On singular arcs in nonsmooth optimal control

In this paper we consider general optimal control problems (OCP) which are characterized by a nonsmooth ordinary state differential equation. However, we allow only mild types of nonsmoothness. More precisely, we assume that the right-hand side of the state equation is piecewise smooth and that the switching points, which separate these pieces, are determined as points, where a state-and possibly control-dependent (smooth) switching function changes sign. For this kind of optimal control problems necessary optimality conditions are developed. Attention is paid to the situation when the switching function vanishes identically along a nontrivial subarc. Such subarcs, which we call singular state subarcs, are investigated with respect to necessary conditions and to junction conditions. In extension to earlier results of the authors, Oberle and Rosendhal (2006), in this paper nonsmooth OCPs are considered with respect to the order of the switching function. Especially, the case of a zero-order switching function is included and examples of order zero, one and two are treated.