A viability theorem for morphological inclusions

The aim of this paper is to adapt the Viability Theorem from differential inclusions (governing the evolution of vectors in a finite dimensional space) to so-called morphological inclusions (governing the evolution of nonempty compact subsets of the Euclidean space). In this morphological framework, the evolution of compact subsets of the Euclidean space is described by means of flows along bounded Lipschitz vector fields (similarly to the velocity method alias speed method in shape analysis). Now for each compact subset, more than just one vector field is admitted - correspondingly to the set-valued map of a differential inclusion in finite dimensions. We specify sufficient conditions on the given data such that for every initial compact set, at least one of these compact-valued evolutions satisfies fixed state constraints in addition. The proofs follow an approximative track similar to the standard approach for differential inclusions in finite dimensions, but they use tools about weak compactness and weak convergence of Banach-valued functions. Finally an application to shape optimization under state constraints is sketched

Shape evolutions under state constraints: A viability theorem

AbstractThe aim of this paper is to adapt the Viability Theorem from differential inclusions (governing the evolution of vectors in a finite-dimensional space) to so-called morphological inclusions (governing the evolution of nonempty compact subsets of the Euclidean space).In this morphological framework, the evolution of compact subsets of RN is described by means of flows along differential inclusions with bounded and Lipschitz continuous right-hand side. This approach is a generalization of using flows along bounded Lipschitz vector fields introduced in the so-called velocity method alias speed method in shape analysis.Now for each compact subset, more than just one differential inclusion is admitted for prescribing the future evolution (up to first order)—correspondingly to the step from ordinary differential equations to differential inclusions for vectors in the Euclidean space.We specify sufficient conditions on the given data such that for every initial compact set, at least one of these compact-valued evolutions satisfies fixed state constraints in addition. The proofs follow an approximative track similar to the standard approach for differential inclusions in RN, but they use tools about weak compactness and weak convergence of Banach-valued functions. Finally the viability condition is applied to constraints of nonempty intersection and inclusion, respectively, in regard to a fixed closed set M⊂RN

Control problems for nonlocal set evolutions

In this paper, we extend fundamental notions of control theory to evolving compact subsets of the Euclidean space. Dispensing with any restriction of regularity, shapes can be interpreted as nonempty compact subsets of the Euclidean space. Their family, however, does not have any obvious linear structure, but in combination with the popular Pompeiu-Hausdorff distance, it is a metric space. Here Aubin's framework of morphological equations is used for extending ordinary differential equations beyond vector spaces, namely to the metric space of nonempty compact subsets of the Euclidean space supplied with Pompeiu-Hausdorff distance. Now various control problems are formulated for compact sets depending on time: open-loop, relaxed and closed-loop control problems – each of them with state constraints. Using the close relation to morphological inclusions with state constraints, we specify sufficient conditions for the existence of compact-valued solutions

Nonsmooth shape evolutions under state constraints: A viability theorem

The aim of this paper is to adapt the Viability Theorem from differential inclusions (governing the evolution of vectors in a finite dimensional space) to so-called morphological inclusions (governing the evolution of nonempty compact subsets of the Euclidean space). In this morphological framework, the evolution of compact subsets of the Euclidean space is described by means of flows along differential inclusions with bounded and Lipschitz continuous right-hand side. This approach is a generalization of using flows along bounded Lipschitz vector fields introduced in the so-called velocity method alias speed method in shape analysis. Now for each compact subset, more than just one differential inclusion is admitted for prescribing the future evolution (up to first order) - correspondingly to the step from ordinary differential equations to differential inclusions for vectors in the Euclidean space. We specify sufficient conditions on the given data such that for every initial compact set, at least one of these compact-valued evolutions satisfies fixed state constraints in addition. The proofs follow an approximative track similar to the standard approach for differential inclusions in the Euclidean space, but they use tools about weak compactness and weak convergence of Banach-valued functions. Finally the viability condition is applied to constraints of nonempty intersection and inclusion, respectively, in regard to a fixed closed set M

Mutational Analysis

This monograph extends the classical concept of ordinary differential equations in the Euclidean space to nonempty sets which are just supplied with a family of "continuous" distance functions. In particular, these sets are not supposed to have any linear structure or to be metric spaces. The main goal is a joint framework for continuous dynamical systems beyond the traditional border of vector spaces so that examples of completely different origins can be coupled in systems. It is motivated by the mutational equations introduced by Jean-Pierre Aubin in the 1990s. Some of the examples discussed here are: nonlocal set evolutions, semilinear evolution equations, nonlinear transport equations for finite Radon measures, functional stochastic differential equations, parabolic differential equations in noncylindrical domains. This monograph is my revised Habilitationsschrift (i.e. thesis for a postdoctoral lecture qualification in Germany) submitted to the Faculty of Mathematics and Computer Science at Heidelberg University in January 2009