Constrained nonsmooth problems of the calculus of variations

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.

A strict inequality for the minimization of the Willmore functional under isoperimetric constraint

Inspired by previous work of Kusner and Bauer–Kuwert, we prove a strict inequality between the Willmore energies of two surfaces and their connected sum in the context of isoperimetric constraints. Building on previous work by Keller, Mondino and Rivière, our strict inequality leads to existence of minimizers for the isoperimetric constrained Willmore problem in every genus, provided the minimal energy lies strictly below 8 ⁢ π {8\pi} . Besides the geometric interest, such a minimization problem has been studied in the literature as a simplified model in the theory of lipid bilayer cell membranes.

Fuzzy Optimal Control Problem of Several Variables

The purpose of this paper is to establish the necessary conditions for a fuzzy optimal control problem of several variables. Also, we define fuzzy optimal control problems involving isoperimetric constraints and higher order differential equations. Then, we convert these problems to fuzzy optimal control problems of several variables in order to solve these problems using the same solution method. The main results of this paper are illustrated throughout three examples, more specifically, a discussion on the strong solutions (fuzzy solutions) of our problems.

Coordinates & Constraints

This short chapter introduces constraints, generalised coordinates and the various spaces of Lagrangian mechanics. Analytical mechanics concerns itself with scalar quantities of a dynamic system, namely the potential and kinetic energies of the particle; this approach is in opposition to Newton’s method of vectorial mechanics, which relies upon defining the position of the particle in three-dimensional space, and the forces acting upon it. The chapter serves as an informal, non-mathematical introduction to differential geometry concepts that describe the configuration space and velocity phase space as a manifold and a tangent, respectively. The distinction between holonomic and non-holonomic constraints is discussed, as are isoperimetric constraints, configuration manifolds, generalised velocity and tangent bundles. The chapter also introduces constraint submanifolds, in an intuitive, graphic format.

Optimal Control Strategies Depending on Interest Level for the Spread of Rumor

Many media channels such as broadcast, newspaper, and social networks diffuse a variety of information which can cause spread of many rumors. There are social damage and economic damage due to the spread of rumors. Thus one needs to establish strategies for controlling the rumors. We first propose rumor model with three control strategies for preventing the spread of rumor, (1) announcing the truth before ignorant receives rumor, (2) punishing spreaders, and (3) deleting information of the rumor in media, and consider optimal control problems to minimize the number of spreaders while minimizing the cost of three control strategies for preventing the spread of rumors. The analysis of optimal control problems is conducted as Pontryagin’s Maximum Principle. Furthermore, adapted optimal control is performed to investigate the effect of three controls under isoperimetric constraints. By using numerical simulations, we compare the number of spreaders before and after applying the three controls and confirm when and how each control should be applied with respect to the interest level of rumor. The lower the interest level of rumor is, the greater the number of spreaders drops after the three controls are applied. In terms of timing of three controls, control (1) should be applied in the early stage of rumor spreading and control (2) is required when the rumors spread the most. After the rumors spread the most, control (3) is needed. Commonly the higher the interest level is, the more controls (1) and (2) are required. On the other hand, control (3) is needed a lot when the interest level is low.

Eulerian formulation of elastic rods

In numerous biological, medical and engineering applications, elastic rods are constrained to deform inside or around tube-like surfaces. To solve efficiently this class of problems, the equations governing the deflection of elastic rods are reformulated within the Eulerian framework of this generic tubular constraint defined as a perfectly stiff normal ringed surface. This reformulation hinges on describing the rod-deformed configuration by means of its relative position with respect to a reference curve, defined as the axis or spine curve of the constraint, and on restating the rod local equilibrium in terms of the curvilinear coordinate parametrizing this curve. Associated with a segmentation strategy, which partitions the global problem into a sequence of rod segments either in continuous contact with the constraint or free of contact (except for their extremities), this re-parametrization not only trivializes the detection of new contacts but also transforms these free boundary problems into classic two-points boundary-value problems and suppresses the isoperimetric constraints resulting from the imposition of the rod position at the extremities of each rod segment.