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In this article, we have derived some sufficient conditions for existence of optimal control for semilinear control system of fractional order (1,2] in Hilbert space. We have discussed existence and uniqueness of mild solution with the help of Banach fixed point theorem for proposed problem. Under certain conditions stated, Lagrange's problem admits at least one optimal control pair. Finally, one example is given to understand theoretical results in better manner. [ABSTRACT FROM AUTHOR]

In the framework of a second‐order semilinear integro‐differential control system in Hilbert spaces, the paper provides sufficient conditions for proving the existence of optimal control. The Banach fixed point theorem is used to investigate the existence and uniqueness of mild solutions for the proposed problem. Additionally, it is shown that, under specific assumptions, there exists at least one optimal control pair for the Lagrange's problem as presented in the article. An example for validation is included in the paper to further support the theoretical findings. [ABSTRACT FROM AUTHOR]

Summary: The problem of polynomial least squares fitting in the standard Lagrange basis is addressed in this work. Although the matrices involved in the corresponding overdetermined linear systems are not totally positive, rectangular totally positive Lagrange‐Vandermonde matrices are used to take advantage of total positivity in the construction of accurate algorithms to solve the considered problem. In particular, a fast and accurate algorithm to compute the bidiagonal decomposition of such rectangular totally positive matrices is crucial to solve the problem. This algorithm also allows the accurate computation of the Moore‐Penrose inverse and the projection matrix of the collocation matrices involved in these problems. Numerical experiments showing the good behaviour of the proposed algorithms are included. [ABSTRACT FROM AUTHOR]

This article concentrates in analyzing optimal controls for stochastic integrodifferential equation (SIDE) in Hilbert space. Necessary parameters are imposed to demonstrate the system that follows a unique variation of parameter formula using Leray Schauder Alternative. Subsequently, the existence of optimal control is investigated for the considered Lagrange control problem. The theoretical example with the mechanical example of ethanol fuelled engine are discussed to validate the results obtained. [ABSTRACT FROM AUTHOR]

The classical Lagrange problem for dynamical systems introduces a Lagrangian action functional defined for any dynamical process that is envisioned to take place over a fixed interval of time with its state at each time lying on an unknown, but prescribed, configuration between two given end points in an n -dimensional state space R n . It is proposed that the fundamental dynamical field equation that characterizes the dynamical process and determines the precise motion between the two given end points is the Euler–Lagrange equation related to the stationarity of the Lagrangian action functional, expressed as the integral of a particularly formulated action density over the fixed time interval, among all admissible configurations that span the two given end points. Thus stated, this variational calculus problem introduces variations of a configuration that carries a dynamical process, and emphasizes the novelty and need to express explicitly how the configuration influences the state of that process. At each time during a dynamical process the state is subjected to an extrinsic force (classically taken to be conservative) which must be transmitted to the configuration that carries the process and, by action-reaction the configuration responds with a configuration contact force on the state of equal magnitude but opposite direction. This allows the Lagrangian action functional for a dynamical process to be interpreted as the difference between the average kinetic energy of the dynamical process that is carried by that configuration and the average configurational work done by the configuration contact force on the moving state as the state traverses that configuration during the fixed time interval. The aim in the Problem of Lagrange is to extremize this difference over all admissible configurations. The implication is that given a time interval and initial and final end points in the space of all states, the dynamical process of physical interest must follow a configuration that optimizes the gap between the average expended kinetic energy and the average expended configurational work. When the optimal condition is met and the dynamical process is so restricted, the difference between these average expenditures of energy and work will be at a local maximum, a local minimum, or a saddle point known as a condition of "least action". Herein, we investigate the optimization implications of this novel interpretation of the action functional for the Problem of Lagrange for dynamical systems for a general, possibly non-conservative, state-dependent extrinsic force field. We show that only a conservative state-dependent extrinsic force field is allowable within the statement of the problem and, thus, reaffirm the predominant classical hypothesis of restricting attention to conservative extrinsic force fields. We close with a section which covers an analogous investigation of the Problem of Lagrange for nonlinear elastodynamics. [ABSTRACT FROM AUTHOR]

In this paper, the non-autonomous fractional evolution inclusions of Clarke subdifferential type in a separable reflexive Banach space are investigated. The mild solution of the non-autonomous fractional evolution inclusions of Clarke subdifferential type is defined by introducing the operators ψ (t , τ) and ϕ (t , τ) and V(t), which are generated by the operator - A (t) and probability density function. Combined the measure of non-compactness, some properties of the Clarke subdifferential with fixed point theorem of κ - condensing multi-valued maps, a new existence result of mild solution is established. Moreover, an existence result of optimal control pair for the Lagrange problem is also derived. The results obtained in this paper extend the study of fractional autonomous evolution equations to the non-autonomous fractional evolution inclusions. Finally, a fractional partial differential inclusion with control is provided to illustrate the applications of the obtained main results. [ABSTRACT FROM AUTHOR]

