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Since the Fifties of the last Century, Control Theory has established itself as a powerful mathematical discipline, particularly fit for applications in various applied research fields, including advanced engineering design, economics, medical sciences and population biology. However, Control Theory is not only limited to a powerful tool for a growing number of applications. In fact, since the emerging of this discipline, the need of rethinking and extending fields such as Calculus of Variations, Differential Geometry, Non smooth Analysis is permanently parallel to investigations devoted to applications. Nowadays Control Theory qualifies as a rich source of basic abstract problems and at the same time, an important viewpoint to investigate purely mathematical issues. This volume collects some recent results, highlighting geometrical and analytical aspects and the connections between them. Applications are in the background and sometimes explicitly treated, in the classical spirit of an indivisible, mutual, interference occurring between abstraction and problem-solving practice.
List of contents
1 Generalized Lax-Hopf formulas for Cournot Maps and Hamilton-Jacobi-McKendrick Equations, Jean-Pierre Aubin and Chen Luxi.- 2 A geometric approach to the optimal control of nonholonomic mechanical systems, Anthony Bloch, Leonardo Colombo, Rohit Gupta and David Martin de Diego.- 3 Lunar perturbation of the metric associated to the averaged orbital transfer, Bernard Bonnard, Helen Henninger and Jeremy Rouot.- 4 Conjugate times and regularity of the minimum time function with differential inclusions, Piermarco Cannarsa and Teresa Scarinci.- 5 Weak solutions for first order mean field games with local coupling, Pierre Cardaliaguet.- 6 omega-limit sets for differential inclusions, Asen L. Dontchev, Mikhail I. Krastanov and Vladimir M. Veliov.- 7 Second-Order Necessary Optimality Conditions for the Mayer Problem Subject to a General Control Constraint, Helene Frankowska and Nikolai P. Osmolovskii.- 8 Optimal Control of Cancer Treatments: Mathematical Models for the Tumor Microenvironment, Heinz Schaettler and Urszula Ledzewicz.
Summary
Since the 1950s control theory has established itself as a major mathematical discipline, particularly suitable for application in a number of research fields, including advanced engineering design, economics and the medical sciences. However, since its emergence, there has been a need to rethink and extend fields such as calculus of variations, differential geometry and nonsmooth analysis, which are closely tied to research on applications. Today control theory is a rich source of basic abstract problems arising from applications, and provides an important frame of reference for investigating purely mathematical issues. In many fields of mathematics, the huge and growing scope of activity has been accompanied by fragmentation into a multitude of narrow specialties. However, outstanding advances are often the result of the quest for unifying themes and a synthesis of different approaches. Control theory and its applications are no exception. Here, the interaction between analysis and geometry has played a crucial role in the evolution of the field. This book collects some recent results, highlighting geometrical and analytical aspects and the possible connections between them. Applications provide the background, in the classical spirit of mutual interplay between abstract theory and problem-solving practice.
