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This article discusses the variational iteration method (VIM) and its significance in solving a wide range of physical problems. The VIM does not require linearization or manipulation of nonlinear terms, making it a powerful mathematical tool. However, the identification of the Lagrange multiplier can be challenging. The article proposes a new method called the He-transform, which is a generalized integral transform that simplifies the identification of the Lagrange multiplier. This method has practical implications in various engineering fields. The authors hope that this method will attract a wide audience and be applied to solve nonlinear oscillators with generalized initial values in the future. [Extracted from the article]

An energy functional for the obstacle problem in linear elasticity is obtained as a variational limit of nonlinear elastic energy functionals describing a material body subject to pure traction load under a unilateral constraint representing the rigid obstacle. There exist loads pushing the body against the obstacle, but unfit for the geometry of the whole system body-obstacle, so that the corresponding variational limit turns out to be different from the classical Signorini problem in linear elasticity. However, if the force field acting on the body fulfils an appropriate geometric admissibility condition, we can show coincidence of minima. The analysis developed here provides a rigorous variational justification of the Signorini problem in linear elasticity, together with an accurate analysis of the unilateral constraint.

We show convergence of minimizers of weighted inertia-energy functionals to solutions of initial value problems for a class of nonlinear wave equations. The result is given for the nonhomogeneous case under a natural growth assumption on the forcing term. [ABSTRACT FROM AUTHOR]

Owing to a variety in nanoscale material applications, the conjunction of two nanostructures is frequently researched for potential new applications. Numerous methods are used to model this conjunction process. One such method, the minimization of elastic energy, only considers the axial curvature when modeling conjoined structures. Another method minimizes the Willmore energy, which depends on both the axial and rotational curvatures. In particular, because the catenoid is an absolute minimizer of Willomre energy, a catenoid section can be utilized to conjoin nanostrucrures. Owing to the similarities among carbon nanostructures, we expanded the use of two different energies to join a boron nitride nanotube with a boron nitride nanotorus. The primary objective of this study was to formulate a basic underlying structure from which any small perturbations can be viewed as departures from an ideal model. Accordingly, elastic energy was used to determine the conjunction region for two-dimensional structures, whereas Willmore energy was used to determine the conjunction region for three-dimensional structures. This approach may be extended to produce other hybrid nanoscale structures. [ABSTRACT FROM AUTHOR]

The differential entropy is a natural analog of the Shannon entropy for discrete distributions in respect to absolutely continuous distributions (with density). In modern studies, many other kinds of entropy have been introduced and analyzed, including various cumulative entropies, which are based not on the density but on the (cumulative) distribution function of random variable. Such characteristics can be used, for example, in computer vision, reliability theory, risk analysis, etc. We consider some generalizations of cumulative entropy, for a wide class of entropy generators. We use the methods of probability theory, calculus of variations and Cauchy-Bunyakovsky-Schwarz inequality. In the class of centered and normalized random variables, exact and conditional bounds are found as well as the distributions on which they are attained. By conditional bounds we understand bounds for one generalized cumulative entropy given the value of another entropy (in the class of random variables with zero mean and unit variance). This problem is analogous to the previously posed and partly solved problem on conditional bounds for expectations of sample maxima when we know the expected maximum of a sample of another size or expected maxima of two smaller samples. [ABSTRACT FROM AUTHOR]

Motivated by some landslide models, and related estimation challenges, this paper presents an optimal estimation method for state and parameter in a special class of so-called ODE–PDE system based on the adjoint method for discrete-time asynchronous measurements. This system is described by a pair of coupled Ordinary Differential Equation (ODE) and Partial Differential Equation (PDE), with a mixed boundary condition for the PDE. The coupling appears both in the ODE and in the Neuman boundary condition of the PDE. For this system, initial conditions or state variables and some empirical parameters are assumed to be unknown and need to be estimated. The Lagrangian multiplier method is used to connect the dynamics of the system and the cost function defined as the least square error between the simulation values and the available measurements. The adjoint state method is applied to the objective functional to get the adjoint system and the gradients with respect to parameters and initial state. The cost functional is optimised, employing the steepest descent method to estimate parameters and initial state. This general approach is illustrated by two application examples corresponding to two different landslide models that validate the presented optimal estimation approach. The first one is about state and parameter estimation in an extended sliding-consolidation landslide model, and the second one is in the viscoplastic sliding-consolidation landslide model. [ABSTRACT FROM AUTHOR]

