Isoperimetric Constraints Research Articles

Optimisation and monotonicity of the second Robin eigenvalue on a planar exterior domain

AbstractWe consider the Laplace operator in the exterior of a compact set in the plane, subject to Robin boundary conditions. If the boundary coupling is sufficiently negative, there are at least two discrete eigenvalues below the essential spectrum. We state a general conjecture that the second eigenvalue is maximised by the exterior of a disk under isochoric or isoperimetric constraints. We prove an isoelastic version of the conjecture for the exterior of convex domains. Finally, we establish a monotonicity result for the second eigenvalue under the condition that the compact set is strictly star-shaped and centrally symmetric.

Isoperimetric Constraint Inference for Discrete-Time Nonlinear Systems Based on Inverse Optimal Control.

In this article, the problem of inferring unknown isoperimetric constraints is considered given optimal state and control trajectories that solve the optimal control problem with isoperimetric constraints. By exploiting Pontryagin's principle, the recovery equations for unknown isoperimetric constraints are established. Under verifiable dimensionality condition and matrix rank condition, the proposed method is guaranteed to infer the unknown isoperimetric constraints exactly. Furthermore, the proposed method is extended to multiple trajectory setting. Finally, the effectiveness of the proposed method is illustrated by two simulation examples with various settings.

Periodic Optimal Control of a Plug Flow Reactor Model with an Isoperimetric Constraint

We study a class of nonlinear hyperbolic partial differential equations with boundary control. This class describes chemical reactions of the type “A→\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A \\rightarrow $$\\end{document} product” carried out in a plug flow reactor (PFR) in the presence of an inert component. An isoperimetric optimal control problem with periodic boundary conditions and input constraints is formulated for the considered mathematical model in order to maximize the mean amount of product over the period. For the single-input system, the optimality of a bang-bang control strategy is proved in the class of bounded measurable inputs. The case of controlled flow rate input is also analyzed by exploiting the method of characteristics. A case study is performed to illustrate the performance of the reaction model under different control strategies.

Periodic on-off operations of the self-cycling ethanol fermentation process: Stability analysis and optimal operation strategies

Fuel ethanol is highlighted for being renewable and carbon neutral, but a multi-scale challenge exists that hampers its role as a future energy substitute, and with the breakthrough of cellulose pretreatment and enzyme hydrolysis technology, fermentation itself has become one of the main limiting factors for process enhancement. To tackle the problem, this paper studies periodic operation strategies to improve process performance and avoid possible instabilities. Firstly, superiority by periodic operation over the corresponding steady-state operation is verified. Then, to settle the problem of possible instabilities under impulsive control, stability of state-impulsive control is conducted and for the studied case, any self-cycling fermentation starts from batch mode with 50 % discharge-and-refill portion exhibits superior repeatability property, which allows continuous fermentation to operate under very slow overall dilution rate to cope with the extreme slow fermentation rate. Finally, on-off operation problem is formulated that is governed by a multidimensional nonlinear system, affine to the manipulated variable that is under the isoperimetric constraint; with the use of Pontryagin maximum principle, the optimal synthesis of the optimal strategies under integral constraint is computed out, and a practically plausible on-off operation strategy for ethanol fermentation is formulated with guaranteed periodic stability and optimality.

The elastica sling

The nonlinear mechanics of a flexible elastic rod constrained at its edges by a pair of sliding sleeves is analyzed. The planar equilibrium configurations of this variable-length elastica are found to have shape defined only by the inclination of the two constraints, while their distance is responsible only for scaling the size. By extending the theoretical stability criterion available for systems under isoperimetric constraints to the case of variable domains, the existence of no more than one stable equilibrium solution is revealed. The set of sliding sleeves’ inclination pairs for which the stability is lost are identified. Such critical conditions allow the indefinite ejection of the flexible rod from the sliding sleeves, thus realizing an elastica sling. Finally, the theoretical findings are validated by experiments on a physical prototype. The present results lead to a novel actuation principle that may find application as a mechanism in energy harvesting, wave mitigation devices, and soft robotic locomotion.

Necessary and Sufficient Optimality Conditions for Fuzzy Variational Problems of Several Dependent Variables in Terms of Fuzzy Granular Derivatives

This paper is intended to investigate fuzzy variational problems of several dependent variables. Firstly, we establish both necessary and sufficient optimality conditions for fundamental fuzzy variational problems and fuzzy variational problems with natural boundary conditions. Then, the necessary and sufficient optimality conditions for fuzzy variational problems with isoperimetric constraints and holonomic constraints are discussed. Some examples are given to illustrate our results.