We consider an optimal control problem with the boundary functional for a Schrödinger equation describing the motion of a charged particle. By using the existence of an optimal solution, we search the necessary optimality conditions for the examined control problem. First, we constitute an adjoint problem by a Lagrange multiplier that is related to constraints of theory on symmetries and conservation laws. The adjoint problem obtained is a boundary value problem with a nonhomogeneous boundary condition. We prove the existence and uniqueness of the solution of the adjoint problem. Then, we demonstrate the differentiability of the objective functional in the sense of Frechet and get a formula for its gradient. Finally, we give a necessary optimality condition in the form of a variational inequality. [ABSTRACT FROM AUTHOR]

In this paper, aiming at the spectrum coexistence scenario between non-geostationary satellite orbit (NGSO) and geostationary satellite orbit (GSO) satellite systems, the NGSO network, a secondary system, reuses the spectrum resources of the GSO primary system through spectrum sensing and sharing technology. Combined with the influence of factors such as the radio wave propagation model and channel characteristics of the sensing links, as well as the difference of channel environment between sensing nodes, a weighted cooperative spectrum sensing strategy named optimal fusion decision for time-invariant channel link (OFD-TIC) method was proposed for NGSO–GSO downlink sensing communication. The OFD-TIC method is designed to minimize the global error detection probability while considering the constraints of spectrum conflict probability and false alarm probability. To achieve this, OFD-TIC method transforms the optimization problem into a Lagrange function and utilizes Newton's iterative method to obtain the weights of sensing node and the parameter thresholds for fusion decision. In the six different spectrum sensing scenarios constructed using satellite network data registered by the International Telecommunication Union (ITU), the performance evaluation of the proposed OFD-TIC method is carried out and the simulation results demonstrate that the proposed method outperforms existing typical algorithms. Under the same signal-to-noise ratios (SNR) and sampling number, the OFD-TIC strategy achieves a higher global detection probability than others. Furthermore, the proposed method exhibits fast convergence to 1 of the global detection probability even at low SNR. [ABSTRACT FROM AUTHOR]

In this article, we consider mechanical billiard systems defined with Lagrange's integrable extension of Euler's two-center problems in the Euclidean space, the sphere, and the hyperbolic space of arbitrary dimension . In the three-dimensional Euclidean space, we show that the billiard systems with any finite combinations of spheroids and circular hyperboloids of two sheets having two foci at the Kepler centers are integrable. The same holds for the projections of these systems on the three-dimensional sphere and in the three-dimensional hyperbolic space by means of central projection. Using the same approach, we also extend these results to the -dimensional cases. [ABSTRACT FROM AUTHOR]

We consider the regularization of classical optimality conditions—the Lagrange principle and the Pontryagin maximum principle—in a convex optimal control problem with an operator equality constraint and functional inequality constraints. The controlled system is specified by a linear functional–operator equation of the second kind of general form in the space , and the main operator on the right-hand side of the equation is assumed to be quasinilpotent. The objective functional of the problem is only convex (perhaps not strongly convex). Obtaining regularized classical optimality conditions is based on the dual regularization method. In this case, two regularization parameters are used, one of which is "responsible" for the regularization of the dual problem, and the other is contained in the strongly convex regularizing Tikhonov addition to the objective functional of the original problem, thereby ensuring the well-posedness of the problem of minimizing the Lagrange function. The main purpose of the regularized Lagrange principle and Pontryagin maximum principle is the stable generation of minimizing approximate solutions in the sense of J. Warga. The regularized classical optimality conditions Are formulated as existence theorems for minimizing approximate solutions in the original problem with a simultaneous constructive representation of these solutions. Are expressed in terms of regular classical Lagrange and Hamilton–Pontryagin functions. "Overcome" the properties of the ill-posedness of the classical optimality conditions and provide regularizing algorithms for solving optimization problems. Based on the perturbation method, an important property of the regularized classical optimality conditions obtained in the work is discussed in sufficient detail; namely, "in the limit" they lead to their classical counterparts. As an application of the general results obtained in the paper, a specific example of an optimal control problem associated with an integro-differential equation of the transport equation type is considered, a special case of which is a certain inverse final observation problem. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a class of backward Riemann-Liouville fractional evolution hemivariational inequalities with dual operators. By using the duality theory on semigroup, resolvent technique, the Cesari property and Schauder's fixed point theorem, we obtain the existence of feasible pairs. Then we prove the existence of optimal feedback control pairs for the Lagrange problem. Finally, we give an example in a Hilbert space to illustrate the results. [ABSTRACT FROM AUTHOR]

