On a Class of Impulsive Control Problems for Continuity Equations

The talk presents a class of singular control problems for the continuity equation driven by a control-affine vector fields subject to a constraint on the L-1-norm of control inputs, ranged in the whole space. Solutions of such distributed systems may occur to be arbitrary close (in a certain natural sense) to discontinuous measure -valued functions, and as a consequence related extremal problems are generically ill-posed. In connection with the addressed model, we discuss the following control-theoretical issues: i) relaxation of the tube of solutions in an appropriate coarse topology; ii) representation of generalized (discontinuous in time) solutions in terms of continuous arcs through a discontinuous time reparameterization of the characteristic ordinary differential equation, and iii) a constructive formula for generalized solutions. For the relaxed model, we state an optimal impulsive ensemble control problem and ensure the existence of a minimizer. Finally, we elaborate a conceptual numeric technique for optimal control and exhibit a case study. (C) 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.