arXiv subjects

Sanja Konjik

Publications and source records attributed to Sanja Konjik.

16 recordsLinked to original sources

arXiv2022

Control theory for nonlinear fractional dispersive systems

We consider a terminal control problem for processes governed by a nonlinear system of fractional ODEs. In order to show existence of the control, we first consider the linear counterpart of the system and reprove a number of classical theorems in the fractional setting (representation of the solution through the Gramian type matrix, Kalman's principle, equivalence of the controllability and observability). We are then in the position to use a fixed point theorem approach and various techniques from the fractional calculus theory to get the desired result.

math.OC↗

arXiv2021

Global Controllability for Quasilinear Non-negative Definite System of ODEs and SDEs

We consider exact and averaged control problem for a system of quasi-linear ODEs and SDEs with a non-negative definite symmetric matrix of the system. The strategy of the proof is the standard linearization of the system by fixing the function appearing in the nonlinear part of the system, and then applying the Leray-Schauder fixed point theorem. We shall also need the continuous induction arguments to prolong the control to the final state which is a novel approach in the field. This enables us to obtain controllability for arbitrarily large initial data (so called global controllability).

math.OC↗

arXiv2018

Distributed order fractional constitutive stress-strain relation in wave propagation modeling

Distributed order fractional model of viscoelastic body is used in order to describe wave propagation in infinite media. Existence and uniqueness of fundamental solution to the generalized Cauchy problem, corresponding to fractional wave equation, is studied. The explicit form of fundamental solution is calculated, and wave propagation speed, arising from solution's support, is found to be connected with the material properties at initial time instant. Existence and uniqueness of the fundamental solutions to the fractional wave equations corresponding to four thermodynamically acceptable classes of linear fractional constitutive models, as well as to power type distributed order model, are established and explicit forms of the corresponding fundamental solutions are obtained.

math-ph↗

arXiv2016

Complex order fractional derivatives in viscoelasticity

We introduce complex order fractional derivatives in models that describe viscoelastic materials. This can not be carried out unrestrictedly, and therefore we derive, for the first time, real valued compatibility constraints, as well as physical constraints that lead to acceptable models. As a result, we introduce a new form of complex order fractional derivative. Also, we consider a fractional differential equation with complex derivatives, and study its solvability. Results obtained for stress relaxation and creep are illustrated by several numerical examples.

math.AP↗

arXiv2016

Wave equation for generalized Zener model containing complex order fractional derivatives

We study waves in a viscoelastic rod whose constitutive equation is of generalized Zener type that contains fractional derivatives of complex order. The restrictions following from the Second Law of Thermodynamics are derived. The initial-boundary value problem for such materials is formulated and solution is presented in the form of convolution. Two specific examples are analyzed.

math.AP↗

arXiv2014

A regularization approach to non-smooth symplectic geometry

We introduce non-smooth symplectic forms on manifolds and describe corresponding Poisson structures on the algebra of Colombeau generalized functions. This is achieved by establishing an extension of the classical map of smooth functions to Hamiltonian vector fields to the setting of non-smooth geometry. For mildly singular symplectic forms, including the continuous non-differentiable case, we prove the existence of generalized Darboux coordinates in the sense of a local non-smooth pull-back to the canonical symplectic form on the cotangent bundle.

math.DG↗

arXiv2013

Symplectic modules over Colombeau-generalized numbers

We study symplectic linear algebra over the ring $\Rt$ of Colombeau generalized numbers. Due to the algebraic properties of $\Rt$ it is possible to preserve a number of central results of classical symplectic linear algebra. In particular, we construct symplectic bases for any symplectic form on a free $\Rt$-module of finite rank. Further, we consider the general problem of eigenvalues for matrices over $\Kt$ ($\K=\R$ or $\C$) and derive normal forms for Hermitian and skew-symmetric matrices. Our investigations are motivated by applications in non-smooth symplectic geometry and the theory of Fourier integral operators with non-smooth symbols.

math.RA↗

arXiv2011

Generalized solutions for the Euler-Bernoulli model with Zener viscoelastic foundations and distributional forces