A new numerical-and-analytical method is developed for solving geometrically and physically nonlinear problems of bending shallow shells of complex shapes made from materials with different resistance to tension and compression. To linearize the initial nonlinear problem, the method of continuous continuation in the parameter associated with the external load was used. For the variational formulation of the linearized problem, a Lagrange functional was constructed, defined at kinematically possible displacement velocities. To find the main unknowns of the problem of nonlinear bending of a hollow shell (displacements, deformations, stresses), the Cauchy problem for a system of ordinary differential equations is formulated. The Cauchy problem was solved by the Runge-Kutta– Merson method with automatic step selection. The initial conditions are found in the solution to the problem of geometrically linear deformation. The right-hand sides of the differential equations at fixed values of the load parameter corresponding to the Runge-Kutta–Merson scheme were obtained from the solution of the variational problem for the Lagrange functional. The variational problems were solved by the Ritz method in combination with the R-function method. The latter makes it possible to present an approximate solution in the form of a formula, which solution structure exactly satisfies all (general structure) or part (partial structure) of the boundary conditions. The problems of nonlinear deformation of a square cylindrical shell and a shell of complex shape with combined fixation conditions are solved. The influence of the direction of external loading, geometric shape, and fixation conditions on the stress-strain state is investigated. It is shown that failure to consider the different behaviors of the material in tension and compression leads to significant errors in calculating the stress-strain state parameters. [ABSTRACT FROM AUTHOR]

The objective of this paper is to investigate multiobjective semiinfinite programming problems with vanishing constraints (in brief, MSIPVC). We firstly propose both the vector Lagrange dual problems and the scalarized Lagrange dual problems for MSIPVC and explore duality relations under convexity assumptions. Then, the saddle point optimality conditions for MSIPVC are considered. Some examples are also given to illuminate the main results. [ABSTRACT FROM AUTHOR]

Given closed convex sets Ci, i=1, ... l, and some nonzero linear maps Ai, i=1, ... l, of suitable dimensions, the multiset split feasibility problem aims at finding a point in ... based on computing projections onto Ci and multiplications by Ai and AiT. In this paper, we consider the associated best approximation problem, i.e., the problem of computing projections onto ...; we refer to this problem as the best approximation problem in multiset split feasibility settings (BA-MSF). We adapt the Dykstra's projection algorithm, which is classical for solving the BA-MSF in the special case when all Ai - I, to solve the general BA-MSF. Our Dykstra-type projection algorithm is derived by applying (proximal) coordinate gradient descent to the Lagrange dual problem, and it only requires computing projections onto Ci and multiplications by Ai and AiT in each iteration. Under a standard relative interior condition and a genericity assumption on the point we need to project, we show that the dual objective satisfies the Kurdyka-Łojasiewicz property with an explicitly computable exponent on a neighborhood of the (typically unbounded) dual solution set when each Ci is C1,α.-cone reducible for some α∈ (0,1]: this class of sets covers the class of C²-cone reducible sets, which include all polyhedrons, second-order cone, and the cone of positive semidefinite matrices as special cases. Using this, explicit convergence rate (linear or sublinear) of the sequence generated by the Dykstra-type projection algorithm is derived. Concrete examples are constructed to illustrate the necessity of some of our assumptions. [ABSTRACT FROM AUTHOR]

This article focuses on Energy Management Problem (EMP) in Muti‐microgrid systems. Each microgrid (MG) has four basic equipments including Renewable Generator (RG), Fuel Generator (FG), Storage Device (SD) and Smart Load (SL). In consideration of equipment capacity, confidence interval for forecasts of RG output, supply–demand balance and other factors, an optimization model with the objective of minimizing operation cost was established. Through Lagrange dual problem and variable substitution, we transform the centralized problem into an equivalent distributed form. Then, a completely distributed algorithm based on Alternating Direction Method of Multipliers (ADMM) and Average Consensus (AC) is proposed to attain a global optimal schedule plan. The algorithm overcomes the defect that each microgrid needs to know the total must‐run load in advance. At the same time, the electricity clearing price is related to the objective function through Lagrange dual variables, which can be obtained while the optimal plan is determined. Finally, the simulation verifies the convergence of the algorithm, and the calculated optimal cost is the same as that of the centralized method, ensuring its effectiveness. Besides, the algorithm also has good performance in plug‐and‐play scenarios. © 2023 Institute of Electrical Engineer of Japan and Wiley Periodicals LLC. [ABSTRACT FROM AUTHOR]

We consider a Lagrange optimal control problem for a Volterra integral equation of fractional potential type. We prove a theorem on the existence of an optimal solution and derive a maximum principle. The proof of the existence theorem is based on the lower closure theorem for orientor fields due to Cesari and Filippov‐type selection theorem due to Rockafellar. The proof of the maximum principle is based on an extremum principle for smooth problems proved in Idczak and Walczak (Games. 2020;11:56). [ABSTRACT FROM AUTHOR]

Application of the time-dependent variational principle to a linear combination of frozen-width Gaussians describing the nuclear wavefunction provides a formalism where the total energy is conserved. The computational downside of this formalism is that trajectories of individual Gaussians are solutions of a coupled system of differential equations, limiting implementation to serial propagation algorithms. To allow for parallelization and acceleration of the computation, independent trajectories based on simplified equations of motion were suggested. Unfortunately, within practical realizations involving finite Gaussian bases, this simplification leads to breaking the energy conservation. We offer a solution for this problem by using Lagrange multipliers to ensure the energy and norm conservation regardless of basis function trajectories or basis completeness. We illustrate our approach within the multi-configurational Ehrenfest method considering a linear vibronic coupling model. [ABSTRACT FROM AUTHOR]

