Investigation of second-order optimality conditions for impulsive control problems under the Frobenius condition

In this work impulsive control problems are investigated in that case in which matrix G that appears in the dynamics multiplying vector-valued control measure μ depends on the state variable, that is, G = G(x, t). The solution concept and the extension procedure in this non-linear case are not as trivial as in the case G = G (t). The key-point is to ensure robustness of the impulsive control system w.r.t. the control measure and regarding the approximations in the weak-∗ topology ('w.r.t.' stands for 'with respect to' here and further). Note that such approximations are required by applications. But this type of robustness is generally lost unless some extra assumptions on the matrix G w.r.t. the x-variable are imposed. It turns out that the weakest possible assumption, that still meets the robustness property, is the so-called Frobenius condition presented and discussed below. Under the Frobenius condition and without a priori regularity assumptions, we derive second-order necessary optimality conditions in a new form. This form and relations with previous results are discussed.