International audienceThis paper reviews the relationships between switching systems defined from a partition of the state space into convex cells, and relay or complementarity dynamical systems, which are other classes of discontinuous systems. Well-posedness results (i.e. results on the existence and the uniqueness of solutions) for different classes of relay and complementarity systems which are also switching systems are presented. The reverse issue (when can a switching system be rewritten equivalently as a relay or a complementarity system) is also tackled. Many examples from Mechanics, Circuits, and Biology illustrate the developments throughout the paper. The paper focuses on systems with continuous solutions (i.e. with no state jumps). Convexity is the central property

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We consider the Mayer optimal control problem with dynamics given by a nonconvex differentialdifference inclusion, whose trajectories are constrained to a closed set. Necessary optimality conditions in the form of the maximum principle are obtained.

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(1) Background: Transposition of the great arteries (TGA) is the most common congenital heart disease, accounting for 5–7% of all cardiac anomalies, with a prevalence of 0.2–0.3 per 1000 live births. (2) Aim: Our main objectives were to evaluate the clinical safety of balloon atrial septostomy in neonates and the possible complications. Furthermore, we tried to establish whether the procedure should be performed in all TGA patients with small atrial septal defects, regardless of oxygen saturation, within a center where corrective surgery cannot be performed on an emergency basis due to the lack of a permanent cardiac surgery team for arterial switch surgery. (2) Methods: We conducted an observational, retrospective, single tertiary-care center study between January 2008 and April 2022, which included 92 neonates with TGA transferred to our institution for specialized treatment. (3) Results: The median age at the time of the Rashkind procedure was four days. The rate of immediate complications after balloon atrial septostomy (BAS) was high (34.3%), but most were transient (metabolic acidosis and arterial hypotension—21.8%). Twenty patients with TGA managed in our hospital underwent definitive and corrective surgical intervention (arterial switch operation) at a median age of 13 days. Most patients (82.6%) were term neonates, but 16 were born preterm. (4) Conclusions: Urgent balloon atrial septostomy is often the only solution to restore adequate systemic perfusion. Bedside balloon atrial septostomy is a safe, effective, and initial palliative intervention in neonates with TGA, which can be performed in the neonatal unit.