In this paper, the application of the strong-form finite block method (FBM) to three-dimensional fracture analysis with functionally graded materials is presented. The main idea of the strong-form FBM is that it transforms the arbitrary physical domain into a normalized domain and utilizes the direct collocation method to form a linear system. Using the mapping technique, partial differential matrices of any order can be constructed directly. Frameworks of the strong-form FBM for three-dimensional problems based on Lagrange polynomial interpolation and Chebyshev polynomial interpolation were developed. As the dominant parameters in linear elastic fracture mechanics, the stress intensity factors with functionally graded materials (FGMs) were determined according to the crack opening displacement criteria. Several numerical examples are presented using a few blocks to demonstrate the accuracy and efficiency of the strong-form FBM. [ABSTRACT FROM AUTHOR]

We consider an optimal control problem constrained by a parabolic partial differential equation with Robin boundary conditions. We use a space–time variational formulation in Lebesgue–Bochner spaces yielding a boundedly invertible solution operator. The abstract formulation of the optimal control problem yields the Lagrange function and Karush–Kuhn–Tucker conditions in a natural manner. This results in space–time variational formulations of the adjoint and gradient equation in Lebesgue–Bochner spaces, which are proven to be boundedly invertible. Necessary and sufficient optimality conditions are formulated and the optimality system is shown to be boundedly invertible. Next, we introduce a conforming uniformly stable simultaneous space–time (tensorproduct) discretization of the optimality system in these Lebesgue–Bochner spaces. Using finite elements of appropriate orders in space and time for trial and test spaces, this setting is known to be equivalent to a Crank–Nicolson time-stepping scheme for parabolic problems. Comparisons with existing methods are detailed. We show numerical comparisons with time-stepping methods. The space–time method shows good stability properties and requires fewer degrees of freedom in time to reach the same accuracy. [ABSTRACT FROM AUTHOR]

In this article, we consider the chemostat system with n≥1$$ n\ge 1 $$ species, one limiting substrate, and mutations between species. Our objective is to globally stabilize the corresponding dynamical system around a desired equilibrium point. Doing so, we introduce auxostat feedback controls which are controllers allowing the regulation of the substrate concentration. We prove that such feedback controls globally stabilize the resulting closed‐loop system near the desired equilibrium point. This result is obtained by combining the theory of asymptotically autonomous systems and an explicit computation of solutions to the limit system. The performance of such controllers is illustrated on an optimal control problem of Lagrange type which consists in maximizing the production of species over a given time period w.r.t. the dilution rate chosen as control variable. [ABSTRACT FROM AUTHOR]

In this article, the main objective is the conversation about the optimal control problem of the semilinear thermoelastic system, in which the control term is placed solely in the thermal equation. We discuss the existence and uniqueness of mild solutions by applying the contraction mapping for the considered system. By assuming some conditions specified Lagrange's problem acknowledges at least one optimal control pair. For proving the main results, we are assuming the Lipschitz condition on the nonlinear term. [ABSTRACT FROM AUTHOR]

Autonomous intelligent transportation systems consistently require effective and secure communication through vehicular networks, enabling autonomous vehicle communication. The reduction of traffic congestion, the alerting of approaching emergency vehicles, and assistance in low visibility traffic are all made possible by effective communication between autonomous vehicles and everything (AV2X). Therefore, a new adaptive AV2X model is proposed in this paper to improve the connectivity of vehicular networks. This proposed model is based on the optimization method and a distributed deep learning model. The presented approach optimizes the inter-vehicle location if required for ensuring effective communication between the autonomous vehicle (AV) and everything (X) using the Lagrange optimization algorithm. Furthermore, the system is evaluated in terms of energy efficiency and achievable data rate based on the optimal inter-vehicle position to show the significance of the proposed approach. To meet the stated goals, the ideal inter-vehicle position is predicted using a distributed deep learning model by learning from mathematically generated data and defined as a restricted optimization problem using the Lagrange optimization technique to improve communication between AV2X under various environmental conditions. To demonstrate the efficiency of the suggested model, the following characteristics are considered: vehicle dispersion, vehicle density, vehicle mobility, and speed. The simulation results show the significance of the proposed model in terms of energy efficiency and achievable data rate compared with other proposed models. [ABSTRACT FROM AUTHOR]

In the paper, the Lagrange problem given by a fractional boundary problem with partial derivatives is considered. The main result is the existence of optimal solutions based on the convexity assumption of a certain set. The proof is based on the lower closure theorem and the appropriate implicit measurable function theorem.