We present several variations of a proof of the position-momentum uncertainty principle that are based on the calculus of variations and does not rely on the Cauchy–Schwartz inequality. We show that the stationary uncertainty wave functions are the Hermite–Gaussian solutions to the quantum harmonic oscillator problem, that the minimum uncertainty wave function is the Gaussian, and that stationary uncertainty wave functions must be their own Fourier transforms. We also provide a calculus of variations proof of the Cauchy–Schwartz inequality. Finally, we discuss the properties of wave functions that are their own Fourier transforms and provide examples of such functions that may be of interest to teachers of undergraduate and graduate level quantum mechanics courses. [ABSTRACT FROM AUTHOR]

In this paper we consider the problem of minimizing functionals of the form E (u) = ∫ B f (x , ∇ u) d x in a suitably prepared class of incompressible, planar maps u : B → R 2 . Here, B is the unit disk and f (x , ξ) is quadratic and convex in ξ . It is shown that if u is a stationary point of E in a sense that is made clear in the paper, then u is a unique global minimizer of E(u) provided the gradient of the corresponding pressure satisfies a suitable smallness condition. We apply this result to construct a non-autonomous, uniformly convex functional f (x , ξ) , depending smoothly on ξ but discontinuously on x, whose unique global minimizer is the so-called N - covering map, which is Lipschitz but not C 1 . [ABSTRACT FROM AUTHOR]

In the present paper, a nonlinear system of sine-Gordon equations that describes the DNA dynamics is considered. A novel unconditionally stable second-order accuracy difference scheme corresponding to the system of sine-Gordon equations is presented. In this work, for the first time in the literature, weak solution of this difference scheme is studied. The existence and uniqueness of the weak solution for the difference scheme are proved in the space of distributions, and the methods of variational calculus are applied. The finite-difference method and the fixed point theory are used in combination to perform numerical experiments that verify the theoretical statements. [ABSTRACT FROM AUTHOR]

An energy functional for the obstacle problem in linear elasticity is obtained as a variational limit of nonlinear elastic energy functionals describing a material body subject to pure traction load under a unilateral constraint representing the rigid obstacle. There exist loads pushing the body against the obstacle, but unfit for the geometry of the whole system body-obstacle, so that the corresponding variational limit turns out to be different from the classical Signorini problem in linear elasticity. However, if the force field acting on the body fulfils an appropriate geometric admissibility condition, we can show coincidence of minima. The analysis developed here provides a rigorous variational justification of the Signorini problem in linear elasticity, together with an accurate analysis of the unilateral constraint. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to investigate the upper bound and H¨older continuity for a general class of parametric elliptical variational-hemivariational inequalities via regularized gap functions. More precisely, we deliver a formulation of the elliptical variational-hemivariational inequalities in the case of the perturbed parameters governed by both the set of constraints and the mappings (for brevity, PEVHI (CM)). Based on the arguments of monotonicity and properties of the Clarke's generalized directional derivative, we establish an upper bound result for the PEVHI (CM) and provide the H¨older continuity of the solution mapping for the PEVHI (CM) under suitable assumptions on the data. [ABSTRACT FROM AUTHOR]

We study the dynamics of particles in cold electron plasma medium based on two dissimilar approaches: the fractional actionlike variational and fractal calculus approaches. In each case, the corresponding Boltzmann and Vlasov–Boltzmann equations were derived. Although the mathematical relationships between fractional calculus and fractal calculus were established in the literature, it was revealed throughout this study that the corresponding physics for its approach is quite different. Each model is characterized by its corresponding Boltzmann and Vlasov–Boltzmann equations which describe dissimilar dynamics and gives rise to unrelated Bohm–Gross formulas for electron plasma waves (dispersion relations) and different group velocities connected to the numerical ranges of the matching fractional parameter. Several consequences were obtained and discussed accordingly. [ABSTRACT FROM AUTHOR]

