Impulsive Control Problems Without the Frobenius Condition

In this chapter, the same dynamical system and problem as in the previous chapter are considered; however, the Frobenius condition may now be violated. It follows that the violation of the Frobenius condition implies that the constructed extension of the problem is not well posed, since the vector-valued Borel measure in this case may generate an entire integral funnel of various trajectories corresponding to the given dynamical control system with measure. Hence, a considerable expansion of the space of impulsive controls is required. Then, the impulsive control is no longer defined simply by the vector-valued measure, but is already a pair, that is: the vector-valued Borel measure plus the so-called attached family of controls of the conventional type associated with this measure. For this extension of a new type, the same strategy is applied as earlier. Namely, the existence theorem is established, Cauchy-like conditions for well-posedness are indicated, and the maximum principle is proved. The chapter ends with ten exercises.