The capital allocation framework presents capital allocation principles as solutions to particular optimisation problems and provides a general solution of the quadratic allocation problem via a geometric proof. However, the widely used haircut allocation principle is not reconcilable with that optimisation setting. Our study complements and generalises the unified capital allocation framework. The goal of the study is to contribute in the following two ways. First, we provide an alternative proof of the quadratic allocation problem based on the Lagrange multipliers method to reach the general solution, which complements the geometric proof. This alternative approach to solve the quadratic optimisation problem is, in our opinion, easier to follow and understand by researchers and practitioners. Second, we show that the haircut allocation principle can be accommodated by the optimisation setting with the quadratic optimisation criterion if one of the original conditions is relaxed. Two examples are provided to illustrate the accommodation of this allocation principle. [ABSTRACT FROM AUTHOR]

This study proposes a new numerical-and-analytical method for solving geometrically nonlinear problems of bending of complex-shaped plates made of functionally graded materials developed. The problem was formulated within the framework of a refined higher-order theory considering the quadratic law of distribution of transverse tangential stresses along the plate thickness. To linearize the nonlinear problem, we used the method of continuous continuation in the parameter associated with the external load. For the variational formulation of the linearized problem, a Lagrange functional was constructed, defined at kinematically possible displacement velocities. To find the main unknowns of the problem of nonlinear plate bending (displacements, strains, and stresses), the Cauchy problem for a system of ordinary differential equations is formulated. The Cauchy problem was solved by the Runge-Kutta–Merson method with automatic step selection. The initial conditions are found from the solution of the problem of geometrically linear deformation. The right-hand sides of the differential equations, at fixed values of the load parameter corresponding to the Runge-Kutta–Merson scheme, were obtained from the solution of the variational problem for the Lagrange functional. The variational problems were solved by the Ritz method in combination with the R-function method. The latter makes it possible to present an approximate solution as a formula. This solution structure exactly satisfies all (general structure) or part (partial structure) of the boundary conditions. Test problems are solved for a homogeneous rigidly fixed and functionally graded hinged square plate subjected to a uniformly distributed load of varying intensity. The results for deflections and stresses obtained by the developed method are compared with the solutions obtained by radial basis functions. The problem of bending of a functionally graded plate of complex shape is solved. The influence of the gradient properties of the material and geometric shape on the stress-strain state is investigated. [ABSTRACT FROM AUTHOR]

In this article, we attempt to characterize optimum solutions for optimization problems with interval-valued functions of interval variables. As the constraint set or the underlying variable spaces of such optimization problems are a set of intervals, we introduce, analyze, and interrelate the notions of normal cone and tangent cone of a set of intervals. In the sequel, their various kinds of properties are defined, such as closedness, weak intersection rule, some algebraic preserving properties, etc. The dual correspondence between the tangent and normal cones is also analyzed. For constrained optimization problems, the normal cone is used to characterize efficient solutions. Furthermore, we derive a necessary condition for efficient solutions for interval optimization problems with the help of Lagrange multipliers, tangent cones, and normal cones. Lastly, an application of the proposed normal cone in solving an interval-valued support vector machines type problem is discussed. [ABSTRACT FROM AUTHOR]

The broadband finite-difference (FD) coefficients computed by a cost function have been widely applied to suppress of numerical dispersion. Under the same conditions, the FD coefficients with small low-wavenumber dispersion error will produce a more accurate numerical solution in the long-time seismic wave simulation. Thus, how to reduce the low-wavenumber dispersion error becomes a crucial problem. According to the research into the zero-point position at the dispersion curve for three types of cost function, we found that the more zero points concentrate on the low-wavenumber region, the less the dispersion error. Therefore, the concentration of zero points is a good way to reduce dispersion error, which can be implemented by modified wavenumbers of zero points. Then, we design a Lagrange dual problem with a restriction based on the modified wavenumbers. The requirements for constructing the Lagrange dual problem are the optimization function and restricted condition, where the former is based on the dispersion relation, and the latter comprises the modified wavenumbers. The solution of this optimization problem, calculated by the dual ascent method, affords a less low-wavenumber dispersion error than the solution yielded by the alternating direction method of multipliers (ADMM). The theoretical analysis and numerical modeling suggest that the proposed method is superior to the existing optimal FD coefficients in reducing numerical error accumulation in low-frequency simulation. [ABSTRACT FROM AUTHOR]

In the paper, the Lagrange problem given by a fractional boundary problem with partial derivatives is considered. The main result is the existence of optimal solutions based on the convexity assumption of a certain set. The proof is based on the lower closure theorem and the appropriate implicit measurable function theorem. [ABSTRACT FROM AUTHOR]

Pressure dependent modeling (PDM) for water distribution systems (WDSs) is now widely accepted as being much more realistic than the previously used demand driven modeling. Steady-state linkflows, q , outflows, c , and heads, h , of a PDM WDS with no controls of flow and pressure in the system can reliably be found as the active set method solution of a linear-equality-constrained nonlinear optimization of the system's content. Introducing linkflow controls, such as flow control valves (FCVs) and check valves can be handled by imposing box constraints on the decision variables q and c in the optimization; these problems can also be found either by an ASM or an interior point method. The heads in these problems are the Lagrange multipliers in the content model, and controlling these cannot be handled simply by imposing constraints on them. In this paper, the problem of modeling pressure-control devices such as pressure-reducing valves (PRVs) is solved by finding the Nash Equilibrium of a model that treats (1) the (global) linkflow constrained content optimization; and (2) the local pressure controls, as players in a competitive, noncooperative game. While this paper details how to model FCVs and PRVs together, this modeling framework is equally applicable to pressure-sustaining valves and variable speed pumps for pressure control without essential modification. An important contribution of this proof-of-concept paper is the development of a comprehensive model that includes flow and pressure controls and which finds a solution without using heuristics. The new method is illustrated on some examples. [ABSTRACT FROM AUTHOR]

