This report examines optimal impulse control problems that arise as relaxation (impulse-trajectory) extensions of degenerate problems. The main focus is on the problem of describing generalized solutions for control systems with a right-hand side that is affine in the control, in two fundamentally different situations.

This report examines optimal impulse control problems that arise as relaxation (impulse-trajectory) extensions of degenerate problems. The main focus is on the problem of describing generalized solutions for control systems with a right-hand side that is affine in the control, in two fundamentally different situations.
1) The presence of a uniform constraint on the $L_1$-norm of the control. In this case, the generalized trajectories have bounded total variation and are naturally interpreted as the system's responses to impulse actions modeled by bounded Borel measures. Various approaches to the closure of the set of solutions in topologies weaker than the topology of uniform convergence are considered. In particular, the most well-known closure in the literature, the weak* topology in the space of functions of bounded variation, is discussed.
2) The absence of a uniform constraint on the $L_1$-norm of the control. For this case, partial results are known concerning the description of generalized solutions from the class of Lebesgue integrable functions and impulse controls defined by generalized functions (distributions). Such descriptions are possible under additional assumptions that ensure the correctness of the transition to impulse modes—for example, when the Frobenius-type commutativity condition is met for the vector fields corresponding to the columns of the matrix under control. However, a complete description of generalized solutions in the general case has not yet been obtained.
The use of functions of bounded p-variation (p>1) (in the sense of N. Wiener) is considered a promising direction for an explicit description of generalized solutions. The report will consider a certain connection between the theory of impulse control and the modern theory of dynamical systems controlled by signals of low regularity (developed within the framework of the theory of rough paths, initiated by Terry Lyons in 1994).