We revisit the notion of parametrization invariance while introducing certain weakened notions of invariance in the calculus of variations. In this work, we employ a straightforward approach in the classical setting and mostly restrict attention to functionals on one-dimensional domains. We establish a connection between parametrization invariant functionals and functionals embodying a weaker notion of invariance of their Lagrangian; we term this notion as T - Lagrangian analogous to the well-known idea of null Lagrangian. However, the Euler–Lagrange operator of a T -Lagrangian vanishes only along the tangential direction in the configuration space. On one-dimensional domain and for first- and second-order theories, we show that functional described by such a Lagrangian is necessarily a parametrization invariant functional modulo null Lagrangian. Keeping the motivation for partial differential equations, we also introduce and explore the notion of N - Lagrangian, with an invariance complementary to the case of T - Lagrangian, whose Euler–Lagrange operator vanishes along normal directions. We find that in a one-dimensional setting, every N -Lagrangian is simply a null Lagrangian. [ABSTRACT FROM AUTHOR]

Francis Ysidro Edgeworth's unduly neglected monograph New and Old Methods of Ethics (1877) advances a highly sophisticated and mathematized account of social well-being in the utilitarian tradition of his 19th-century contemporaries. This article illustrates how his usage of the 'calculus of variations' was combined with findings from empirical psychology and economic theory to construct a consequentialist axiological framework. A conclusion is drawn that Edgeworth is a methodological predecessor to several important methods, ideas, and issues that continue to be discussed in contemporary social well-being studies. • Provides an account of Francis Ysidro Edgeworth's psychological and economic views, and how his interlocutors influenced him. • Discusses his methodology in the monograph New and Old Methods of Ethics. • Illustrates his usage of mathematical methods in physics for the application of social well-being research. • Contributes to the history of well-being research by way of 1870's economic thinking. • Present methods and issues in social well-being methodology have precedent in the 1870's. [ABSTRACT FROM AUTHOR]

The uncertainty relation is considered to be one of key features of quantum mechanics which distinguishes quantum and classical systems. Recently, we developed a new formulation of the uncertainty relation based on the generalized scheme of variational principle, the stochastic variational method (SVM). In this method, the uncertainty relation is related to the nondifferentiability of observables and thus can be obtained even in classical stochastic systems. This new formulation resolves the famous paradox in quantum mechanics, the angular uncertainty relation without introducing artificial assumptions. In this paper, we show that the fluctuations of position and momentum for a nonrelativistic viscous fluid element satisfies the uncertainty relation analogous to the corresponding quantum mechanical one. Such a fluctuation is sensitive to the temperature gradient at the freezeout surface and can affect the collective flow anisotropy in relativistic heavy‐ion collisions. [ABSTRACT FROM AUTHOR]

In this paper, a new version of a higher-order strongly convex function is introduced which is named (ω-σ)-higherorder strongly convex function. Sufficient conditions for the existence of minimum for (ω - σ)-higher order strongly convex function is provided. The vector version of (ω-σ)-higher order strongly convex function is given and by using KKM theory an existence results for a solution of it is proved. Moreover, the compactness of the solution set of the vector version of (ω -σ)-higher order strongly convex function is investigated. The results of this article improve and extend the corresponding results presented in this area. [ABSTRACT FROM AUTHOR]

A PDE‐constrained model predictive control (MPC) algorithm for open‐channel systems based on the Saint‐Vevant(S‐V) equations is investigated in this paper. The S‐V equations, which precisely model the dynamics of open‐channel systems, are quasi‐linear hyperbolic partial differential equations (PDEs) without analytical solutions. Directly applying the S‐V equations to an MPC controller design becomes sophisticated. In this work, the calculus of variation is used to obtain the adjoint equations and the adjoint analysis method is utilized to deduce the gradients of the MPC optimization problem. Particularly, the physical constraints involving both the state and control variables are also considered. A gradient‐based optimization algorithm in combination with the numerical computation of Preissmann implicit scheme is proposed to solve the constrained MPC optimization problem. The control performances of the developed PDE‐constrained MPC algorithm with respect to the controlled water levels and gate openings are compared with those of the MPC controller designed for the linearized model. All the simulation tests are carried out on an aqueduct reach in Yehe Irrigation District in Hebei Province, China. The results show that the proposed PDE‐constrained MPC algorithm is a promising method in dealing with the constraints in terms of hyperbolic PDEs, control variables and state variables simultaneously. [ABSTRACT FROM AUTHOR]

