Numerical method for shape optimization using BEM

In general, optimal control problems rely on numerically rather than analytically solving methods, due to their nonlinearities. The direct method, one of the numerically solving methods, is mainly to transform the optimal control problem into a nonlinear optimization problem with finite dimensions, via discretizing the objective functional and the forced dynamical equations directly. However, in the procedure of the direct method, the classical discretizations of the forced equations will reduce or affect the accuracy of the resulting optimization problem as well as the discrete optimal control. In view of this fact, more accurate and efficient numerical algorithms should be employed to approximate the forced dynamical equations. As verified, the discrete variational difference schemes for forced Birkhoffian systems exhibit excellent numerical behaviors in terms of high accuracy, long-time stability and precise energy prediction. Thus, the forced dynamical equations in optimal control problems, after being represented as forced Birkhoffian equations, can be discretized according to the discrete variational difference schemes for forced Birkhoffian systems. Compared with the method of employing traditional difference schemes to discretize the forced dynamical equations, this way yields faithful nonlinear optimization problems and consequently gives accurate and efficient discrete optimal control. Subsequently, in the paper we are to apply the proposed method of numerically solving optimal control problems to the rendezvous and docking problem of spacecrafts. First, we make a reasonable simplification, i.e., the rendezvous and docking process of two spacecrafts is reduced to the problem of optimally transferring the chaser spacecraft with a continuously acting force from one circular orbit around the Earth to another one. During this transfer, the goal is to minimize the control effort. Second, the dynamical equations of the chaser spacecraft are represented as the form of the forced Birkhoffian equation. Then in this case, the discrete variational difference scheme for forced Birkhoffian system can be employed to discretize the chaser spacecraft's equations of motion. With further discretizing the control effort and the boundary conditions, the resulting nonlinear optimization problem is obtained. Finally, the optimization problem is solved directly by the nonlinear programming method and then the discrete optimal control is achieved. The obtained optimal control is efficient enough to realize the rendezvous and docking process, even though it is only an approximation of the continuous one. Simulation results fully verify the efficiency of the proposed method for numerically solving optimal control problems, if the fact that the time step is chosen to be very large to limit the dimension of the optimization problem is noted.

In this paper, we focus on a new class of optimal control problems governed by a simple integral cost functional and isoperimetric-type constraints (constant level sets of some simple integral functionals). By using the notions of a variational differential system and adjoint equation, necessary optimality conditions are established for a feasible solution in the considered optimization problem. More precisely, under simplified hypotheses and using a modified Legendrian duality, we establish a maximum principle for the considered optimization problem.

This paper is devoted to applications of modern methods of variational analysis to constrained optimization and control problems generally formulated in infinite-dimensional spaces. The main focus is on the study of problems with nonsmooth structures, which require the usage of advanced tools of generalized differentiation. In this way we derive new necessary optimality conditions in optimization problems with functional and operator constraints and then apply them to optimal control problems governed by discrete-time inclusions in infinite dimensions. The principal difference between finite-dimensional and infinite-dimensional frameworks of optimization and control consists of the “lack of compactness” in infinite dimensions, which leads to imposing certain “normal compactness” properties and developing their comprehensive calculus, together with appropriate calculus rules of generalized differentiation. On the other hand, one of the most important achievements of the paper consists of relaxing the latter assumptions for certain classes of optimization and control problems. In particular, we fully avoid the requirements of this type imposed on target endpoint sets in infinite-dimensional optimal control for discrete-time inclusions.

Modeling private highways in networks with entry–exit based toll charges

The network routing problem is one of important problems to consider behaviors of multiple flows on a network with traffic. Some network routing problems can be classified into the selfish routing game in which the behaviors of the flows are competitive and the optimal routing problem in which the behaviors of the flows are cooperative. This study focuses on the optimal routing problem. The optimal routing problem is formulated as an optimization problem with a linear equality constraint and lower limit constraints. The learning problem of the support vector machine (SVM) is formulated as an optimization problem with a linear equality constraint and upper and lower limit constraints. The sequential minimal optimization (SMO) is one of the fast solvers for the learning problem. SMO can be applied to solve the optimal routing problem. This study proposes a new fast optimal routing search method using SMO. In the numerical experiments, the effectiveness of the proposed method on the computational cost is verified with the comparison to the search method using the replicator dynamics.

