Searcharxiv⌕ Search

arXiv subjects

Agnieszka B. Malinowska

Publications and source records attributed to Agnieszka B. Malinowska.

52 records · Page 3Linked to original sources
arXiv2010

Fractional variational calculus in terms of a combined Caputo derivative

We generalize the fractional Caputo derivative to the fractional derivative ${^CD^{α,β}_γ}$, which is a convex combination of the left Caputo fractional derivative of order $α$ and the right Caputo fractional derivative of order $β$. The fractional variational problems under our consideration are formulated in terms of ${^CD^{α,β}_γ}$. The Euler-Lagrange equations for the basic and isoperimetric problems, as well as transversality conditions, are proved.

math.OC↗
arXiv2010

Euler-Lagrange equations for composition functionals in calculus of variations on time scales

In this paper we consider the problem of the calculus of variations for a functional which is the composition of a certain scalar function $H$ with the delta integral of a vector valued field $f$, i.e., of the form $H(\int_{a}^{b}f(t,x^σ(t),x^Δ(t))Δt)$. Euler-Lagrange equations, natural boundary conditions for such problems as well as a necessary optimality condition for isoperimetric problems, on a general time scale, are given. A number of corollaries are obtained, and several examples illustrating the new results are discussed in detail.

math.OC↗
arXiv2010

The contingent epiderivative and the calculus of variations on time scales

The calculus of variations on time scales is considered. We propose a new approach to the subject that consists in applying a differentiation tool called the contingent epiderivative. It is shown that the contingent epiderivative applied to the calculus of variations on time scales is very useful: it allows to unify the delta and nabla approaches previously considered in the literature. Generalized versions of the Euler-Lagrange necessary optimality conditions are obtained, both for the basic problem of the calculus of variations and isoperimetric problems. As particular cases one gets the recent delta and nabla results.

math.OC↗
arXiv2010

The Hahn Quantum Variational Calculus

We introduce the Hahn quantum variational calculus. Necessary and sufficient optimality conditions for the basic, isoperimetric, and Hahn quantum Lagrange problems, are studied. We also show the validity of Leitmann's direct method for the Hahn quantum variational calculus, and give explicit solutions to some concrete problems. To illustrate the results, we provide several examples and discuss a quantum version of the well known Ramsey model of economics.

math.OC↗
arXiv2010

A unified approach to the calculus of variations on time scales

In this work we propose a new and more general approach to the calculus of variations on time scales that allows to obtain, as particular cases, both delta and nabla results. More precisely, we pose the problem of minimizing or maximizing the composition of delta and nabla integrals with Lagrangians that involve directional derivatives. Unified Euler-Lagrange necessary optimality conditions, as well as sufficient conditions under appropriate convexity assumptions, are proved. We illustrate presented results with simple examples.

math.OC↗
arXiv2010

A Fractional Calculus of Variations for Multiple Integrals with Application to Vibrating String

We introduce a fractional theory of the calculus of variations for multiple integrals. Our approach uses the recent notions of Riemann-Liouville fractional derivatives and integrals in the sense of Jumarie. Main results provide fractional versions of the theorems of Green and Gauss, fractional Euler-Lagrange equations, and fractional natural boundary conditions. As an application we discuss the fractional equation of motion of a vibrating string.

math.OC↗
arXiv2010

Leitmann's direct method of optimization for absolute extrema of certain problems of the calculus of variations on time scales

The fundamental problem of the calculus of variations on time scales concerns the minimization of a delta-integral over all trajectories satisfying given boundary conditions. This includes the discrete-time, the quantum, and the continuous/classical calculus of variations as particular cases. In this note we follow Leitmann's direct method to give explicit solutions for some concrete optimal control problems on an arbitrary time scale.

math.OC↗
arXiv2009

The delta-nabla calculus of variations

The discrete-time, the quantum, and the continuous calculus of variations have been recently unified and extended. Two approaches are followed in the literature: one dealing with minimization of delta integrals; the other dealing with minimization of nabla integrals. Here we propose a more general approach to the calculus of variations on time scales that allows to obtain both delta and nabla results as particular cases.

math.OC↗
arXiv2009

Strong Minimizers of the Calculus of Variations on Time Scales and the Weierstrass Condition

We introduce the notion of strong local minimizer for the problems of the calculus of variations on time scales. Simple examples show that on a time scale a weak minimum is not necessarily a strong minimum. A time scale form of the Weierstrass necessary optimality condition is proved, which enables to include and generalize in the same result both continuous-time and discrete-time conditions.

math.OC↗
arXiv2007

A Computational Approach to Essential and Nonessential Objective Functions in Linear Multicriteria Optimization

The question of obtaining well-defined criteria for multiple criteria decision making problems is well-known. One of the approaches dealing with this question is the concept of nonessential objective function. A certain objective function is called nonessential if the set of efficient solutions is the same both with or without that objective function. In this paper we put together two methods for determining nonessential objective functions. A computational implementation is done using a computer algebra system.

math.OC↗
arXiv2006

Nonessential Functionals in Multiobjective Optimal Control Problems

We address the problem of obtaining well-defined criteria for multiobjective optimal control systems. Necessary and sufficient conditions for an optimal control functional to be nonessential are proved. The results provide effective tools for determining nonessential objectives in vector-valued optimal control problems.

math.OC↗