Second-Order Necessary Conditions for Optimal Control Problems with Fixed Terminal Time and Free Terminal State

In this paper, we focus on piecewise continuous control strategies and derive second-order necessary conditions for optimal control problems with a fixed terminal time and free terminal state under the sole additional assumption that the state dynamics and the cost functional are two-times continuously differentiable with respect to the state variable. In analogy to nonlinear programming problems with two-times continuously differentiable objective functions, these conditions can reduce the set of control strategies which need to be investigated when searching for the optimal control strategy. This is especially true for problems with bang-bang extremals where our derived results can be shown to be consistent with those in literature for control-affine systems.