Impulsive control systems with trajectories of bounded $p$-variation

The paper deals with impulse-trajectory relaxations of control-affine systems under the assumption that $L_1$-norms of the control functions are not uniformly bounded. Generalized solutions of such control systems may be of infinite total variation. Most of the known results related to impulse-trajectory relaxations of control-affine systems are mainly devoted to the case of state with bounded variation and controls of the type of bounded Borel measures and do not concern the considered case. The main issue of the study are constrictive techniques for impulse-trajectory relaxations within the class of functions of bounded $p$-variation ($p>1$) (in the sense N. Wiener) and explicit representations of relaxed systems.
Based on a sort of discontinuous time reparametrization, we propose a new approach to trajectory extension of systems with generalized solutions of bounded $p$-variation, $p>1$. This approach includes some space-time extension of the original system and its transformation to an auxiliary one with continuous solutions of bounded $p$-variation. In this paper, we dwell on the case of scalar control inputs, while a similar space-time transformation can be applied for control-affine systems with vector-valued inputs, even, in the absence of the involution property (or a more conventional commutativity assumption) of the vector fields.
For the case $p\in[1,2)$ and scalar control input, we obtain an explicit representation of the extended control system by a specific discrete-continuous integral equation involving Young integral.