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1
Method of averaging for the system of functional-differential inclusions
100%
Janiak T. , Łuczak-Kumorek E.
|
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nr 2
137-151
EN
The basic idea of this paper is to give the existence theorem and the method of averaging for the system of functional-differential inclusions of the form ⎧$ẋ(t) ∈ F(t,x_t,y_t)$ (0) ⎨ ⎩$ẋ(t) ∈ G(t,x_t,y_t)$ (1)
2
Lower semicontinuous differential inclusions
100%
Donchev T.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
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1998
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tom 18
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nr 1-2
19-25
EN
In the paper we consider lower semicontinuous differential inclusions with one sided Lipschitz and compact valued right hand side in a Banach space with uniformly convex dual. We examine the nonemptiness and some qualitative properties of the solution set.
3
Content available remote Existence and Topological Properties of Solution Sets for Differential Inclusions with Delay
80%
Gomaa A. M.
Commentationes Mathematicae
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2008
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tom Vol. 48, [Z] 1
45-58
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..
4
Content available remote Local analysis of hybrid systems on polyhedral sets with state-dependent switching
80%
Leth J. , Wisniewski R.
International Journal of Applied Mathematics and Computer Science
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2014
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tom 24
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nr 2
341-355
EN
This paper deals with stability analysis of hybrid systems. Various stability concepts related to hybrid systems are introduced. The paper advocates a local analysis. It involves the equivalence relation generated by reset maps of a hybrid system. To establish a tangible method for stability analysis, we introduce the notion of a chart, which locally reduces the complexity of the hybrid system. In a chart, a hybrid system is particularly simple and can be analyzed with the use of methods borrowed from the theory of differential inclusions. Thus, the main contribution of this paper is to show how stability of a hybrid system can be reduced to a specialization of the well established stability theory of differential inclusions. A number of examples illustrate the concepts introduced in the paper.
5
Content available remote Forward invariant sets, homogeneity and small-time local controllability
80%
Krastanov M.
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tom 32
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nr 1
287-300
EN
The property of forward invariance of a subset of $R^n$ with respect to a differential inclusion is characterized by using the notion of a perpendicular to a set. The obtained results are applied for investigating the dependence of the small-time local controllability of a homogeneous control system on parameters.
6
Some remarks on boundary value problem for differential inclusions
80%
Kisielewicz M.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
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1997
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tom 17
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nr 1-2
43-49
EN
Some sufficient conditions for the existence of solutions to boundary value problem for differential inclusions are given.
7
Co-density and other properties of the solution set of differential inclusions with noncompact right-hand side
80%
Goncharov V.
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nr 2
103-120
EN
There are studied two classes of differential inclusions with right-hand side admitting noncompact values in a Banach space. Co-density, lower semicontinuity in initial point and relaxation property of the solution set have been obtained.
8
Content available A constructive method for solving stabilization problems
80%
Azhmyakov V.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
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2000
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tom 20
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nr 1
51-62
EN
The problem of asymptotic stabilization for a class of differential inclusions is considered. The problem of choosing the Lyapunov functions from the parametric class of polynomials for differential inclusions is reduced to that of searching saddle points of a suitable function. A numerical algorithm is used for this purpose. All the results thus obtained can be extended to cover the discrete systems described by difference inclusions.
9
Content available On differential equations and inclusions with mean derivatives on a compact manifold
80%
Azarina S. , Gliklikh Y.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
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2007
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tom 27
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nr 2
385-397
EN
We introduce and investigate a new sort of stochastic differential inclusions on manifolds, given in terms of mean derivatives of a stochastic process, introduced by Nelson for the needs of the so called stochastic mechanics. This class of stochastic inclusions is ideologically the closest one to ordinary differential inclusions. For inclusions with forward mean derivatives on manifolds we prove some results on the existence of solutions.
10
Content available On the existence of viable solutions for a class of second order differential inclusions
80%
Cernea A.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
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2002
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tom 22
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nr 1
67-78
EN
We prove the existence of viable solutions to the Cauchy problem x'' ∈ F(x,x'), x(0) = x₀, x'(0) = y₀, where F is a set-valued map defined on a locally compact set $M ⊂ R^{2n}$, contained in the Fréchet subdifferential of a ϕ-convex function of order two.
11
Content available Controllability theorem for nonlinear dynamical systems
80%
Kisielewicz M.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
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2002
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tom 22
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nr 2
225-232
EN
Some sufficient conditions for controllability of nonlinear systems described by differential equation ẋ = f(t,x(t),u(t)) are given.
12
Content available Boundary value problems for n-th order differential inclusions with four-point integral boundary conditions
70%
Ahmad B. , Ntouyas S. K.
Opuscula Mathematica
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2012
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tom Vol. 32, no. 2
205-226
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.
13
Content available remote On sup-measurability of multifunctions with some density properties
70%
Kwiecińska G.
Demonstratio Mathematica
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2012
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tom Vol. 45, nr 4
821-833
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.
14
Content available remote A deterministic approach to the Skorokhod problem
70%
Bettiol P.
Control and Cybernetics
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2006
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tom Vol. 35, no 4
787-802
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.
15
Content available A Tikhonov-type theorem for abstract parabolic differential inclusions in Banach spaces
70%
Gudovich A. , Kamenski M. , Nistri P.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
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2001
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tom 21
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nr 2
207-234
EN
We consider a class of singularly perturbed systems of semilinear parabolic differential inclusions in infinite dimensional spaces. For such a class we prove a Tikhonov-type theorem for a suitably defined subset of the set of all solutions for ε ≥ 0, where ε is the perturbation parameter. Specifically, assuming the existence of a Lipschitz selector of the involved multivalued maps we can define a nonempty subset $Z_L(ε)$ of the solution set of the singularly perturbed system. This subset is the set of the Hölder continuous solutions defined in [0,d], d > 0 with prescribed exponent and constant L. We show that $Z_L(ε)$ is uppersemicontinuous at ε = 0 in the C[0,d]×C[δ,d] topology for any δ ∈ (0,d].
16
Content available A study of second order differential inclusions with four-point integral boundary conditions
70%
Ahmad B. , Ntouyas S.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
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2011
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tom 31
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nr 2
137-156
EN
In this paper, we discuss the existence of solutions for a four-point integral boundary value problem of second 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.
17
Content available On differential inclusions of velocity hodograph type with Carathéodory conditions on Riemannian manifolds
70%
Gliklikh Y. , Obukhovski A.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
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2004
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tom 24
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nr 1
41-48
EN
We investigate velocity hodograph inclusions for the case of right-hand sides satisfying upper Carathéodory conditions. As an application we obtain an existence theorem for a boundary value problem for second-order differential inclusions on complete Riemannian manifolds with Carathéodory right-hand sides.
18
Content available remote On the theorem of Filippov-Pliś and some applications
70%
Donchev T. , Farkhi E.
Control and Cybernetics
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2009
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tom Vol. 38, no 4A
1251-1271
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.
19
Mathematical model and optimal control of flow induced vibration of pipelines
60%
Ahmed N.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
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1999
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tom 19
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nr 1-2
67-84
EN
In this paper we consider a dynamic model for flow induced vibration of pipelines. We study the questions of existence and uniqueness of solutions of the system. Considering the flow rate as the control variable, we present three different necessary conditions of optimality. The last one with state constraint involves Differential Inclusions. The paper is concluded with an algorithm for computing the optimal controls.
20
Content available remote Viability theory : an applied mathematics tool for achieving dynamic systems' sustainability
60%
Krawczyk J. B. , Pharo A. S.
Mathematica Applicanda
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2013
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tom Vol. 41, No. 1
97--126
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.
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