Impulsive Control Problems Under the Frobenius Condition

In this chapter, the matrix-multiplier G for the impulsive control is enriched with a dependence on the state variable x. This naturally leads to some ambiguity in the choice of the state trajectory, since it is assumed that the state trajectory may exhibit jumps. Therefore, generally speaking, different types of integral w.r.t. measure will lead us to different solution concepts. Herein, we have settled on the type of integration which implies the stability of the solution w.r.t. approximations by absolutely continuous measures. The uniqueness and stability of the solution in this case are guaranteed by the well-known Frobenius condition. The extension of the original problem is treated w.r.t. this type of solution which is stable in the weak-* topology. The main result of this chapter is the second-order necessary conditions of optimality without a priori assumptions of normality, which are obtained under the assumption that the Frobenius condition for the columns of matrix G is satisfied. The chapter ends with 11 exercises.