The article discusses the challenges of colonizing Mars. One option to overcome much of the difficulties may be the creation of an inhabited base on Phobos in the first phase. Three different flight schemes are considered. Mathematical calculations are made for the most realistic one of them. The trajectory of flight to Phobos with starting since year 2020 for year 2030 is optimized. The attraction of the Sun, the Earth and Mars is considered on the whole trajectory. The Earth, Mars and Phobos positions correspond to ephemerides. The control is realized by impulses at the start and at the end of the trajectory. The characteristic velocity is minimized. On the basis of the Lagrange principle the problem of optimization is reduced to the boundary value problem. The boundary value problem is solved numerically with the use of the shooting method. • A Lagrange's extremal is constructed, taking into account gravity of the planets. • The interplanetary trajectory is optimized taking into account ephemeris and planets. • An inhabited base on Phobos will allow robots to begin colonizing Mars. • Flight from Phobos base to Mars surface and back burns 93% of mass. • Starting colonization of Mars from Phobos solves a large number of problems. [ABSTRACT FROM AUTHOR]

The numerical approximation of incompressible fluid–structure interaction problems with Lagrange multiplier is generally based on strongly coupled schemes. This delivers unconditional stability, but at the expense of solving a computationally demanding coupled system at each time step. For the case of the coupling with immersed thin-walled solids, we introduce a class of semi-implicit coupling schemes that avoids strongly coupling without compromising stability and accuracy. A priori energy and error estimates are derived. The theoretical results are illustrated through numerical experiments in an academic benchmark. [ABSTRACT FROM AUTHOR]

A novel continuum-based fast projection scheme is proposed for cloth simulation. Cloth geometry is described by NURBS, and the dynamic response is modeled by a displacement-only Kirchhoff-Love shell element formulated directly on NURBS geometry. The fast projection method, which solves strain limiting as a constrained Lagrange problem, is extended to the continuum version. Numerical examples are studied to demonstrate the performance of the current scheme. The proposed approach can be applied to grids of arbitrary topology and can eliminate unrealistic over-stretching efficiently if compared to spring-based methodologies. [ABSTRACT FROM AUTHOR]

In this work, we introduce a variational multi-scale (VMS) method for the numerical approximation of parabolic problems, where sub-grid scales are approximated from the eigenpairs of associated elliptic operator. The abstract method is particularized to the one-dimensional advection-diffusion equations, for which the sub-grid components are exactly calculated in terms of a spectral expansion when the advection velocity is approximated by piecewise constant velocities on the grid elements.We prove error estimates that in particular imply that when Lagrange finite element discretisations in space are used, the spectral VMS method coincides with the exact solution of the implicit Euler semi-discretisation of the advection-diffusion problem at the Lagrange interpolation nodes. We also build a feasible method to solve the evolutive advection-diffusion problems by means of an offline/online strategy with reduced computational complexity.We perform some numerical tests in good agreement with the theoretical expectations, that show an improved accuracy with respect to several stabilised methods. [ABSTRACT FROM AUTHOR]

We discuss first order optimality conditions for geometric optimization problems with Neumann boundary conditions and boundary observation. The methods we develop here are applicable to large classes of state systems or cost functionals. Our approach is based on the implicit parametrization theorem and the use of Hamiltonian systems. It establishes equivalence with a constrained optimal control problem and uses Lagrange multipliers under a simple constraint qualification. In this setting, general functional variations are performed, that combine classical topological and boundary variations, in a natural way. [ABSTRACT FROM AUTHOR]

The optimal control solutions for Hilfer fractional neutral integrodifferential evolution equations with infinite delay in Banach space are investigated in this study. To begin, we use fractional computations, Holder inequality, and Banach fixed point approaches to examine conclusions for fractional integrodifferential evolution equations. Following that, we created the fractional integrodifferential evolution system's continuous dependency. We also looked at optimal controls for the Lagrange problem. Finally, an application is given to extract the primary theory. [ABSTRACT FROM AUTHOR]

