A Pontryagin Maximum Principle in Wasserstein Spaces for Constrained Optimal Control Problems

In this paper, we prove a Pontryagin Maximum Principle for constrained optimal control problems in the Wasserstein space of probability measures. The dynamics, is described by a transport equation with non-local velocities and is subject to end-point and running state constraints. Building on our previous work, we combine the classical method of needle-variations from geometric control theory and the metric differential structure of the Wasserstein spaces to obtain a maximum principle stated in the so-called Gamkrelidze form.Comment: 35 page

Lagrange multipliers theorem and saddle point optimality criteria in mathematical programming

We prove a version of Lagrange multipliers theorem for nonsmooth functionals defined on normed spaces. Applying these results, we extend some results about saddle point optimality criteria in mathematical programming

Openness of mappings

V práci studujeme zobecněné verze metrické regularity, například nelineární a směrová regularita. Rovněž studujeme podobné zobecnění metrické subregularity a semiregularity a odvozujeme postačují podmínky pro tyto vlastnosti v případě jednoznačných zobrazení v konečné dimenzi. Prvním cílem práce je definovat metrickou regularitu, metrickou subregularitu a metrickou semiregularity jednoznačných i mnohoznačných zobrazení. Formulujeme několik ekvivalentních vlastností a také uvedeme postačující i nutné podmínky pro jejich platnost. Dále se zabýváme stabilitou zmíněných vlastností vzhledem k jednoznačné i mnohoznačné perturbaci. Druhým cílem je poskytnout postačující podmínky pro směrovou semiregularitu a semiregularitu s vazbou jednoznačných zobrazení v konečné dimenzi založených na aproximaci lineárním zobrazením a svazkem lineárních zobrazení. Zaměříme se na výpočet modulů (semi)regularity lineárních zobrazení. Posledním cílem je zobecnit kritéria Ioffeho typu do kvazimetrických prostorů a tím získat kritéria pro nelineární a směrové verze uvedených vlastností.ObhájenoIn this thesis, we study criteria for generalized notions of metric regularity for single-valued and set-valued mappings, such as nonlinear and directional versions and the combination of both. We also study similar generalizations of metric subregularity and semiregularity and we focus on the criteria for constrained and directional semiregularity of single-valued mappings in finite dimensional spaces. The first aim of this thesis is to discuss metric regularity, metric subregularity, and metric semiregularity of both single-valued and set-valued mappings. Several equivalent properties are formulated and the sufficient as well as the necessary conditions are presented. Further, we discuss the stability of these properties with respect to single-valued and set-valued perturbations. The second aim is to provide sufficient conditions for directional and constrained semiregularity of single-valued mappings in finite dimensional spaces via an approximation by a linear mapping and by a bunch of linear mappings. We also focus on the computation of directional (semi)regularity modulus of linear mappings. The last aim is to extend Ioffe-type criteria to quasi-metric spaces and thus to achieve criteria for nonlinear and directional versions of the mentioned properties