In this paper, we obtain necessary and sufficient conditions for the existence of variational principles for a given first-order differential-difference operator equation with a special form of the linear operator Pλ(t) depending on t and the nonlinear operator Q. Under the corresponding conditions the functional is constructed. These conditions are obtained thanks to the well-known criterion of potentiality. Examples show how the inverse problem of the calculus of variations is constructed for given differential-difference operators. [ABSTRACT FROM AUTHOR]

Non-zonal methods of hybrid Reynolds Averaged Navier–Stokes (RANS) equations-Large Eddy Simulation (LES) with subfilter transport equations enable to simulate turbulent flows out of spectral equilibrium on relatively coarse grid resolution. They can operate from RANS to LES depending on a control parameter linked to the grid step size. Variational analysis gives a useful framework to specify the functional dependence of this parameter to the grid size. This is the case for the partially integrated transport modelling method originally established in spectral space for homogeneous turbulence. The partitioning control function then monitors the ratio of the subfilter part to the total turbulent energy and the same process is developed in the present work for the thermal or transported scalar variance as well. So, we demonstrate here that this control function mechanism can be derived by variational calculus from a mathematical physics formalism developed both for first-order and second-order moment closures. We show that the result is entirely consistent with the spectral model derivation made in previous papers and that this approach can be transposed to almost any subfilter transport model developed in hybrid RANS-LES methodology in full generality. As a result, it will be evidenced also that resolved turbulent diffusion terms play a significant role in the acting mechanisms of turbulence and cannot be therefore neglected, even if these terms do not explicitly appear in LES subfilter closure because they are computed by the simulation itself and not modelled by the subfilter model. In addition, numerical simulations have been performed for illustrating the theoretical results of the variational analysis. [ABSTRACT FROM AUTHOR]

We extend known results concerning directional transience of nearest-neighbor random walks in random environments to allow for bounded jumps. Zerner and Merkl (2001) proved a 0-1 law for directional transience for planar random walks in random environments. We extend the result to non-planar i.i.d. random walks in random environments on Z2 with bounded jumps. Sabot and Tournier (2017) characterized directional transience for a given direction for nearest-neighbor random walks in Dirichlet environments on Zd, d = 1. We extend this characterization to random walks in Dirichlet environments with bounded jumps. [ABSTRACT FROM AUTHOR]

We prove local Lipschitz regularity for local minimisers of W 1 , 1 (Ω) ∋ v ↦ ∫ Ω F (D v) d x where Ω ⊆ R N , N ≥ 2 and F : R N → R is a quasiuniformly convex integrand in the sense of Kovalev and Maldonado (Ill J Math 49:1039–1060, 2005), i. e. a convex C 1 -function such that the ratio between the maximum and minimum eigenvalues of D 2 F is essentially bounded. This class of integrands includes the standard singular/degenerate functions F (z) = | z | p for any p > 1 and arises as the closure, with respect to a natural convergence, of the strongly elliptic integrands of the Calculus of Variations. [ABSTRACT FROM AUTHOR]

