Stochastic Invariance for Differential Inclusions

The aim of this paper is to combine two ways for representing uncertainty through stochastic differential inclusions: a 'stochastic uncertainty", driven by a Wiener process, and a 'contingent uncertainty", driven by a set-valued map. The paper is also devoted to the invariance of closed under stochastic differential inclusions with a Lipschitz right-hand side, characterized in terms of stochastic tangent sets to closed subsets.     

Towards Fully Discretized Differential Inclusions

Euler schemes for the calculation of the solution set for a differential inclusion are investigated. Under Lipschitz and convexity conditions on the set-valued map the usual O(h)-approximation, h denoting the time step, is preserved, if one uses only boundary points in each step. An O( )-approximation is achieved, if one uses only extremal points. So, in case that the extremal sets are finite, a full discretization of the differential inclusion is performed.     

Existence of Periodic Solutions for Differential Inclusions

§ 0 .Introduction　 During the past decades,periodic optimal control problems have been paidconsiderable attention;see,for example [1 ]— [9] and the references therein. In orderto deal with such periodic optimal control problems,one often considers the existence ofperiodic solutions for the nonlinear differential systemx′=f(t,x,u)　′=ddt ,where x is the real state n-vector attime t∈ R and u is the control parameter m-vector attime tvarying in some compactsetΩ of Rm,and hence the function…     

Generalized Differential Inclusions in Banach Spaces

We study a new type of solutions to differential inclusions in Banach spaces, which we call directional solutions. The idea is based on the observation that for a differentiable function and a closed set

Banach空间中微分包含的解

利用不动点方法,证明了Banach-空间中右端项不连续的微分包含解的存在性.     

含有非线性半群的非凸值泛函微分包含

本文研究Banach空间中含有非线性半群的非凸值泛函微分包含，证明了积分解的存在性，得到一个新的存在性定理．     

A Kind of Nonconvex—valued Functional Differential Inclusions Problems

51.IntroductionLetXbeaBanachspaceandI=[O,7'](7'>O),L=[-a,Oj,J=[-a,7']bethreeinter-valsofR.WesetF:IXC[L;Xj-2"\diisaset-valuesmap.Weintroducethetimelagopera-tor7'.):CLJ;Xj-CLLiXj,definedby7'(t)x(s)=X(t s)foreveryseL-LetusassumethatA:Dom(A)-2"\diisaset-values))l-accretiveoperator.Inthispaper,w3discusstheini-tialvalueprob1emsasfoIlows:Sincethedevelopmentoftl1eControlTheoryandEconomics,theinitialva1ueproblem(1.1)havebeendiscussinginrecentyears,someimportantresultshavebeenobtainecl,for…     

On the Solution Sets of Constrained Differential Inclusions with Applications

We study the existence and the structure of solutions to differential inclusions with constraints. We show that the set of all viable solutions to the Cauchy problem for a Carathéodory-type differential inclusion in a closed domain is an R -set provided some mild boundary conditions expressed in terms of functional constraints defining the domain are satisfied. Presented results generalize most of the existing ones. Some applications to the existence of periodic solutions as well as equilibria are given.     

Existence Results for Functional Differential Inclusions with Infinite Delay

The aim of the present paper is to investigate the existence of solutions to functional differential inclusions with infinite delay in Banach spaces. A relevant set of phase space axioms is proposed. The main tools used in this paper are certain fixed point theorems based on the setcontraction theory.     

Linear Relations, Differential Inclusions, and Degenerate Semigroups

A generalized linear differential equation in a Banach space is studied. The construction of a phase space and solutions with the help of the spectral theory of linear operators, ergodic theorems, and degenerate semigroups of linear operators is carried out.     

一类三阶脉冲微分包含解的存在性

研究了一类三阶脉冲泛函微分包含解的存在性,应用不动点定理和多值分析理论获得了两个新结果,举例说明了所得的结果.     

Reynold''s Transport Theorem for Differential Inclusions

The well-known Reynold's Transport Theorem deals with the integral over a time-dependent set (that is evolving along a smooth vector field) and specifies its semiderivative with respect to time. Here reachable sets of differential inclusions are considered instead. Dispensing with any assumptions about the regularity of the compact initial set, we give sufficient conditions on the differential inclusion for the absolute continuity (of the integral) with respect to time and its weak derivative is formulated as a Hausdorff integral over the topological boundary.     

微分包含的单调轨道

该文研究微分包含单调轨道的存在性.基于一个新的逼近方法,作者证明了无限维空间中的一个存在性,其特殊情形包含了已有的若于生存定理.     

On Stability and Boundedness for Lipschitzian Differential Inclusions: The Converse of Lyapunov's Theorems

For Lipschitzian differential inclusions, we prove that the existence of suitable Lyapunov functions is necessary for uniform stability and uniform boundedness of solutions.     

On Periodic Viable Solutions of Differential Inclusions

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Nonconvex Differential Inclusions with Nonlinear Monotone Boundary Conditions

Existence results for problems with monotone nonlinear boundary conditions obtained in the previous publications by the author for functional differential equations are transferred to the case of nonconvex differential inclusions with the help of the selection theorem due to A. Bressan and G. Colombo.     

Formulating Measure Differential Inclusions in Infinite Dimensions

Measure differential inclusions were introduced by J. J. Moreau to study sweeping processes, and have since been used to study rigid body dynamics and impulsive control problems. The basic formulation of an MDI is d / d (t) K(t) where is a vector measure, an unsigned measure, and K() is a set-valued map with closed, convex values and is hemicontinuous. Note that need not be absolutely continuous with respect to . Stewart extended Moreau's original concept (which applied only to cone-valued K()) to general convex sets, and gave strong and weak formulations of d / d (t) K(t) where K(t) R ⁿ . Here the strong and weak formulations of Stewart are extended to infinite-dimensional problems where K(t) X where X is a separable reflexive Banach space; they are shown to be equivalent under mild assumptions on K().     

On Distribution Semigroups with a Singularity at Zero and Bounded Solutions of Differential Inclusions

We study distribution semigroups with a singularity at zero and their generators, and establish a relationship between this semigroup and a degenerate semigroup of linear operators on the open right half-line. The study makes an intensive use of spectral theory of linear relations. Applications to the existence problem for bounded solutions of linear differential inclusions are obtained.     

Needle Variations and Almost Lower Semicontinuous Differential Inclusions

It is shown that the construction of needle variations can be carried out for almost lower semicontinuous differential inclusions rather than for the case of ordinary single-valued continuously differentiable vector fields usually considered in the literature. The construction leads to needle variations whose flows are in general set-valued but still have good differentiability properties. The variations are constructed by using single-valued selections that are not necessarily continuous with respect to the state variable, but have instead a much weaker 'integral continuity' property, somewhat more general that the 'directional continuity' considered in previous work by A. Cambini and S. Querci, A. Pucci, and A. Bressan. The existence of many such selections is proved by slightly adapting an argument due to Bressan, extending it from the lower semicontinuous to the almost lower semicontinuous case, and strengthening it to yield not only directional continuity at all points but also full continuity at a specified point.