Semiconcavity of the bilateral minimal time function: The linear case

For a linear control system, we provide conditions under which the bilateral minimal time function T(⋅,⋅)T(⋅,⋅) is semiconcave near a given point (α,β)(α,β). A semiconvexity result of Nour [C. Nour, The bilateral minimal time function, J. Convex Anal. 13 (1) (2006) 61–80, Theorem 4.7] allows us to deduce that T(⋅,⋅)T(⋅,⋅) is then also C1,1C1,1-smooth near (α,β)(α,β). The nonlinear case, which remains open, is discussed in the concluding remarks.