Optimal control is an important branch of applied mathematics with numerous applications in appliedsciences, including macroeconomics and ambiental studies, engineering and operations research. It can beseen as an extension of calculus of variations or as an infinite dimensional optimization, where the objective isto find an optimal control law over a system that satisfies a certain extremality criterion.

This 4--year bilateral project between TU Wien (headed by Aris Daniilidis) and University of Brest (headed byMarc Quincampoix), the team of which includes two researchers from INSA-Rennes, is inscribed in this rich area of mathematics and emanates from the working assumptions that Pontryagin maps of nonconvex control systems enjoy some metric regularity property and that abstract descend moduli relate to Hamilton--Jacobi Equations.

Ingrained by the complementary competences of the involved teams, which is reinforced by the additional value of a strong synergy among the principal investigators (Aris Daniilidis, Marc Quincampoix and Olivier Ley), already witnessed by previous collaborations, the project proposes significant advances with respect to the current state of the art: it aspires stability results setting-up a strong mathematical framework that allow the efficient implementation of numerical schemes in nonconvex systems. These schemes are based on indirect methods and techniques from hierarchical optimization. In parallel, it endeavors to explore control problems of infinite horizon, putting forward a rigorous study of alternative ways to average the running cost up to infinity as well as to determine the value function of the control problems in the non-ergodic case, where classical results from the weak KAM theory typically fail to apply. Last, but not least, the project aspires to build a bridge between the viscosity theory of Hamilton-Jacobi equations and the moduli of abstract descent, a recently developed scheme that encompasses the De Giorgi local slope, the global slope and the average slope operators. This latter is motivated from recent developments on function determination and seems to relate to uniqueness issues of the value function and the weak KAM theory.

This proposal is both of theoretical and applied nature and involves the joint supervision of two PhD students and one post-doc.