The most appropriate procedures in the inventory organization area are inventory arrangements based on ABC investigation, a well-known technique for establishing the objects in a different collection, giving their status and principles. This research Bi- A mathematical goal to advance the inventory group founded on the ABC. The Planned model instantly improves the amenity level, the amount of inventory grouping, and the number of due things. An Arithmetical model is available in this study to categorize inventory objects, considering significant revenue and rate decrease catalogues. The model aims to maximize the net gain of available items. Economic and inventory constraints are also taken into account. The Benders decay and Lagrange reduction procedures respond to classical arithmetical stands. The outcomes of the two answers are then equated. TOPSIS and numerical examinations estimate the planned answers and choose the best. Later, numerous sensitivity studies on the classic were completed, which assists inventory control executives in regulating the outcome of inventory administration rates configured for optimum verdict production and element grouping. The Arithmetical diagram was run for ten different arithmetic instances, and the results of the two suggested explanations were statistically equated using a t-test. As a result, the TOPSIS technique was appropriate; the Lagrangean approach was chosen as the more fabulous technique. [ABSTRACT FROM AUTHOR]

In this paper we consider a multigrid approach for solving elliptic equations over non‐matching grids with domain decomposition methods. The domain is partitioned into subdomains with different mesh levels that do not match at the interface. The proposed algorithm searches for the global solution over different levels by projecting the residuals on the overlap region. This method is used in conjunction with a domain decomposition solver which only requires, in each iteration step, the solutions of several small local subproblems over finite element blocks. This algorithm is shown to converge to the solution of the corresponding Lagrange multiplier problem for non‐matching grids. The convergence properties of the algorithms are analyzed and numerical examples are presented. When the multigrid and domain decomposition approaches are combined, the method is shown to be reliable and easy to implement. Furthermore the local nature of the solver allows for a straightforward implementation on multiple parallel computers and graphics processing unit (GPU) clusters. [ABSTRACT FROM AUTHOR]

The concept of generalized augmented Lagrange multiplier was introduced originally for nonlinear programming problems. In this paper we further study this concept in the framework of convex cone constrained optimization problems. In particular we establish the relation of optimal solution sets between the primal problem and augmented Lagrange minimization problem. Some discussions on related concepts such as saddle points and exact penalty are given as well. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the nonsmooth composite optimization problems over networks, which consist of a smooth term and a nonsmooth term. Both equality constraints and box constraints for the decision variables are also considered. Based on the multi-agent networks, the objective problems are split into a series of agents on which the problems can be solved in a decentralized manner. By establishing the Lagrange function of the problems, the first-order optimal condition is obtained in the primal-dual domain. Then, we propose a decentralized algorithm with the proximal operators. The proposed algorithm has uncoordinated stepsizes with respect to agents or edges, where no global parameters are involved. By constructing the compact form of the algorithm with operators, we complete the convergence analysis with the fixed-point theory. With the constrained quadratic programming problem, simulations verify the effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we study a general minimization vector problem which is expressed in terms of a perturbation mapping defined on a product of locally convex Hausdorff topological vector spaces with values in another locally convex topological vector space. Several representations of the epigraph of the conjugate of the perturbation mapping are given, and then, variants vector Farkas lemmas associated with the system defined by this mapping are established. A dual problem and another so-called loose dual problem of the mentioned problem are defined and stable strong duality results between these pairs of primal–dual problems are established. The results just obtained are then applied to a general class of composed constrained vector optimization problems. For this class of problems, two concrete perturbation mappings are proposed. These perturbation mappings give rise to variants of dual problems including the Lagrange dual problem and several kinds of Fenchel–Lagrange dual problems of the problem under consideration. Stable strong duality results for these pairs of primal–dual problems are derived. Several classes of concrete vector (and scalar) optimization problems are also considered at the end of the paper to illustrate the significance of our approach. [ABSTRACT FROM AUTHOR]

The paper investigates the accuracy of the model predictive control (MPC) method for finding on-line approximate optimal feedback control for Bolza-type problems on a fixed finite horizon. The predictions for the dynamics, the state measurements, and the solution of the auxiliary open-loop control problems that appear at every step of the MPC method may be inaccurate. The main result provides an error estimate of the MPC-generated solution compared with the optimal open-loop solution of the "ideal" problem, where all predictions and measurements are exact. The technique of proving the estimate involves an extension of the notion of strong metric subregularity of set-valued maps and utilization of a specific new metric in the control space, which makes the proof nonstandard. The result is specialized for two problem classes: coercive problems and affine problems. [ABSTRACT FROM AUTHOR]

The paper is devoted to the Lagrange problem in the bounded region for first-order partial differential inclusions (PDIs). For this, using discretisation method and locally adjoint mappings (LAMs), in the form of Euler–Lagrange type inclusions and conjugate boundary conditions, sufficient optimality conditions are obtained. The transition to a continuous problem with PDIs is possible using a specially proved equivalence theorem. To demonstrate this approach, some semilinear problems and polyhedral optimisation with first-order partial differential inclusions are considered. Furthermore, the numerical results also are provided. [ABSTRACT FROM AUTHOR]

This article investigates the issue of optimal control and approximate controllability results for fractional integrodifferential evolution equations with infinite delay of r∈(1,2) in Banach space. In the beginning, we analyze approximate controllability results for fractional integrodifferential evolution equations using the fractional calculations, cosine families, and Banach fixed point theorem. After, we developed the continuous dependence of the fractional integrodifferential evolution equations by using the Henry–Gronwall inequality. Furthermore, we tested the existence of optimal controls for the Lagrange problem. Lastly, an application is presented to illustrate the theory of the main results. [ABSTRACT FROM AUTHOR]

