On “discontinuous” continuity equation and impulsive ensemble control

The note is devoted to a class of singular ensemble control problems: we consider the continuity equation driven by a control-affine vector field subject to the constraint on the L1-norm of the input function. Since no geometric constraints on control are imposed, solutions to the continuity equation can, potentially, tend somehow to discontinuous measure-valued functions. We elaborate a representation of such discontinuous, generalized solutions through an ordinary control equation being a time reparameterization of the characteristic ordinary differential equation (ODE) of the original continuity equation. By virtue of standard results from impulsive control theory of ODEs, we perform a constructive formula for generalized solutions in terms of a measure differential equation. Finally, we state an optimal impulsive control problem for the singular continuity equation, and prove the existence of its solution. The obtained results can be useful for modeling and analysis of impulsive control problems with uncertain initial data.