Regularity of minimizers for higher order variational problems in one independent variable

2011 ◽  
Vol 35 (2) ◽  
pp. 172-177 ◽  
Author(s):  
Christos Gavriel ◽  
Sofia Lopes ◽  
Richard Vinter
Keyword(s):  
Variational Problems ◽  
Higher Order ◽  
Regularity Of Minimizers ◽  
Independent Variable

AFFINE DUALITY AND LAGRANGIAN AND HAMILTONIAN SYSTEMS

2011 ◽  
Vol 08 (03) ◽  
pp. 669-697 ◽  
Author(s):  
OLGA KRUPKOVÁ ◽  
DAVID J. SAUNDERS
Keyword(s):  
Hamiltonian Systems ◽  
Arbitrary Order ◽  
Variational Problems ◽  
Higher Order ◽  
Jet Bundle ◽  
Jet Bundles ◽  
First Order ◽  
Variational Form ◽  
Independent Variable

We use affine duals of jet bundles to describe how Legendre maps may be used to provide Hamiltonian representations of variational problems in a single independent variable. Such a problem may be given as a Lagrangian (of first-order or of higher-order), or alternatively as a locally variational form on a jet bundle of arbitrary order with no preferred Lagrangian.

scholarly journals Variational Problems with Time Delay and Higher-Order Distributed-Order Fractional Derivatives with Arbitrary Kernels

Mathematics ◽  
2021 ◽  
Vol 9 (14) ◽  
pp. 1665
Author(s):  
Fátima Cruz ◽  
Ricardo Almeida ◽  
Natália Martins
Keyword(s):  
Time Delay ◽  
Fractional Derivatives ◽  
Variational Problems ◽  
Higher Order ◽  
Sufficient Optimality Conditions ◽  
Fractional Operator ◽  
Different Types ◽  
Generalized Fractional Derivatives ◽  
Necessary And Sufficient ◽  
Distributed Order

In this work, we study variational problems with time delay and higher-order distributed-order fractional derivatives dealing with a new fractional operator. This fractional derivative combines two known operators: distributed-order derivatives and derivatives with respect to another function. The main results of this paper are necessary and sufficient optimality conditions for different types of variational problems. Since we are dealing with generalized fractional derivatives, from this work, some well-known results can be obtained as particular cases.

scholarly journals Unified formalism for higher-order variational problems and its applications in optimal control

2014 ◽  
Vol 11 (04) ◽  
pp. 1450034 ◽  
Author(s):  
Leonardo Colombo ◽  
Pedro Daniel Prieto-Martínez
Keyword(s):  
Optimal Control ◽  
Equations Of Motion ◽  
Variational Problems ◽  
Higher Order ◽  
Point Of View ◽  
Regular Case ◽  
Underactuated Mechanical Systems ◽  
Interesting Application ◽  
Principal Bundles ◽  
Order Constraints

In this paper, we consider an intrinsic point of view to describe the equations of motion for higher-order variational problems with constraints on higher-order trivial principal bundles. Our techniques are an adaptation of the classical Skinner–Rusk approach for the case of Lagrangian dynamics with higher-order constraints. We study a regular case where it is possible to establish a symplectic framework and, as a consequence, to obtain a unique vector field determining the dynamics. As an interesting application we deduce the equations of motion for optimal control of underactuated mechanical systems defined on principal bundles.

scholarly journals Higher-order variational problems on lie groups and optimal control applications

2014 ◽  
Vol 6 (4) ◽  
pp. 451-478 ◽  
Author(s):  
Leonardo Colombo ◽  
◽  
David Martín de Diego ◽  
Keyword(s):  
Optimal Control ◽  
Lie Groups ◽  
Variational Problems ◽  
Higher Order ◽  
Control Applications

scholarly journals Lagrangian for Circuits with Higher-Order Elements

Entropy ◽  
2019 ◽  
Vol 21 (11) ◽  
pp. 1059 ◽  
Author(s):  
Zdenek Biolek ◽  
Dalibor Biolek ◽  
Viera Biolkova
Keyword(s):  
Variational Principle ◽  
Equations Of Motion ◽  
Sufficient Conditions ◽  
Higher Order ◽  
The State ◽  
Even Order ◽  
Independent Variable ◽  
Necessary And Sufficient ◽  
Derivatives Of ◽  
State Functions

The necessary and sufficient conditions of the validity of Hamilton’s variational principle for circuits consisting of (α,β) elements from Chua’s periodical table are derived. It is shown that the principle holds if and only if all the circuit elements lie on the so-called Σ-diagonal with a constant sum of the indices α and β. In this case, the Lagrangian is the sum of the state functions of the elements of the L or +R types minus the sum of the state functions of the elements of the C or −R types. The equations of motion generated by this Lagrangian are always of even-order. If all the elements are linear, the equations of motion contain only even-order derivatives of the independent variable. Conclusions are illustrated on an example of the synthesis of the Pais–Uhlenbeck oscillator via the elements from Chua’s table.

scholarly journals The motions of algebraic differential equations

1984 ◽  
Vol 25 (1) ◽  
pp. 93-96
Author(s):  
Lee A. Rubel
Keyword(s):  
Differential Equations ◽  
Higher Order ◽  
Algebraic Differential Equations ◽  
First Order ◽  
Order Algebraic ◽  
Independent Variable

We confine ourselves, for simplicity, to first-order algebraic differential equations (ADE's), although analogous considerations may be made for higher-order ADE's:P(x, y(x), y'(x)) = 0. (*)A motion of (*) is a change of independent variable that takes solutions to solutions, that is, a suitable map <p of the underlying interval I into itself so that if y is a solution of (*) then y ° φ is a solution of (*), i.e.

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