Multinational corporations operate across locations with different risk profiles. We examine how multinational corporations address the optimal allocation of capital across multiple locations and analyse the transition path to the intertemporal equilibrium. Our model considers returns, risks and adjustment costs to reflect the dynamics of allocating capital assets across locations over time, as well as the mix of assets across locations in equilibrium. Variational calculus is employed to show that the model confirms standard expectations that where a location's rates of return on assets increase, or adjustment costs decrease, equilibrium capital allocation and transitional capital flows to that location will increase. Symmetrically, rising (falling) risk increases (decreases) the proportion of the capital asset holdings of a location. The crucial insight is that for the transitional dynamics to intertemporal equilibrium, the optimal relative capital flow response to changes in risk can generate relative portfolio allocations that may initially move in the opposite direction to that implied by the stock equilibrium. Specifically, an increase in risk for the high-risk location may initially result in an increase in the relative capital asset flow to the high-risk location relative to the low-risk location. Empirical research must account for the possibility of non-monotonicity in asset allocation flows to avoid misspecification. Moreover, policy makers will have to anticipate possible pressure for reversal resulting from short-term worsening capital flows. These reflections are mirrored in recent research calls for separating structural and transition effects of institutional change on the investment decisions by multinational corporations. [ABSTRACT FROM AUTHOR]

In this paper, we will prove a theorem for a functional where we prove that the necessary condition for the extreme of a functional is that the variation of a functional is equal to zero and we will give an example of its application to the catenary problem. [ABSTRACT FROM AUTHOR]

In the calculus of variations, an important role is played by the Minkowski duality, or the Legendre transform of convex functions. We consider weakly regular, conformally flat Riemannian metrics of nonnegative curvature defined on the n-dimensional unit sphere. For this class of metrics, an analog of the Legendre transformation is introduced and studied in detail. [ABSTRACT FROM AUTHOR]

In the conventional formulation of general relativity, gravity is represented by the metric curvature of Riemannian geometry. There are also alternative formulations in flat affine geometries, wherein the gravitational dynamics is instead described by torsion and nonmetricity. These so-called general teleparallel geometries may also have applications in material physics, such as the study of crystal defects. In this work, we explore the general teleparallel geometry in the language of differential forms. We discuss the special cases of metric and symmetric teleparallelisms, clarify the relations between formulations with different gauge fixings and without gauge fixing, and develop a method of recasting Riemannian into teleparallel geometries. As illustrations of the method, exact solutions are presented for the generic quadratic theory in 2, 3 and 4 dimensions. [ABSTRACT FROM AUTHOR]

This work studies a mathematical model involving a dynamic contact between two elasto-viscoplastic piezoelectric bodies with damage. The contact is modeled on a combination of a normal compliance and a normal damped response law associated with friction. We derive a variational formulation of the problem and we prove the existence and uniqueness result of the weak solution. The proof is based on the classical existence and uniqueness result for parabolic inequalities, differential equations and fixed-point arguments. This paper examines a friction problem between two piezoelectric bodies that are elastic- viscoplastic. The study uses both numerical and analytical methods. The contact surface is assumed to have a thin layer of lubricant, and a damped response contact condition is considered. The problem also takes into account electrical and frictional effects. The paper proves the existence of a unique weak solution to the problem by using variational inequalities and a fixed point argument. Additionally, a fully discrete approximation is investigated using the Euler scheme and finite element method for the spatial variable and time derivatives. Error estimates are then calculated, leading to convergence results under certain regularity conditions. [ABSTRACT FROM AUTHOR]

In this work our main objective is to establish various (high frequency-) uniqueness criteria. Initially, we consider p -Dirichlet type functionals on a suitable class of measure preserving maps u : B ⊂ R 2 → R 2 , B being the unit disk, and subject to suitable boundary conditions. In the second part we focus on a very similar situations only exchanging the previous functionals by a suitable class of p -growing polyconvex functionals and allowing the maps to be arbitrary. In both cases a particular emphasis is laid on high pressure situations, where only uniqueness for a subclass, containing solely of variations with high enough Fourier-modes, can be obtained. [ABSTRACT FROM AUTHOR]

