A theory of singular and quasi-bounded temperatures associated with the heat equation is developed. Guided by the harmonic case [1] an operator S is defined on the set M of non-negative functions on an open set of Rn+l which admit superthermic majorants. Quasi-bounded and singular functions in M are defined from the operator S . The basic properties of S are discussed. Non-negative temperatures...

Bolza problem governed by PDE control systems with unbounded controls is considered. The motivating example is fluid structure interaction model with boundary-interface controls. The aim of the work is to provide optimal feedback synthesis associated with well defined gain operator constructed from the Riccati equation. The dynamics considered is of mixed parabolic-hyperbolic type which prevent...

In this paper, we give the solution of the inverse Sturm–Liouville problem on two partially coinciding spectra. In particular, in this case we obtain Hochstadt's theorem concerning the structure of the difference q(x) − ˜ q(x) for the singular Sturm Liouville problem defined on the finite interval (0, π) having the singularity type 1 4 sin 2 x at the points 0 and π.

We extend Krupnik’s criterion of self-adjointness of the Cauchy singular integral operator to the case of finitely connected domains. The main aim of the paper is to present a new approach for proof of the criterion. Let G+ be a finitely connected domain bounded by the rectifiable curve C = ∂G+, G− = C \ clos G+ and ∞ ∈ G−. Suppose also that w(z), z ∈ C is a nonnegative weight such that w(z) 6≡...

Abstract: Pole assignment by feedback control of the second order coupled singular distributed parameter systems is discussed via functional analysis and operator theory in Hilbert space. The solutions of the problem and the constructive expression of the solutions are given by the generalized inverse one of bounded linear operator. This research is theoretically important for studying the stab...

We construct examples of potentials V (x) satisfying |V (x)| ≤ h(x) 1+x , where the function h(x) is growing arbitrarily slowly, such that the corresponding Schrödinger operator has imbedded singular continuous spectrum. This solves one of the fifteen “twenty-first century” problems for Schrödinger operators posed by Barry Simon in [22]. The construction also provides the first example of a Sch...

We study general conditions under which the computations of the index of a perturbed Dirac operator Ds = D + sZ localize to the singular set of the bundle endomorphism Z in the semi-classical limit s → ∞. We show how to use Witten’s method to compute the index of D by doing a combinatorial computation involving local data at the nondegenerate singular points of the operator Z. In particular, we...

We introduce the cone operator, as a tool for the homotopy and excision axioms. Linear chains As a first step, we study the following situation. Suppose Z is a convex subspace of R. Given any n + 1 points v0, v1, . . . , vn in Z (not necessarily linearly independent), we have the linear map λ = [v0, v1, . . . , vn]: ∆ n −−→ Z (1) from the standard n-simplex ∆ ⊂ R with vertices ei (for 0 ≤ i ≤ n...

We consider methods of reduction of differential operator equations with the Fredholm operator in the main expression to regular problems. Relation between the initial conditions choice problem and the Jordan structure of operator coefficients of equations is shown. The theorem of existence and uniqueness of the initial problem is proved. The method of fundamental operators is used for construc...