On a switching control problem with càdlàg costs

This work addresses a switching control problem under which the cost associated with changes of regimes is allowed to have discontinuities in time. Our main contribution show several characterizations optimal function as well existence ε-optimal policies. As by-product, we also study and uniqueness solutions system backward stochastic differential equations whose barriers (or obstacles) are discontinuous (in fact càdlàg type) depend itself on unknown solution. In last part paper, case when an underlying diffusion dynamic system. this special case, payoff becomes weak solution HJB PDEs obstacles quasi-variational type. paper somehow continuation papers [B. Djehiche, S. Hamadène, A. Popier, A finite horizon multiple problem, SIAM J. Control Optim. 48 (2009), pp. 2751–2770; Hamadène M.A. Morlais, Viscosity systems interconnected Obstacle Appl. Math. 67 (2013), 163–196.] that consider continuous costs.