In medium-term scheduling, the end-of-term storage energy maximization model is proposed to create conditions for the safety, stability and economic operation of the hydropower system after control term, which satisfies the system load demand undertaken by the cascaded system in a given scheduling period. This paper presents a simplified solution method based on the Lagrangian relaxation method (LR) to solve the end-of-term storage energy maximization model. The original Lagrange dual problem with multiple Lagrange multipliers is converted to that with only one Lagrange multiplier by an entropy-based aggregate function method, which relaxes the complex cascaded hydropower system load balance constraints. The subgradient method and successive approximation of dynamic programming (DPSA) are adopted to update the Lagrange multiplier iteratively and solve the subproblem of the Lagrange dual problem, respectively. The Wujiang cascaded hydropower system is studied, and the result shows that the simplified solution method for the end-of-term storage energy maximization model both improves solving efficiency and ensures solving accuracy to a great extent. [ABSTRACT FROM AUTHOR]

This paper focuses on reduced-order modeling for contact mechanics problems treated by Lagrange multipliers. The high nonlinearity of the dual solutions lead to poor classical data compression. A hyper-reduction approach based on a reduced integration domain (RID) is considered. The dual reduced basis is the restriction to the RID of the full-order dual basis, which ensures the hyper-reduced model to respect the non-linearity constraints. However, the verification of the solvability condition, associated with the well-posedness of the solution, may induce an extension of the primal reduced basis without guaranteeing accurate dual forces. We highlight the strong link between the condition number of the projected contact rigidity matrix and the precision of the dual reduced solutions. Two efficient strategies of enrichment of the primal POD reduced basis are then introduced. However, for large parametric variation of the contact zone, the reachable dual precision may remain limited. A clustering strategy on the parametric space is then proposed in order to deal with piece-wise low-rank approximations. On each cluster, a local accurate hyper-reduced model is built thanks to the enrichment strategies. The overall solution is then deeply improved while preserving an interesting compression of both primal and dual bases. [ABSTRACT FROM AUTHOR]

Sparse code multiple access (SCMA), as a codebook-based non-orthogonal multiple access (NOMA) technique, has received research attention in recent years. The codebook design problem for SCMA has also been studied to some extent since codebook choices are highly related to the system’s error rate performance. In this paper, we approach the SCMA codebook design problem by formulating an optimization problem to maximize the minimum Euclidean distance (MED) of superimposed codewords under power constraints. While SCMA codebooks with a larger MED are expected to obtain a better BER performance, no optimal SCMA codebook in terms of MED maximization, to the authors’ best knowledge, has been reported in the SCMA literature yet. In this paper, a new iterative algorithm based on alternating maximization with exact penalty is proposed for the MED maximization problem. The proposed algorithm, when supplied with appropriate initial points and parameters, achieves a set of codebooks of all users whose MED is larger than any previously reported results. A Lagrange dual problem is derived which provides an upper bound of MED of any set of codebooks. Even though there is still a nonzero gap between the achieved MED and the upper bound given by the dual problem, simulation results demonstrate clear advantages in error rate performances of the proposed set of codebooks over all existing ones not only in AWGN channels but also in some downlink scenarios that fit in 5G/NR applications, making it a good codebook candidate thereof. The proposed set of SCMA codebooks, however, are not shown to outperform existing ones in uplink channels or in the case where non-consecutive OFDMA subcarriers are used. The correctness and accuracy of error curves in the simulation results are further confirmed by the coincidences with the theoretical upper bounds of error rates derived for any given set of codebooks. [ABSTRACT FROM AUTHOR]

We study the optimal feedback control for fractional evolution equations with a nonlinear perturbation of the time-fractional derivative term involving Caputo fractional derivatives with arbitrary kernels. Firstly, we derive a mild solution in terms of the semigroup operator generated by resolvents and a kernel from the general Caputo fractional operators and establish the existence and uniqueness of mild solutions for the feedback control systems. Then, the existence of feasible pairs by applying Filippov's theorem is obtained. In addition, the existence of optimal control pairs for the Lagrange problem has been investigated. [ABSTRACT FROM AUTHOR]

This paper aims to establish the Karush-Kuhn-Tucker type optimality criteria for linear fractional optimization problems with fuzzy parameters. To evolve the desired criteria first, the fractional optimization problem is transformed into the non-fractional optimization problem with fuzzy parameters. Then Hukuhara differentiability for the differentiation of functions with fuzzy parameters and Hausdorff metric to expound the distance between the fuzzy numbers is invoked. Optimality criteria are then elicited for the non-fractional optimization problems by introducing Lagrange multipliers and Riemann integration theory. In order to validate the developed theory, two numerical optimization problems are also verified. [ABSTRACT FROM AUTHOR]