Approximation schemes for a quadratic type generalized isoperimetric constraint fractional variational problems

Approximation schemes for a quadratic type generalized isoperimetric constraint fractional variational problems

Optimal vaccination control for COVID-19 in a metapopulation model: a case of the Philippines

We investigate a contextual problem of how to distribute a limited supply of vaccines over a period of time in a country where different regions have its own vaccination capacities. Considering that daily vaccination will affect future disease progression, we aim to find a distribution strategy over time that can minimize the total infection and implementation costs. Lagrangian and Eulerian migrations connect our multi-patch COVID-19 model, and vaccination is added as a control measure. An optimal control problem with an isoperimetric constraint is formulated and solved using the Adapted Forward–Backward Sweep Method. In distributing 5 million vaccines in 50 days, simulations showed that the optimal control strategy could lead to a difference of reducing two hundred thousand infections in just one region.

Optimal vaccine allocation for the control of sexually transmitted infections.

The burden of sexually transmitted infections (STIs) poses a challenge due to its large negative impact on sexual and reproductive health worldwide. Besides simple prevention measures and available treatment efforts, prophylactic vaccination is a powerful tool for controlling some viral STIs and their associated diseases. Here, we investigate how prophylactic vaccines are best distributed to prevent and control STIs. We consider sex-specific differences in susceptibility to infection, as well as disease severity outcomes. Different vaccination strategies are compared assuming distinct budget constraints that mimic a scarce vaccine stockpile. Vaccination strategies are obtained as solutions to an optimal control problem subject to a two-sex Kermack-McKendrick-type model, where the control variables are the daily vaccination rates for females and males. One important aspect of our approach relies on conceptualizing a limited but specific vaccine stockpile via an isoperimetric constraint. We solve the optimal control problem via Pontryagin's Maximum Principle and obtain a numerical approximation for the solution using a modified version of the forward-backward sweep method that handles the isoperimetric budget constraint in our formulation. The results suggest that for a limited vaccine supply ([Formula: see text]-[Formula: see text] vaccination coverage), one-sex vaccination, prioritizing females, appears to be more beneficial than the inclusion of both sexes into the vaccination program. Whereas, if the vaccine supply is relatively large (enough to reach at least [Formula: see text] coverage), vaccinating both sexes, with a slightly higher rate for females, is optimal and provides an effective and faster approach to reducing the prevalence of the infection.

Optimal Control with Isoperimetric Constraint for Chemotherapy of Tumors

Optimal Control with Isoperimetric Constraint for Chemotherapy of Tumors

Embedded Delaunay tori and their Willmore energy

A family of embedded rotationally symmetric tori in the Euclidean 3-space consisting of two opposite signed constant mean curvature surfaces that converge as varifolds to a double round sphere is constructed. Using complete elliptic integrals, it is shown that their Willmore energy lies strictly below 8π. Combining such a strict inequality with previous works by Keller–Mondino–Rivière and Mondino–Scharrer allows to conclude that for every isoperimetric ratio there exists a smoothly embedded torus minimising the Willmore functional under isoperimetric constraint, thus completing the solution of the isoperimetric-constrained Willmore problem for tori. Similarly, we deduce the existence of smoothly embedded tori minimising the Helfrich functional with small spontaneous curvature. Moreover, it is shown that the tori degenerate in the moduli space which gives an application also to the conformally-constrained Willmore problem. Finally, because of their symmetry, the Delaunay tori can be used to construct spheres of high isoperimetric ratio, leading to an alternative proof of the known result for the genus zero case.

Constrained controlled optimization problems involving second-order derivatives

In this paper, we investigate the nonlinear Lagrange dynamics associated with two classes of constrained controlled optimization problems involving second- order derivatives. More precisely, we formulate and prove necessary conditions of optimality for the considered variational control problems governed by simple integral functionals and second-order ordinary differential equation (ODE)/ isoperimetric constraints. Moreover, we propose an algorithm to synthesize the concrete steps to be followed for solving constrained controlled optimization problems.

