Minimum-time functions (Reference: ESR4) *****this position has been filled*****

• Title: Minimum time functions
• Location: University of Padova
• Advisor: Giovanni Colombo
• Secondment: Université Pierre et Marie Curie (Paris 6)
• Contact: Fabio Ancona 
• Reference: ESR4
• Key Words: Nonsmooth analysis, geometric measure theory, set valued analysis

• Description of the subject:   In the theory of partial differential equations of Hamilton-Jacobi type arising in  Optimal Control, semiconcavity can be expected to hold for some classes of viscosity solutions or value functions, under certain assumptions such as convexity and coercivity of the Hamiltonian or a strong type of controllability. Semiconcave functions are known to have several significant regularity properties, such as the almost everywhere twice differentiability and the availability of precise estimates on the set of nondifferentiability points. Such properties are inherited from analogous properties of convex functions. The semiconcavity of a function may be seen as the combination of two distinct properties: 1) local Lipschitz continuity and 2) a kind of external sphere condition on the hypograph, which we call in the sequel proximal smoothness. Property 1) is a consequence of the coercivity/controllability, while 2) is associated with the structure of the problem. For minimum time problems, 2) is implied by a certain interplay between the controlled dynamics and the target.

     In this project area, we shall investigate regularity properties of the value function for minimum time problems, and solutions of related partial differential equations. We shall, in particular, examine whether proximal smoothness of the epi/hypograph of value functions or of viscosity solutions holds, under conditions which do not imply the Lipschitz continuity of these functions. It is likely that such investigations will require consideration of higher order controllability conditions, in the spirit of the Kalman Rank Condition or of Chow’s theorem.