In this article, we present a new strategy to determine an unmanned aerial vehicle trajectory that minimizes its flight time in presence of avoidance areas and obstacles. The method combines classical results from optimal control theory, i.e. the Euler-Lagrange Theorem and the Pontryagin Minimum Principle, with a continuation technique that dynamically adapts the solution curve to the presence of obstacles. We initially consider the two-dimensional path planning problem and then move to the three-dimensional one, and include numerical illustrations for both cases to show the efficiency of our approach.

We describe immune-proteome structures using libraries of protein fragments that define a structural immunological alphabet. We propose and validate such an alphabet as i) composed of letters of five consecutive amino acids, pentapeptide units being sufficient minimal antigenic determinants in a protein, and ii) characterized by low-similarity to human proteins, so representing structures unknown to the host and potentially able to evoke an immune response. In this context, we have thoroughly sifted through the entire human proteome searching for non-redundant protein motifs. Here, for the first time, a complete sequence redundancy dissection of the human proteome has been conducted. The non-redundant peptide islands in the human proteome have been quantified and catalogued according to the amino acid length. The library of uniquely occurring n-peptide sequences that was obtained is characterized by a logarithmic decrease of the number of non-redundant peptides as a function of the peptide length. This library represents a highly specific catalogue of molecular protein signatures, the possible use of which in cancer/autoimmunity research is discussed, with a major focus on non-redundant dodecamer sequences.