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EN
Sustainability is an issue of paramount importance, as scientists and politicians seek to understand what it means, practically and conceptually, to be sustainable. This paper’s aim is to introduce viability theory, a relatively young branch of mathematics which provides a conceptual framework that is very well suited to such problems. Viability theory can be used to answer important questions about the sustainability of systems, including those studied in macroeconomics, and can be used to determine sustainable policies for their management. The principal analytical tool of viability theory is the viability kernel which describes the set of all state-space points in a constrained system starting from which it is possible to remain within the system’s constraints indefinitely. Although, in some circumstances, kernel determination can be performed analytically, most practical results in viability theory rely on graphical approximations of viability kernels, which for nonlinear and high-dimensional problems can only be approached numerically. This paper provides an outline of the core concepts of viability theory and an overview of the numerical approaches available for computing approximate viability kernels. VIKAASA, a specialised software application developed by the authors and designed to compute such approximate viability kernels is presented along-side examples of viability theory in action in the spheres of bio-economics and macroeconomics.
PL
Zrównoważony rozwój jest terminem często używanym lecz naukowcy i politycy nie są zgodni ani co do jego znaczenia, ani jak praktycznie i teoretycznie zapewnić taki rozwój. Niniejsza praca ma na celu wprowadzenie czytelnika do teorii wiabilności30, stosunkowo młodej gałęzi matematyki ciągłej, której narzędzia nadają się do opisu problemów zrównoważonego rozwoju. W szczególności, teoria wiabilności może być wykorzystana do określania strategii zrównoważonego rozwoju systemów ekonomicznych, w tym makroekonomicznych. Głównym narzędziem analitycznym teorii wiabilności jest jądro wiabilności, którym jest zbiór wszystkich punktów przestrzeni stanu, z jakich mogą się dokonać ewolucje systemu, które nigdy nie przekroczą zadanych z góry ograniczeń. Chociaż w pewnych okolicznościach opis jądra może być otrzymany analitycznie, większość praktycznych rezultatów w teorii wiabilności uzyskuje się przez analizę graficznych przybliżeń jąder wiabilności, które w przypadku nieliniowych i wysokowymiarowych problemów mogą być uzyskane jedynie drogą obliczeniową. Niniejsza praca przedstawia podstawowe pojęcia teorii wiabilności oraz przegląd dostępnych metod numerycznych do obliczania przybliżeń jąder. VIKAASA, specjalistyczne oprogramowanie opracowane przez autorów, umożliwia otrzymywanie takich przybliżeń. W pracy, użycie VIKAASY jest zilustrowane przykładami z zakresu bio- i makroekonomii.
EN
In this paper, we discuss the existence of solutions for a four-point integral boundary value problem of n-th order differential inclusions involving convex and non-convex multivalued maps. The existence results are obtained by applying the nonlinear alternative of Leray Schauder type and some suitable theorems of fixed point theory.
3
Content available remote On sup-measurability of multifunctions with some density properties
EN
The paper is concerned with sup measurability of a multifunction F defined on the product (…) of metric spaces with some differentiation bases. We introduce the lower (…) property and the upper (…) property of multifunction, where (…), and we prove sup measurabilty of F when it has the upper (…) property at (x, y) and F(x, ź) has the lower (…) property at y for every (…). Some application of this theorem to the existence of solutions of differential inclusions (…) is given.
4
Content available remote On the theorem of Filippov-Pliś and some applications
EN
In the paper some known and new extensions of the famous theorem of Filippov (1967) and a theorem of Plis (1965) for differential inclusions are presented. We replace the Lipschitz condition on the set-valued map in the right-hand side by a weaker onesided Lipschitz (OSL), one-sided Kamke (OSK) or a continuity-like condition (CLC). We prove new Filippov-type theorems for singularly perturbed and evolution inclusions with OSL right-hand sides. In the CLC case we obtain two extended theorems, one of which implies directly the relaxation theorem. We obtain also a theorem in Banach spaces for OSK multifunctions. Some applications to exponential formulae are surveyed.
EN
We consider the problem [formula] in a Banach space E, where belongs to the Banach space, CE([-d, 0]), of all continuous functions from [-d, 0] into E. A multifunction F from [0, b] × CE([-d, 0]) into the set, Pfc(E), of all nonempty closed convex subsets of E is weakly sequentially hemi-continuous, tx(s) = x(t + s) for all s 2 [-d, 0] and {A(t) : 0 6 t 6 b} is a family of densely defined closed linear operators generating a continuous evolution operator S(t, s). Under a generalization of the compactness assumptions, we prove an existence result and give some topological properties of our solution sets that generalizes earlier theorems by Papageorgiou, Rolewicz, Deimling, Frankowska and Cichon..
EN
The problem of existence of adjoint functions to boundary solutions is considered – it depends on the geometry of the attained set at the end point. This is applied to prove the smoothness of boundary solutions in the case of strictly convex right-hand side of di.erential inclusion which in turn permitts to show the smoothness of barrier solutions on semipermeable surfaces.
7
Content available remote Autonomous differential inclusion sharing the families of trajectories
EN
We give a sufficient condition for equality of sets of trajectories of two differential inclusions with right-hand sides Borel measurable with respect to the state variable, not necessarily bounded and possibly containing the origin.
8
Content available remote A deterministic approach to the Skorokhod problem
EN
We prove an existence and uniqueness result for the solutions to the Skorokhod problem on uniformly prox-regular sets through a deterministic approach. This result can be applied in order to investigate some regularity properties of the value function for differential games with reflection on the boundary.
EN
This paper develops the theory of solution tubes to differential inclusions (uncertain systems) within a prescribed collection of sets. The notion is defined as a minimal invariant tube with values in the collection. Under certain requirements for the collection we prove existence and Lipschitz-like stability of the solution tubes. The theory is relevant to problems of systems estimation in the context of control or differential games.
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