Volume 3, Issue 1

In the paper we consider nonlinear, nonoscillatory controlled objects of second order. Main Theorem affirms that for these controlled objects there exist (in the controllability region) the time-optimal synthesis of Feldbaum's type. In the beginning of the paper, Felfbaum's n -interval Theorem is proved for linear controlled objects of n -th order with real eigenvalues (and without the requirement that the eigenvalues are pairwise distinct).

First order forced nonlinear neutral differential equations are studied and sufficient conditions are derived for all solutions to be oscillatory.

Let T , X , Y be topological spaces and F : T × X ↦ n ( Y ) be a set–valued function. We consider the Nemytskii operator generated by F which associates with every set–valued function G : T ↦ n ( X ) the superposition F (·, G (·)) : T ↦ n ( Y ). Conditions under which this superposition is lower or upper semicontinuous are presented.

We consider the uniqueness property for various invariant measures. Primarily, we discuss this property for the standard Lebesgue measure on the n –dimensional Euclidean space R n (sphere S n ) and for the standard Borel measure on the same space (sphere), which is the restriction of the Lebesgue measure to the Borel σ –algebra of R n ( S n ). The main goal of the paper is to show an application of the well known theorems of Ulam and Ershov to the uniqueness property of Lebesgue and Borel measures.

We consider the problem to minimize an integral functional denned on the space of absolutely continuous functions and measurable controls with values in an infinite–dimensional complex Banach space. The states are governed by abstract first order semilinear differential equations and are subject to periodic or anti–periodic type boundary conditions. We derive necessary conditions for optimality and introduce the notion of a dual field of extremals to obtain sufficient conditions for optimality. Such a dual field of extremals is constructed and a dual optimal synthesis is proposed. The paper is an extension of an earlier paper written for real Banach spaces. This extension covers optimal control problems which are governed by equations like the Schrodinger equation and other equations arising in Quantum mechanics.

In this work we study when an arbitrary linear random operator between Banach spaces must behave as a continuous operator on some measurable set with positive measure. We deal with the continuity in probability as well as the continuity in r –mean.

We use the method of norms on possibilities to answer a question of Kunen and construct a ccc σ –ideal on 2 ω with various closure properties and distinct from the ideal of null sets, the ideal of meager sets and their intersection.

Using the semigroup approach we prove that the heat equation on a bounded domain in R d driven by general noise has a stationary solution iff a certain functional of the jump part of the noise has a finite expectation.

In this note we construct a complete orthonormal system (ONS) of uniformly bounded functions, such that for any function ƒ ∈ L 2 [0,1] its Fourier series with respect to the system, taken in decreasing order of magnitude of the coefficients, converges almost everywhere.