:

  1. Basic theory: motivating examples, basic definitions, generalized differentials, inf/sup convolutions.
  2. Singularities of semiconcave functions: rectifiable sets, rectifiability criteria, singular sets of semiconcave functions, rectifiability of singular sets, propagation of singularities of semiconcave functions.
  3. Semiconcave functions in HJ equations and control theory: semiconcavity and viscosity solutions of HJ equations, propagation of singularities of semiconcave solutions, distance function, semiconcavity of the value function of optimal control problems, estimates for the closure of the singular set.
  4. Semiconcavity and geometric properties of attainable sets: the minimum time problem for a general target, inner ball property and perimeter estimates for attainable.

More information about the speaker is available at his website: https://www.mat.uniroma2.it/~cannarsa/

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Piermarco Cannarsa

Piermarco Cannarsa is a full professor of mathematical analysis at the University of Rome "Tor Vergata". His scientific interests include partial differential equations, control theory and nonsmooth analysis. His current research is concerned with regularity of viscosity solutions to Hamilton-Jacobi-Bellman equations, singularities of (semi)concave functions in both finite and infinite dimensions, and controllability of nonlinear evolution equations.

Cannarsa authored more than 60 papers in international journals, and the book Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control, published by Birkhäuser in 2004. He served as Associate Editor for the SIAM Journal on Control and Optimization. He is currently in the editorial board of the Rendiconti di Matematica.