The optimization problem consists in finding an element that minimizes or maximizes the goal (or cost) functional on the set of admissible controls. The paper is devoted to the optimization problem for a pseudoparabolic equation with control in coefficients of the variational equality as well as on the right-hand side. We consider optimal control problem (parabolic initial boundary value problem) of a viscoelastic plate. The bending of a plate is described by means of the Kirchhoff model. We assume that a nonhomogeneous and anisotropic viscoelastic plate occupying a domain Ω × (—H, H) of the space ℝ3 is loaded be a transversal distributed force perpendicular to the middle plane of the plate. The viscoelastic reinforced plate with welded bars rests on a unilateral elastic foundation. The role of the control variables is played by the thickness of the viscoelastic plate and the stiffness characteristics of the elastic Winkler medium, respectively. We consider the desired moments field (the bending moments and torque) in time T as the cost functional. The control variables have to belong to a set of Lipschitz-continuous functions and to a set of continuously differentiable functions, respectively. The state problem is modelled by a pseudoparabolic variational equality, where the control variables influence both the coefficients of the linear strictly monotone operator and the Winkler functional. We derive there some apriori estimates of the state function which are useful for the problem of the existence of the optimal thickness function and the Winkler modulus, respectively, and for unicity of the optimal controls. On the basis of a general existence theorem for a class of pseudoparabolic optimization problems with the state variational equation, we prove the existence of at least one solution of the optimal control problem.

Historically, fluidic devices such as switches, amplifiers, and oscillators, have an advantage, compared with electronic devices, in terms of maintenance-free operation and operating life. Therefore, prior to the great progress in electronic technologies that has occurred during the past several decades, the structure and function of fluidic devices were the subject of extensive research. In particular, the structures of these fluidic devices are often composed of complicated flow channel layouts. Recently, fluidic devices are again attracting significant attention, stimulated by progress in the development of MEMS technologies. In this study, to develop an energy-efficient structure for a MEMS-scale fluidic device, we apply a topology optimization method to an optimal design problem for a steady state incompressible viscous flow field. We use a level set-based topology optimization method incorporating the concept of the phase field method for the topology optimization so that clear boundaries between the solid and fluid domains are expressed in the optimal configurations. To define the topology optimization problem for a fluid regime, the expressions of the primary and adjoint problems are formulated concretely, to minimize viscous energy dissipation. Moreover, to ensure the intended design outflow rate at a designated outlet, the optimization problem includes an outflow rate inequality constraint in this paper. Following the concept of the standard adjoint variable method, a stable optimization process that satisfies the outflow rate inequality constraint is achieved. Two numerical examples, one for two-terminal and the other for multi-terminal flow, are provided to demonstrate the usefulness of the proposed level set-based topology optimization method.

Purpose of work is to research and design process optimization and optimal operational management for a two-stage technological line. Methodology. To achieve the goal, we apply a comprehensive approach, which consists of the formalization and analysis of operational management tasks and technological lines designing.The methods of probability theory are used to determine the optimal level between the maximum bandwidths of the stages of the technological lines and for processing mathematical models of random variables. Research results. The need for a system-wide approach to the management of two-stage technological lines is substantiated, which involves joint consideration of design and operational management tasks. Three optimization problems are considered, which differ in their formulation by initial data, goals and optimization parameters.This is the problem of design optimization when creating equipment, the problem of optimal design of technological schemes and the problem of optimal operational control.These three types of tasks must be considered comprehensively, in mutual connection to achieve a global optimum in terms of the economic criterion.The formalization of such a complex problem for two-stage technological schemes was carried out in a general form and approaches to its solution. The methodology for solving such problems is proposed, based on information about the specific consumption of the resource and the characteristics of the flow of the processed material. Scientific novelty. The formulation of the complex optimization problem for the functioning of two-stage technological lines, including the choice of the equipment and a type of operational management, and a solving approach for the considered problem was provided. Practical significance of work results. The proposed approach makes it possible to find the optimal solution to the problem of management and design of two-stage technological lines from the point of view of the economic criterion.

