arXiv subjects

Agnieszka B. Malinowska

Publications and source records attributed to Agnieszka B. Malinowska.

At least 19 recordsLinked to original sources

arXiv2017

Non-invasive control of the fractional Hegselmann-Krause type model

In this paper, the fractional order Hegselmann-Krause type model with leadership is studied.We seek an optimal control strategy for the system to reach a consensus in such a way that the control mechanism is included in the leader dynamics. Necessary optimality conditions are obtained by the use of a fractional counterpart of Pontryagin Maximum Principle. The effectiveness of the proposed control strategy is illustrated by numerical examples.

math.DS↗

arXiv2016

Fractional differential equations with dependence on the Caputo-Katugampola derivative

In this paper we present a new type of fractional operator, the Caputo-Katugampola derivative. The Caputo and the Caputo-Hadamard fractional derivatives are special cases of this new operator. An existence and uniqueness theorem for a fractional Cauchy type problem, with dependence on the Caputo--Katugampola derivative, is proven. A decomposition formula for the Caputo-Katugampola derivative is obtained. This formula allows us to provide a simple numerical procedure to solve the fractional differential equation.

math.CA↗

arXiv2016

Variational methods for the solution of fractional discrete/continuous Sturm-Liouville problems

The fractional Sturm-Liouville eigenvalue problem appears in many situations, e.g., while solving anomalous diffusion equations coming from physical and engineering applications. Therefore to obtain solutions or approximation of solutions to this problem is of great importance. Here, we describe how the fractional Sturm-Liouville eigenvalue problem can be formulated as a constrained fractional variational principle and show how such formulation can be used in order to approximate the solutions. Numerical examples are given, to illustrate the method.

math.NA↗

arXiv2014

A Time-Scale Variational Approach to Inflation, Unemployment and Social Loss

Both inflation and unemployment inflict social losses. When a tradeoff exists between the two, what would be the best combination of inflation and unemployment? A well known approach in economics to address this question consists to write the social loss as a function of the rate of inflation $p$ and the rate of unemployment $u$, with different weights, and then, using known relations between $p$, $u$, and the expected rate of inflation $π$, to rewrite the social loss function as a function of $π$. The answer is achieved by applying the theory of the calculus of variations in order to find an optimal path $π$ that minimizes the total social loss over a given time interval. Economists dealing with this question use a continuous or a discrete variational problem. Here we propose to use a time-scale model, unifying available results in the literature. Moreover, the new formalism allow us to obtain new insights to the classical models when applied to real data of inflation and unemployment.

math.OC↗

arXiv2014

Noether's Theorem for Control Problems on Time Scales

We prove a generalization of Noether's theorem for optimal control problems defined on time scales. Particularly, our results can be used for discrete-time, quantum, and continuous-time optimal control problems. The generalization involves a one-parameter family of maps which depend also on the control and a Lagrangian which is invariant up to an addition of an exact delta differential. We apply our results to some concrete optimal control problems on an arbitrary time scale.

math.OC↗

arXiv2014

Fractional Variational Principle of Herglotz

The aim of this paper is to bring together two approaches to non-conservative systems -- the generalized variational principle of Herglotz and the fractional calculus of variations. Namely, we consider functionals whose extrema are sought, by differential equations that involve Caputo fractional derivatives. The Euler--Lagrange equations are obtained for the fractional variational problems of Herglotz-type and the transversality conditions are derived. The fractional Noether-type theorem for conservative and non-conservative physical systems is proved.

math.OC↗

arXiv2014

The second Noether theorem on time scale

We extend the second Noether theorem to variational problems on time scales. Our result provides as corollaries the classical second Noether theorem, the second Noether theorem for the $h$-calculus and the second Noether theorem for the $q$-calculus.

math.OC↗

arXiv2014

An Inverse Problem of the Calculus of Variations on Arbitrary Time Scales

We consider an inverse extremal problem for variational functionals on arbitrary time scales. Using the Euler-Lagrange equation and the strengthened Legendre condition, we derive a general form for a variational functional that attains a local minimum at a given point of the vector space.

