The Japanese city presents a certain number of peculiarities in the organization of its physical space (weak zoning regulations, fast piecemeal destruction/reconstruction of buildings and blocks, high compacity, incremental reorganization). Compared to countries where urban fabrics are more perennial and easily distinguishable (old centers, modern planned projects, residential areas, etc.), in Japanese metropolitan areas we often observe higher heterogeneity and more complex spatial patterns. Even within such a model, it should be possible to recognize the internal organization of the physical city. The aim of this paper is thus to study the spatial structure of the contemporary Japanese city, generalizing on the case study of Osaka and Kobe. In order to achieve this goal, we will need to identify urban forms at different local scales (building types, urban fabrics) and to analyze them at a wider scale to delineate morphological regions and their structuring of the overall layout of the contemporary Japanese city. Several analytical protocols are used together with field observations and literature. The results, and more particularly the building and urban fabrics types and their location within the Osaka-Kobe metropolitan area, are interpreted in the light of Japanese history and model of urbanization. A synoptic graphical model of an urban morphological structure based upon Osaka is produced and proposed as an interpretative pattern for the Japanese metropolitan city in general.

We consider a constrained optimal control problem and an extension of it, in which the set of strict-sense trajectories is enlarged. Extension is a common procedure in optimal control, used to derive necessary and sufficient optimality conditions for the original problem from the extended one, which usually admits a minimizer and has a more regular structure. However, this procedure fails if the two problems have different infima. It is therefore relevant to identify such situations. Following on from earlier work by Warga but adopting perturbation techniques developed in non-smooth analysis, we investigate the relation between the occurrence of an infimum gap and abnormality of necessary conditions. For a notion of local minimizer based on control distance and an extension including the impulsive one, we prove that (i) a local extended minimizer which is not a local minimizer of the original problem and (ii) a local strict-sense minimizer which is not a local minimizer of the extended problem, both satisfy the extended maximum principle in abnormal form. The main novelty is result (ii), as until now it had only been shown that a strict-sense minimizer which is not an extended minimizer is abnormal for an `averaged version’ of the maximum principle.