In this paper, a distributed computing framework that exploits high-performance compute clusters and employs a Distributed Multiple-Swarm Particle Swarm Optimization (DMSPSO) algorithm is proposed for solving two-point boundary-value problems in astrodynamics. The DMSPSO algorithm performs a search of the solution space by attempting to minimize the weighted sum of squares of the boundary condition residuals, generating many guesses for the unknown variables that nearly satisfy the boundary conditions. These guesses are then used to solve the boundary-value problem with a gradient-based algorithm that runs in parallel with the DMSPSO search. The proposed framework is applied to solve two different boundary-value problems in astrodynamics. The first is formulated to compute energy-optimal transfers between a Geostationary Transfer Orbit and an L1 halo orbit using the Calculus of Variations approach, and the second is formulated to compute periodic orbits in the Circular Restricted Three Body Problem model. It is demonstrated that the methodology provides a large speedup over a conventional Particle Swarm Optimization algorithm, can be applied to solve sensitive boundary value problems, and facilitates the discovery of many (if not all) solutions of a given two-point boundary-value problem. [ABSTRACT FROM AUTHOR]

The article considers a high-order optimal control problem and its dual problems described by high-order differential inclusions. In this regard, the established Euler–Lagrange type inclusion, containing the Euler–Poisson equation of the calculus of variations, is a sufficient optimality condition for a differential inclusion of a higher order. It is shown that the adjoint inclusion for the first-order differential inclusions, defined in terms of a locally adjoint mapping, coincides with the classical Euler–Lagrange inclusion. Then the duality theorems are proved. [ABSTRACT FROM AUTHOR]

The behavior and evolution of stepped circular bi-laminates with edge damage are studied for structures subjected to uniform thermal load. The problem is formulated as a moving intermediate boundaries problem in the calculus of variations, where the boundary of an evolving region of damage emanating from the edge of the smaller substructure, as well as the boundary of a progressing/regressing region of sliding contact adjacent to the intact region of the composite structure are each allowed to vary along with the displacements. This yields the associated transversality conditions that define the locations of the propagating boundaries that correspond to equilibrium configurations of the evolving composite structure, as well as the equations of equilibrium and the associated interior and exterior boundary conditions. Various configurations of contact of the detached segments of the composite structure and the associated behavior are considered, and the influence and progression of contact on the overall evolution of the composite structure are assessed. Closed-form analytical solutions to the geometrically nonlinear problem are obtained, and expressions for the critical buckling load are developed. The explicit forms of the total energy release rate along the delamination front, as well as of the conditions for propagation of the contact zone boundary, are obtained from the analytical solutions and the corresponding transversality conditions. Results of numerical simulations based on the analytical solutions are presented and are seen to unveil a rich evolution process involving contact progression/recession and metamorphosis, buckling, and detachment progression during the prebuckling phase, during sling-shot buckling, and during the postbuckling phase, depending on the material properties of the sublaminates, the geometry of the sublaminates, the initial size of the damage, and the strength of the interfacial bond. Characteristic behavior of damage propagation is found to be quite robust and is seen to include stable propagation, stable followed by unstable progression, arrest, and catastrophic propagation. [ABSTRACT FROM AUTHOR]

In this paper, we use the calculus of variations to derive a sensitivity analysis for ordinary differential equations with events. This sweeping gradient method (SGM) requires a forward sweep to evaluate the original model and a backwards sweep of the adjoint to compute the sensitivity. The method is applied to canonical optimal control problems with numerical examples, including the sampled linear quadratic regulator and the optimal time-switching and state-switching for minimum-time transfer of the double integrator. We show that the application of the SGM for these examples matches the gradient determined analytically. Numerical examples are produced using gradient-based optimization algorithms. The emphasis of this work is on modeling considerations for the effective application of this method. [ABSTRACT FROM AUTHOR]

This paper is devoted to showing a discrete adaptive finite element approximation result for the isotropic two-dimensional Griffith energy arising in fracture mechanics. The problem is addressed in the geometric measure theoretic framework of generalized special functions of bounded deformation, which corresponds to the natural energy space for this functional. It is proved to be approximated in the sense of T-convergence by a sequence of discrete integral functionals defined on continuous piecewise affine functions. The main feature of this result is that the mesh is part of the unknown of the problem, and it gives enough flexibility to recover isotropic surface energies. [ABSTRACT FROM AUTHOR]