We study the initial-boundary value problem for an Euler-Bernoulli beam model with discontinuous bending stiffness laying on a viscoelastic foundation and subjected to an axial force and an external load both of Dirac-type. The corresponding model equation is fourth order partial differential equation and involves discontinuous and distributional coefficients as well as a distributional right-hand side. Moreover the viscoelastic foundation is of Zener type and described by a fractional differential equation with respect to time. We show how functional analytic methods for abstract variational problems can be applied in combination with regularization techniques to prove existence and uniqueness of generalized solutions.

math.AP↗

arXiv2011

Variational problems with fractional derivatives: Invariance conditions and Nöther's theorem

A variational principle for Lagrangian densities containing derivatives of real order is formulated and the invariance of this principle is studied in two characteristic cases. Necessary and sufficient conditions for an infinitesimal transformation group (basic Nöther's identity) are obtained. These conditions extend the classical results, valid for integer order derivatives. A generalization of Nöther's theorem leading to conservation laws for fractional Euler-Lagrangian equation is obtained as well. Results are illustrated by several concrete examples. Finally, an approximation of a fractional Euler-Lagrangian equation by a system of integer order equations is used for the formulation of an approximated invariance condition and corresponding conservation laws.

math.FA↗

arXiv2011

Waves in fractional Zener type viscoelastic media

Classical wave equation is generalized for the case of viscoelastic materials obeying fractional Zener model instead of Hooke's law. Cauchy problem for such an equation is studied: existence and uniqueness of the fundamental solution is proven and solution is calculated.

math.AP↗

arXiv2011

Variational Problems with Fractional Derivatives: Euler-Lagrange Equations

We generalize the fractional variational problem by allowing the possibility that the lower bound in the fractional derivative does not coincide with the lower bound of the integral that is minimized. Also, for the standard case when these two bounds coincide, we derive a new form of Euler-Lagrange equations. We use approximations for fractional derivatives in the Lagrangian and obtain the Euler-Lagrange equations which approximate the initial Euler-Lagrange equations in a weak sense.

math.FA↗

arXiv2011

Generalized Hamilton's Principle with Fractional Derivatives

We generalize Hamilton's principle with fractional derivatives in Lagrangian $L(t,y(t),{}_0D_t^\al y(t),α)$ so that the function $y$ and the order of fractional derivative $α$ are varied in the minimization procedure. We derive stationarity conditions and discuss them through several examples.

math.FA↗

arXiv2007

Foundations of the calculus of variations in generalized function algebras

We propose the use of algebras of generalized functions for the analysis of certain highly singular problems in the calculus of variations. After a general study of extremal problems on open subsets of Euclidean space in this setting we introduce the first and second variation of a variational problem. We then derive necessary (Euler-Lagrange equations) and sufficient conditions for extremals. The concept of association is used to obtain connections to a distributional description of singular variational problems. We study variational symmetries and derive an appropriate version of Nöther's theorem. Finally, a number of applications to geometry, mechanics, elastostatics and elastodynamics are presented.

math.FA↗

arXiv2007

Symmetries of Conservation Laws

The aim of this paper is to apply techniques of symmetry group analysis in solving two systems of conservation laws: a model of two strictly hyperbolic conservation laws and a zero pressure gas dynamics model, which both have no global solution, but whose solution consists of singular shock waves. We show that these shock waves are solutions in the sense of 1-strong association. Also, we compute all projectable symmetry groups and show that they are 1-strongly associated, hence transform existing solutions in the sense of 1-strong association into other solutions.

math.AP↗

arXiv2005

Group invariants in algebras of generalized functions

We study invariance properties of Colombeau generalized functions under actions of smooth Lie transformation groups. Several characterization results analogous to the smooth setting are derived and applications to generalized rotational invariance are given.

math.FA↗

arXiv2005

Generalized Group Actions in a Global Setting

We study generalized group actions on differentiable manifolds in the Colombeau framework, extending previous work on flows of generalized vector fields and symmetry group analysis of generalized solutions. As an application, we analyze group invariant generalized functions in this setting.

math.FA↗