Elastic disk with isoperimetric Cosserat coating

A circular elastic disk is coated with an elastic beam, absorbing shear and normal forces without deformation and linearly reacting to a bending moment with a change in curvature. The inextensibility of the elastic beam introduces an isoperimetric constraint, so that the length of the initial circumference of the disk is constrained to remain fixed during the loading of the disk/coating system. The mechanical model for this system is formulated, solved for general loading, and particularized to the case of two equal and opposite traction distributions, each applied on a small boundary segment (thus modeling indentation of a coated fiber). The stress fields, obtained via complex potentials, are shown to evidence a nice correspondence with photoelastic experiments, ad hoc designed and performed. The presented results are useful for the design of coated fibers at the micro and nano scales.

Effects of Isoperimetric Constraint for Reducing HSV-2 Infection Using Optimal Control Strategy

Optimal control is helpful for testing and comparing different vaccination strategies of a certain disease. Genital herpes is one of the most prevalent sexually-transmitted diseases globally. In this paper, we have proposed an optimal control problem applied to HSV-2 model after introducing the constraint and state variables. Optimal control problem is formulated based on ordinary differential equation and isoperimetric constraint in the vaccine supply is also included. Mathematical analysis such as the characterization of optimal control using Pontryagin’s maximum principle is studied. Generally optimal control theory is used for finding the optimal way for implementing the strategies, minimizing the number of infectious and latent individuals and keeping the cost of implementation as low as possible. Here the optimality system is derived and solved numerically using a Runge-Kutta fourth order method and this is an iterative method. Using numerical simulation we observe that how the optimal vaccination schedule is altered by imposing isoperimetric constraint. Finally, on applying the isoperimetric constraint on the optimal control problem of HSV-2 epidemic model, we observe that optimal vaccination schedule with isoperimetric constraint indicates successful short-term control of the disease.
 GUB JOURNAL OF SCIENCE AND ENGINEERING, Vol 7, Dec 2020 P 62-68

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

Abstract 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.

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.

On Asymptotics of Small-Time Reachable Sets for Nonlinear Systems with Isoperimetric Constraints

We study the problem of approximate representation of reachable sets at small time intervals for nonlinear control-affine systems with isoperimetric control constraints. By isoperimetric constraints we mean integral constraints on the control input and state of the system. It is shown that under certain conditions imposed on the controllability Gramian of the linearized system the reachable set turns out to be convex and asymptotically close in shape to an ellipsoid in the state space for a sufficiently small time interval. This ellipsoid is a reachable set of the system linearized along the trajectory generated by zero control input.

Optimal area variation for maximum stiffness isostatic beams under parametric linear distributed loads

The stiffness maximization of elastic straight Euler-Bernoulli beams under the action of linearly distributed loads is addressed. The goal is achieved by minimizing the average compliance, which is given by the value of internal elastic energy distributed over the length of the beam. Studies in the literature suggest considering this approach since it provides, unlike the minimization of the maximum deflection, a constant bending stress behavior along the beam axis. An isoperimetric constraint on the material volume is also considered and optimal solutions are derived by means of calculus of variations. Three types of boundary conditions are discussed, namely cantilever, simply supported and guided-simply supported beams. Introducing a well known relation between the cross sectional area and moment of inertia, closed-form solutions for several cross sections commonly used in engineering are derived. Finally, a sensitivity analysis with respect to the load parameters is addressed within a numerical example.

Continuum mechanics of a cellular tissue model

We consider a two-dimensional cellular vertex model, modeling the mechanics of epithelial tissues. The energy of a planar configuration penalizes deviations in each cell from a reference perimeter P0 and a reference area A0. We study the variational limit of this model as the cell size tends to zero, obtaining a continuum variational model. For P02/A0 below a critical threshold, which corresponds to an isoperimetric constraint, the system is residually-stressed—there are no zero-energy states. For P02/A0 above this threshold, the zero-energy states are highly degenerate, allowing in particular for the formation of microstructures, which are not captured by formal long-wavelength expansions.

Essential Boundedness and Singularity in Optimal Control

Sufficient optimality conditions for optimal control problems involving isoperimetric and mixed inequality and equality constraints are derived. The main novelty of our approach is the fact that, for such problems, discontinuous and singular solutions can be detected. In other words, our result can deal with solutions for which the proposed optimal control is not continuous, but only essentially bounded, and the classical, crucial strengthened Legendre-Clebsch condition is no longer imposed.