The paper formulates and solves an optimization problem concerning the study of the deviation of the Lipschitz class functions from the superposition of orthogonal polynomials. The problem is based on a positive operator as a generalization of the Abel–Poisson integral and the biharmonic Poisson integral. The main result of the research is presented in the form of an asymptotic equality for the deviation of Lipschitz functions from the constructed generalized Poisson-type operator. The applicability of the operator under consideration in the optimal decision theory is determined by its belonging to the class of positive operators as optimal solutions to boundary value problems. In addition, the optimality of the generalized Poisson-type operator is significantly enhanced by the presence of approximation properties for the Lipschitz class functions, which can act as mathematical models in optimal design problems. Keywords: Lipschitz class functions, optimization problem, asymptotic equality, generalized operator.

Most structural optimization problems that have been investigated use approaches designed for complete data. Nevertheless, some parametric probabilistic problems of optimal structural design with limiting state and buckling constraints have been formulated and investigated in [AB73, ABC84, AC79, Bra84, DFH77, ETB84, AGRZ84, EKS86, Mos77, Mar95, BV07]. Nondeterministic approaches have been used for some optimal design problems when the external conditions (external forces) were taken as a random variables. Fewer probabilistic studies have been devoted to the important class of problems of quasi-brittle elastic body optimization on the basis of modern fracture mechanics criteria [ABC84, BRS99, BRS03a, BRS03b, LM96]. In accordance with fracture mechanics representations it is necessary to consider all possibilities for the appearance of cracks, and to take into account that the crack position, size and mode (opening cracks, shear cracks, etc.) are unknown beforehand. Therefore, the formulation and solution of these optimization problems requires the application of methods which can take into account incomplete information. Minimax methods of game theory (guaranteed approach), probabilistic methods, etc., can be used for this purpose.

Optimal control for multistage uncertain random dynamic systems with multiple time delays

Optimal channel switching in multiuser systems under average capacity constraints

Existing methods for nonlinear robust control often use scenario-based approaches to formulate the control problem as nonlinear optimization problems. Increasing the number of scenarios improves robustness, while increasing the size of the optimization problems. Mitigating the size of the problem by reducing the number of scenarios requires knowledge about how the uncertainty affects the system. This paper draws from local reduction methods used in semi-infinite optimization to solve robust optimal control problems with parametric uncertainty. We show that nonlinear robust optimal control problems are equivalent to semi-infinite optimization problems and can be solved by local reduction. By iteratively adding interim globally worst-case scenarios to the problem, methods based on local reduction provide a way to manage the total number of scenarios. In particular, we show that local reduction methods find worst case scenarios that are not on the boundary of the uncertainty set. The proposed approach is illustrated with a case study with both parametric and additive time-varying uncertainty. The number of scenarios obtained from local reduction is 101, smaller than in the case when all 214+3x192 boundary scenarios are considered. A validation with randomly drawn scenarios shows that our proposed approach reduces the number of scenarios and ensures robustness even if local solvers are used.

Decomposition method for solving two optimal control problems and one optimization problem in batch fermentation is proposed. The problems are formulated based on a nonstructured mathematical model with slowly varying parameters and a finite cost criterion of maximum end production. Dependence of the model parameters on one physical or chemical parameter, which could easily be used as a control input is introduced analytically in the model equations and three model descriptions are obtained by nonlinear difference equations. Sensitivity functions of state trajectories towards slowly varying coefficients are introduced to account for model uncertainties. Based on them extended optimal control and optimization problems are formulated.

In the previous papers of Frenod & Menard & Safa [1] and Frenod & Safa [2] we used the continuous-in-time financial model developed by Frenod & Chakkour [3], which describes working of loan and repayment, in an optimal control theory framework to effectively conduct project objectives. The goal was to determine the optimal loan scheme taking into account the objectives of the project, the income and the spending. In this article, we enrich this continuous-in-time financial model to take into account possibility of saving. Then we propose two new optimization problems involving this model. The first one consists in finding a loan scheme minimizing the cost of the loan for a project, together with the time to achieve its objectives. The second problem consists, from given loan, saving and withdrawal schemes, to finding optimal variants of them. We have built a mathematical framework for the optimal control problem which consists to solving a constraint optimization problem. We have used Simplex algorithm to solve the first optimization problem and the quadratic programming for the second one. The simulations’ results are consistent with the theoretical part and show the performance and stability of our approach.