In this paper, a general expression formula for the Boubaker scaling (BS) operational matrix of the derivative is constructed. Then it is used to study a new parameterization direct technique for treating calculus of the variation problems approximately. The calculus of variation problems describe several important phenomena in mathematical science. The first step in our suggested method is to express the unknown variables in terms of Boubaker scaling basis functions with unknown coefficients. Secondly, the operational matrix of the derivative together with some important properties of the BS are utilized to achieve a non-linear programming problem in terms of the unknown coefficients. Finally, the unknown parameters are obtained using the quadratic programming technique. Some numerical examples are included to confirm the accuracy and applicability of the suggested direct parameterization method. [ABSTRACT FROM AUTHOR]

This article considers fractional optimal control problems (FOCPs) including both integer‐order and Atangana–Baleanu Caputo derivatives. First, the existence and uniqueness of the solution of a fractional Cauchy problem is given. Then, applying calculus of variations and Lagrange multiplier method, we present necessary optimality conditions of FOCPs and sufficient optimality conditions are also given under some assumptions. Next, a collection method is developed to derive numerical solutions by using shifted Legendre polynomials. Finally, error estimate of numerical solutions is also provided, and numerical examples further show the accuracy and feasibility of our method. [ABSTRACT FROM AUTHOR]

There are no common methods of the search for optimal solutions to problems of economic dynamics. Application of variational methods is possible for a quite narrow class of problems. In this paper, we discuss application of the Pontryagin maximum principle, which significantly expands the class of problems considered and allows one to obtain numerical solutions. [ABSTRACT FROM AUTHOR]

This paper is devoted to the analysis of mathematical models of chemotherapy for malignant tumors growing according to the Gompertz law or the generalized logistic law. The influence of the therapeutic agent on the tumor dynamics is determined by a therapy function depending on the time-varying concentration of the drug in the patient's body. The case of a non-monotonic therapy function with two maxima is studied. It reflects the use of two different therapeutic agents. The state variables of the dynamics are the tumor volume and the amount of the therapeutic agent able to suppress malignant cells (concentration of the drug in the body). The treatment protocol (the rate of administration of the therapeutic agent) is the control in the dynamics. The optimal control problem for this models is considered. It is the problem of the construction of treatment protocols that provide the minimal tumor volume at the end of the treatment. The solution of this problem was obtained by the authors in previous works via the optimal control theory. The form of the considered therapy functions provides a specific structure for the optimal controls. The managerial insights of this structure are discussed. In this paper, the structure of the viability set is described for the model according to the generalized logistic law. It is the set of the initial states of the model for which one can find a treatment protocol that guarantees that the tumor volume remains within the prescribed limits throughout the treatment. The description of the viability set's structure is based on the optimal control theory and the theory of Hamilton–Jacobi equations. An inverse problem of therapy is also considered, namely the problem of reconstruction of the treatment protocol and identification of the unknown parameter of the intensity of the tumor growth. Reconstruction is carried out by processing information about the observations of the tumor volume dynamics and the measurements of the drug concentration in the body. A solution to this problem is obtained through the use of a method based on the calculus of variations. The results of the numerical simulations are presented herein. [ABSTRACT FROM AUTHOR]

We study the determination of some rigid inclusions immersed in an isotropic elastic medium from overdetermined boundary data. We propose an accurate approach based on the topological sensitivity technique and the reciprocity gap concept. We derive a higher-order asymptotic formula, connecting the known boundary data and the unknown inclusion parameters. The obtained formula is interesting and useful tool for developing accurate and robust numerical algorithms in geometric inverse problems. [ABSTRACT FROM AUTHOR]

We prove for the first time that classical Morse theory applies to functionals of the form 풥 ⁢ (u) = 1 2 ⁢ ∫ Ω A α ⁢ β i ⁢ j ⁢ (x) ⁢ ∂ ⁡ u i ∂ ⁡ x α ⁢ ∂ ⁡ u j ∂ ⁡ x β ⁢ 푑 x + ∫ Ω G ⁢ (x , u) ⁢ 푑 x where u : Ω → ℝ N , Ω ⊂ ℝ n compact with C ∞ boundary ∂ ⁡ Ω , u | ∂ ⁡ Ω = φ , and we argue that this is the largest class to which Morse theory applies. [ABSTRACT FROM AUTHOR]

