Calculus of variations on time scales and applications to economics

We consider some problems of the calculus of variations on time scales. On the beginning our attention is paid on two inverse extremal problems on arbitrary time scales. Firstly, using the Euler-Lagrange equation and the strengthened Legendre condition, we derive a general form for a variation functional that attains a local minimum at a given point of the vector space. Furthermore, we prove a necessary condition for a dynamic integro-differential equation to be an Euler-Lagrange equation. New and interesting results for the discrete and quantum calculus are obtained as particular cases. Afterwards, we prove Euler-Lagrange type equations and transversality conditions for generalized infinite horizon problems. Next we investigate the composition of a certain scalar function with delta and nabla integrals of a vector valued field. Euler-Lagrange equations in integral form, transversality conditions, and necessary optimality conditions for isoperimetric problems, on an arbitrary time scale, are proved. In the end, two main issues of application of time scales in economic, with interesting results, are presented. In the former case we consider a firm that wants to program its production and investment policies to reach a given production rate and to maximize its future market competitiveness. The model which describes firm activities is studied in two different ways: using classical discretizations; and applying discrete versions of our result on time scales. In the end we compare the cost functional values obtained from those two approaches. The latter problem is more complex and relates to rate of inflation, p, and rate of unemployment, u, which inflict a social loss. Using known relations between p, u, and the expected rate of inflation π, we rewrite the social loss function as a function of π. We present this model in the time scale framework and find an optimal path π that minimizes the total social loss over a given time interval.

Integrated coastal geoengineering approach for maritime environments

This dissertation introduces several methodological approaches which integrate a proposed coastal management model in an interdisciplinary perspective. The research presented herein is displayed as a set of publications comprising different thematic outlooks. The thesis develops an integrated coastal geoengineering approach which is intrinsically linked to the studied maritime environments. From sandy coasts and marine works to rocky platforms and sea cliffs, this study includes field work between Caminha – Figueira da Foz (NW Portugal) and Galicia (NW Spain). The research also involves an analysis and geological-geotechnical characterisation of natural rock (armourstone) and artificial units (concrete blocks) applied to coastal structures. The main goal is to contribute to the characterisation and re-evaluation of georesources and to determine armourstone suitability and availability from its source (quarry). It was also important to diagnose the geomaterials in situ concerning their degradation/deterioration level on the basis of the current status of the coastal protection works in order to facilitate more efficient monitoring and maintenance, with economic benefits. In the rocky coast approach the coastal blocks were studied along the platform, but also the geoforms were studied from a coastal morphodynamics point of view. A shoreline evolution analysis was developed for sandy coasts through Digital Shoreline Analysis System (DSAS) extension. In addition, the spatial and statistical analysis applied to sea cliffs allowed the establishment of susceptibility zones to erosion and hazardous areas. All of these studies have different purposes and results however, there is a common denominator – GIS mapping. Hence, apart from the studied coastal environment, there is an integrated system which includes a sequence of procedures and methodologies that persisted during the research period. This is a step forward in the study of different coastal environments by using almost the same methodologies. This will allow the characterisation, monitoring and assessment of coastal protection works, rocky coasts, and shore platforms. With such data, it is possible to propose or recommend strategies for coastal and shoreline management based on several justifications in terms of social, economic, and environmental questions, or even provide a GIS-based planning support system reinforced by geocartographic decisions. Overall the development of the applied cartography embraces six stages which will allow the production of detailed maps of the maritime environment: (1) high-resolution aerial imagery surveys; (2) visual inspection and systematic monitoring; (3) applied field datasheet; (4) in situ evaluation; (5) scanline surveying; and (6) GIS mapping. This thesis covers fundamental matters that were developed over the course of scientific publication and as a consequence they represent the results obtained and discussed. The subjects directly related to the thesis architecture are: (i) cartography applied to coastal dynamics (including an art historical analysis as a tool to comprehend the coastal evolution and the littoral zone); (ii) georesources assessment (the role of cartography in georesources zoning, assessment and armourstone durability); (iii) coastal geoengineering applications and monitoring (Espinho pilot site in NW Portugal as an experimental field); (iv) rocky coast and shore platform studies and characterisation; (v) sandy and mixed environment approaches; (vi) coastal geosciences GIS mapping and photogrammetric surveying (coastal geoengineering); and (vii) shoreline change mapping and coastal management strategies (the CartGalicia Project as an example – NW Spain). Finally, all of these thematic areas were crucial to generate the conceptual models proposed and to shape the future of integrated coastal coastal geoengineering management.