On the regularity of optimal controls
The authors show that regularity proofs for closed-loop optimal controls require results on piecewise regularity with bounds on the number of switchings for open-loop optimal controls. They survey known results of this kind [\textit{A. Bressan}, SIAM J. Control Optimization 23, 38-48 (1985; Zbl 0557.49011); the second author, J. Differ. Equations 31, 31-52 (1979; Zbl 0407.93010) and ibid. 25, 433-465 (1987)] and announce new ones for the time-optimal control problem for systems of type \[ \Sigma: \dot x=f(x)+g(x)u,\quad | u| \leq 1,\quad x\in R^ n,\quad n=2,3. \] The results are obtained by characterizing the local structure of time-optimal trajectories near a reference point p through the Lie bracket configuration at p of the family of vector fields generated by \(\Sigma\).
scientific article; zbMATH DE number 4036572
scientific article