The center of interest in this work are variational problems with integral functionals depending on nonlocal gradients with finite horizon that correspond to truncated versions of the Riesz fractional gradient. We contribute several new aspects to both the existence theory of these problems and the study of their asymptotic behavior. Our overall proof strategy builds on finding suitable translation operators that allow to switch between the three types of gradients: classical, fractional, and nonlocal. These provide useful technical tools for transferring results from one setting to the other. Based on this approach, we show that quasiconvexity, which is the natural convexity notion in the classical calculus of variations, gives a necessary and sufficient condition for the weak lower semicontinuity of the nonlocal functionals as well. As a consequence of a general Γ -convergence statement, we obtain relaxation and homogenization results. The analysis of the limiting behavior for varying fractional parameters yields, in particular, a rigorous localization with a classical local limit model. [ABSTRACT FROM AUTHOR]

Variational principles for evolutionary systems arise in many settings, both in those describing the physical world and in man-made algorithms for data science and optimization tasks. Variational principles are available for Hamiltonian systems in classical mechanics, gradient flows for dissipative systems, as well as in time-incremental minimization techniques for more general evolutionary problems. Additional challenges arise via the interplay of two or more functionals (e.g. a free energy and a dissipation potential), thus encompassing a large variety of applications in the modeling of materials and fluids, in biology, and in multi-agent systems. Variational principles and associated evolutions are also at the core of the modern approaches to machine learning tasks, since many of them are posed as minimizing an objective functional that models the problem. The discrete and random nature of these problems and the need for accurate computation in high dimension present a set of challenges that require new mathematical insights. Variational methods for evolution allow for the usage of the rich toolbox provided by the calculus of variations, metric-space geometry, partial differential equations, and other branches of applied analysis. The variational methods for evolution have seen a rapid growth over the last two decades. This workshop continued the successful line of meetings (2011, 2014, 2017, and 2020), while evolving and branching into new directions. We have brought together a wide scope of mathematical researchers from calculus of variations, partial differential equations, numerical analysis, and stochastics, as well as researchers from data science and machine learning, to exchange ideas, foster interaction, develop new avenues, and generally bring these communities closer together. [ABSTRACT FROM AUTHOR]

In this paper, an optimal switch-time control problem is solved for a class of impulsive switched autonomous systems. The considered systems jump at the switching times, and the sequence of active subsystems is pre-specified. The control variables consist of the impulse times and a set of scalars which determine the jump amplitudes. Moreover, the subsystems do not require a refractory period, which can bring more generality. Using the calculus of variation, the partial derivatives of the cost with respect to the control variables are derived, based on which the optimality conditions are given. Meanwhile, the obtained formulas can be used in some gradient descent algorithms to locate the optimal control variables. Finally, the viability of the proposed method is illustrated through two numerical examples. [ABSTRACT FROM AUTHOR]

We study a continuous-time expected utility maximisation problem where the investor at maturity receives the value of a contingent claim in addition to the investment payoff from the financial market. The investor knows nothing about the claim other than its probability distribution; hence the name "intractable claim". In view of the lack of necessary information about the claim, we consider a robust formulation to maximise her utility in the worst scenario. We apply the quantile formulation to solve the problem, express the quantile function of the optimal terminal investment income as the solution of certain variational inequalities of ordinary differential equations, and obtain the resulting optimal trading strategy. In the case of exponential utility, the problem reduces to a (non-robust) rank-dependent utility maximisation with probability distortion whose solution is available in the literature. The results can also be used to determine the utility indifference price of the intractable claim. [ABSTRACT FROM AUTHOR]

This study presents a practical method for solving fractional order delay variational problems. The fractional derivative is given in the Caputo sense. The suggested approach is based on the Laplace transform and the shifted Legendre polynomials by approximating the candidate function by the shifted Legendre series with unknown coefficients yet to be determined. The proposed method converts the fractional order delay variational problem into a set of (n ＋ 1) algebraic equations, where the solution to the resultant equation provides us the unknown coefficients of the terminated series that have been utilized to approximate the solution to the considered variational problem. Illustrative examples are given to show that the recommended approach is applicable and accurate for solving such kinds of problems.