We prove that every separable Banach space containing an isomorphic copy of $\ell _{1}$ can be equivalently renormed so that the new bidual norm is octahedral. This answers, in the separable case, a question in Godefroy [Metric characterization of first Baire class linear forms and octahedral norms, Studia Math. 95 (1989), 1–15]. As a direct consequence, we obtain that every dual Banach space, with a separable predual and failing to be strongly regular, can be equivalently renormed with a dual norm to satisfy the strong diameter two property.

The paper deals with an optimal control problem with a scalar first-order state constraint and a scalar control.In presence of (nonessential) touch points,the arc structure of the trajectory is not stable.Under some reasonable assumptions,we show that boundary arcs are structurally stable, and that touch point can either remain so, vanish or be transformed into a single boundary arc. Assuming a weak second-order optimality condition (equivalent to uniform quadratic growth), stability and sensitivity results are given. The main tools are the study of a quadratic tangent problem and the notion of strong regularity.Those results enable us to design a new continuation algorithm,presented at the end of the paper, that handles automatically changes in the structure of the trajectory.

The (F, dn) method is investigated with respect to perfectness, strong regularity, and summing bounded divergent sequences. In the process the columns of the inverse matrix are characterized in terms of {dn}.