scholarly journals Constrained nonsmooth problems of the calculus of variations

2021 ◽  
Author(s):  
Maksim Dolgopolik
Keyword(s):  
Sobolev Space ◽  
Optimality Conditions ◽  
Calculus Of Variations ◽  
Integral Functional ◽  
Nonsmooth Problems ◽  
Multidimensional Problems ◽  
Nonsmooth Optimisation ◽  
Isoperimetric Constraints ◽  
Additional Constraints ◽  
Better Than

The paper is devoted to an analysis of optimality conditions for nonsmooth multidimensional problems of the calculus of variations with various types of constraints, such as additional constraints at the boundary and isoperimetric constraints. To derive optimality conditions, we study generalised concepts of differentiability of nonsmooth functions called codifferentiability and quasidifferentiability. Under some natural and easily verifiable assumptions we prove that a nonsmooth integral functional defined on the Sobolev space is continuously codifferentiable and compute its codifferential and quasidifferential. Then we apply general optimality conditions for nonsmooth optimisation problems in Banach spaces to obtain optimality conditions for nonsmooth problems of the calculus of variations. Through a series of simple examples we demonstrate that our optimality conditions are sometimes better than existing ones in terms of various subdifferentials, in the sense that our optimality conditions can detect the non-optimality of a given point, when subdifferential-based optimality conditions fail to disqualify this point as non-optimal.

On conservation laws and necessary conditions in the calculus of variations

2002 ◽  
Vol 132 (6) ◽  
pp. 1361-1371 ◽  
Author(s):  
G. Francfort ◽  
J. Sivaloganathan
Keyword(s):  
Conservation Laws ◽  
Calculus Of Variations ◽  
Conservation Law ◽  
Necessary Conditions ◽  
Lagrange Equation ◽  
Integral Functional ◽  
Variational Symmetry

It is well known from the work of Noether that every variational symmetry of an integral functional gives rise to a corresponding conservation law. In this paper, we prove that each such conservation law arises directly as the Euler-Lagrange equation for the functional on taking suitable variations around a minimizer.

scholarly journals Optimal Control with Time Delays via the Penalty Method

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Mohammed Benharrat ◽  
Delfim F. M. Torres
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Optimality Conditions ◽  
Calculus Of Variations ◽  
Convergence Theorem ◽  
Unknown Function ◽  
Time Delays ◽  
Penalty Method ◽  
Necessary Optimality Conditions

We prove necessary optimality conditions of Euler-Lagrange type for a problem of the calculus of variations with time delays, where the delay in the unknown function is different from the delay in its derivative. Then, a more general optimal control problem with time delays is considered. Main result gives a convergence theorem, allowing us to obtain a solution to the delayed optimal control problem by considering a sequence of delayed problems of the calculus of variations.

scholarly journals Optimality conditions for the calculus of variations with higher-order delta derivatives

2011 ◽  
Vol 24 (1) ◽  
pp. 87-92 ◽  
Author(s):  
Rui A.C. Ferreira ◽  
Agnieszka B. Malinowska ◽  
Delfim F.M. Torres
Keyword(s):  
Optimality Conditions ◽  
Calculus Of Variations ◽  
Higher Order ◽  
Delta Derivatives

Optimality Principles for Fuzzy Dual Uncertain Systems

2018 ◽  
pp. 47-72
Author(s):  
Félix Mora-Camino ◽  
Hakim Bouadi ◽  
Roger Marcelin Faye ◽  
Lunlong Zhong
Keyword(s):  
Optimality Conditions ◽  
Calculus Of Variations ◽  
Optimization Problems ◽  
Lagrange Equation ◽  
Necessary Optimality Conditions ◽  
Continuous System ◽  
Propagation Of Uncertainty ◽  
Performance Indexes ◽  
Definition Of ◽  
Optimality Principles

This chapter considers the extension of the calculus of variations to the optimization of a class of fuzzy systems where the uncertainty of variables and parameters is represented by symmetrical triangular membership functions. The concept of fuzzy dual numbers is introduced, and the consideration of the necessary differentiability conditions for functions of dual variables leads to the definition of fuzzy dual functions. It is shown that when this formalism is adopted to represent performance indexes for uncertain optimization problems, the calculus of variations can be used to establish necessary optimality conditions as an extension to this case of the Euler-Lagrange equation. Then the chapter discusses the propagation of uncertainty when the fuzzy dual formalism is adopted for the state representation of a time continuous system. This leads to the formulation of a fuzzy dual optimization problem for which necessary optimality conditions, corresponding to an extension of Pontryagine's optimality principle, are established.

Some remarks concerning quasiconvexity and strong convergence

1987 ◽  
Vol 106 (1-2) ◽  
pp. 53-61 ◽  
Author(s):  
L. C. Evans ◽  
R. F. Gariepy
Keyword(s):  
Sobolev Space ◽  
Weak Convergence ◽  
Strong Convergence ◽  
Calculus Of Variations ◽  
Heuristic Principle ◽  
Sequence Of Functions

SynopsisWe show that the weak convergence of a sequence of functions in a Sobolev space plus the convergence of appropriately quasiconvex “energies” imply, in fact, strong convergence. This assertion makes rigorous, for example, the heuristic principle that “quasiconvexity damps out oscillations in the gradients” of minimising sequences in the calculus of variations.

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