math.OC↗

arXiv2013

Fractional Calculus of Variations of Several Independent Variables

We prove multidimensional integration by parts formulas for generalized fractional derivatives and integrals. The new results allow us to obtain optimality conditions for multidimensional fractional variational problems with Lagrangians depending on generalized partial integrals and derivatives. A generalized fractional Noether's theorem, a formulation of Dirichlet's principle and an uniqueness result are given.

math-ph↗

arXiv2013

Optimal Control for a Steady State Dead Oil Isotherm Problem

We study the optimal control of a steady-state dead oil isotherm problem. The problem is described by a system of nonlinear partial differential equations resulting from the traditional modelling of oil engineering within the framework of mechanics of a continuous medium. Existence and regularity results of the optimal control are proved, as well as necessary optimality conditions.

math.OC↗

arXiv2013

Variational Methods for the Fractional Sturm--Liouville Problem

This article is devoted to the regular fractional Sturm--Liouville eigenvalue problem. Applying methods of fractional variational analysis we prove existence of countable set of orthogonal solutions and corresponding eigenvalues. Moreover, we formulate two results showing that the lowest eigenvalue is the minimum value for a certain variational functional.

math.OC↗

arXiv2013

A Generalized Fractional Calculus of Variations

We study incommensurate fractional variational problems in terms of a generalized fractional integral with Lagrangians depending on classical derivatives and generalized fractional integrals and derivatives. We obtain necessary optimality conditions for the basic and isoperimetric problems, transversality conditions for free boundary value problems, and a generalized Noether type theorem.

math.OC↗

arXiv2013

Noether's theorem for fractional variational problems of variable order

We prove a necessary optimality condition of Euler-Lagrange type for fractional variational problems with derivatives of incommensurate variable order. This allows us to state a version of Noether's theorem without transformation of the independent (time) variable. Considered derivatives of variable order are defined in the sense of Caputo.

math.OC↗

arXiv2012

Green's Theorem for Generalized Fractional Derivatives

We study three types of generalized partial fractional operators. An extension of Green's theorem, by considering partial fractional derivatives with more general kernels, is proved. New results are obtained, even in the particular case when the generalized operators are reduced to the standard partial fractional derivatives and fractional integrals in the sense of Riemann-Liouville or Caputo.

math.CA↗

arXiv2012

Variable Order Fractional Variational Calculus for Double Integrals

We introduce three types of partial fractional operators of variable order. An integration by parts formula for partial fractional integrals of variable order and an extension of Green's theorem are proved. These results allow us to obtain a fractional Euler-Lagrange necessary optimality condition for variable order two-dimensional fractional variational problems.

math.OC↗

arXiv2012

Generalized Transversality Conditions in Fractional Calculus of Variations

Problems of calculus of variations with variable endpoints cannot be solved without transversality conditions. Here, we establish such type of conditions for fractional variational problems with the Caputo derivative. We consider: the Bolza-type fractional variational problem, the fractional variational problem with a Lagrangian that may also depend on the unspecified end-point $φ(b)$, where $x=φ(t)$ is a given curve, and the infinite horizon fractional variational problem.

math.OC↗

arXiv2012

Towards a combined fractional mechanics and quantization

A fractional Hamiltonian formalism is introduced for the recent combined fractional calculus of variations. The Hamilton-Jacobi partial differential equation is generalized to be applicable for systems containing combined Caputo fractional derivatives. The obtained results provide tools to carry out the quantization of nonconservative problems through combined fractional canonical equations of Hamilton type.

math-ph↗

arXiv2012

Fractional variational calculus of variable order

We study the fundamental problem of the calculus of variations with variable order fractional operators. Fractional integrals are considered in the sense of Riemann-Liouville while derivatives are of Caputo type.

math.OC↗