Second order optimality conditions for piecewise regular extremals in Optimal Control
We study second-order optimality conditions for optimal control problems with integral cost. We consider extremals obtained by concatenating finitely many regular arcs and prove both a necessary condition for weak local optimality and a sufficient condition for strong local optimality within this class. The key object is the Jacobi curve, a curve of Lagrangian subspaces encoding the second variation along the reference extremal. In the regular case, this curve is smooth and can be identified with the tangent spaces to a suitable field of extremals. For piecewise regular extremals, the field of extremals is only piecewise smooth, and the associated Jacobi curve has discontinuities at the switching times. This makes the relation between vertical intersections, conjugate points, and optimality more delicate. We develop this discontinuous Jacobi-curve framework and show how it yields effective second-order tests. The case of one-dimensional free-control systems is studied in detail as an illustrative class in which piecewise regular extremals arise naturally.