Interior variations in dynamic optimization problems

In this article, the author examines the properties of interior variations and indicates how to use them in order to formulate the necessary condition of optimality for problems of dynamic optimization, in particular, problems of variational calculus and of optimal control. For optimal control problems, an optimization technique based on interior variations and polynomial approximations is suggested and then illustrated by an explanatory example.     

Optimal and approximate boundary controls of an elastic body with quasistatic evolution of damage

In this paper, we study an optimal control problem for the mixed boundary value problem for an elastic body with quasistatic evolution of an internal damage variable. We suppose that the evolution of microscopic cracks and cavities responsible for the damage is described by a nonlinear parabolic equation. A density of surface traction p acting on a part of boundary of an elastic body Ω is taken as a boundary control. Because the initial boundary value problem of this type can exhibit the Lavrentieff phenomenon and non‐uniqueness of weak solutions, we deal with the solvability of this problem in the class of weak variational solutions. Using the convergence concept in variable spaces and following the direct method in calculus of variations, we prove the existence of optimal and approximate solutions to the optimal control problem under rather general assumptions on the quasistatic evolution of damage. Copyright © 2014 John Wiley & Sons, Ltd.     

Optimal Goodwill Path to Introduce a New Product

We consider the problem of determining an optimal goodwill path for the introduction of a new product in a market, while looking for the maximum foreseen profit. The foreseen revenue depends on the product introduction time and on the goodwill level at the same time. We focus on the advertising costs associated with the goodwill evolution and assume that the cost function possesses some rather general features which are shared by the cost functions of the Nerlove-Arrow type models. The dynamic optimization problem is discussed in the calculus of variations framework. A few examples associated with special cost functions are discussed in detail.     

A Study on Abnormality in Variational and Optimal Control Problems

In this paper, a variational problem is considered with differential equality constraints over a variable interval. It is stressed that the abnormality is a local character of the admissible set; consequently, a definition of regularity related to the constraints characterizing the admissible set is given. Then, for the local minimum necessary conditions, a compact form equivalent to the well-known Euler equation and transversality condition is given. By exploiting this result and the previous definition of regularity, it is proved that nonregularity is a necessary and sufficient condition for an admissible solution to be an abnormal extremal. Then, a necessary and sufficient condition is given for an abnormal extremal to be weakly abnormal. The analysis of the abnormality is completed by considering the particular case of affine constraints over a fixed interval: in this case, the abnormality turns out to have a global character, so that it is possible to define an abnormal problem or a normal problem. The last section is devoted to the study of an optimal control problem characterized by differential constraints corresponding to the dynamics of a controlled process. The above general results are particularized to this problem, yielding a necessary and sufficient condition for an admissible solution to be an abnormal extremal. From this, a previously known result is recovered concerning the linearized system controllability as a sufficient condition to exclude the abnormality.     

Sufficient conditions for a minimum of the free-final-time optimal control problem

The sufficient conditions for a minimum of the free-final-time optimal control problem are the strengthened Legendre-Clebsch condition and the conjugate point condition. In this paper, a new approach for determining the location of the conjugate point is presented. The sweep method is used to solve the linear two-point boundary-value problem for the neighboring extremal path from a perturbed initial point to the final constraint manifold. The new approach is to solve for the final condition Lagrange multiplier perturbation and the final time perturbation simultaneously. Then, the resulting neighboring extremal control is used to write the second variation as a perfect square and obtain the conjugate point condition. Finally, two example problems are solved to illustrate the application of the sufficient conditions.     

Variational Calculus and Approximate Solution of Optimal Control Problems

Variational calculus is a differential process whereby Taylor series expansions can be developed on a term-by-term basis. Therefore, it can be used to obtain the equations which must be solved for the various-order terms arising from the application of regular perturbation theory to problems involving a small parameter. Variational calculus is summarized and applied to the approximate analytical solution of the optimal control problem. First, the various-order equations are obtained directly for a particular problem. Then, assuming that the zeroth-order solution is almost good enough, the equations for the first-order correction are obtained for the general optimal control problem and applied to the particular problem. The first-order solution is the same as the neighboring extremal for the given value of the parameter.     

The Legendre condition of the fractional calculus of variations

Fractional operators play an important role in modelling nonlocal phenomena and problems involving coarse-grained and fractal spaces. The fractional calculus of variations with functionals depending on derivatives and/or integrals of noninteger order is a rather recent subject that is currently in fast development due to its applications in physics and other sciences. In the last decade, several approaches to fractional variational calculus were proposed by using different notions of fractional derivatives and integrals. Although the literature of the fractional calculus of variations is already vast, much remains to be done in obtaining necessary and sufficient conditions for the optimization of fractional variational functionals, existence and regularity of solutions. Regarding necessary optimality conditions, all works available in the literature concern the derivation of first-order fractional conditions of Euler–Lagrange type. In this work, we obtain a Legendre second-order necessary optimality condition for weak extremizers of a variational functional that depends on fractional derivatives.     

The maximum principle,Bellman's equation,and Carathéodory's work

One of the most important and deep results in optimal control theory is the maximum principle attributed to Hestenes (1950) and in particular to Boltyanskii, Gamkrelidze, and Pontryagin (1956). Another prominent result is known as the Bellman equation, which is associated with Isaacs' and Bellman's work (later than 1951). However, precursors of both the maximum principle and the Bellman equation can already be found in Carathéodory's book of 1935 (Ref. 1a), the first even in his earlier work of 1926 which is given in Ref. 2. This is not a widely acknowledged fact. The present tutorial paper traces Carathéodory's approach to the calculus of variations, once called the royal road in the calculus of variations, and shows that famous results in optimal control theory, including the maximum principle and the Bellman equation, are consequences of Carathéodory's earlier results.This paper is in honor of the seventieth birthday of Professor Angelo Miele. It was exactly 70 years ago, in the months of August and September 1922, around the time of the birth of Professor Miele, when Constantin Carathéodory wrote a paper in Italian entitled Sui Campi di Estremali Uscenti da un Punto e Riempienti Tutto lo Spazio—On Extremal Fields Emanating from a Point and Covering All the Space (Ref. 3). This paper was inspired by the great Italian mathematician Leonida Tonelli, one of Angelo Miele's academic teachers.     

Formulation of Euler-Lagrange equations for fractional variational problems

This paper presents extensions to traditional calculus of variations for systems containing fractional derivatives. The fractional derivative is described in the Riemann-Liouville sense. Specifically, we consider two problems, the simplest fractional variational problem and the fractional variational problem of Lagrange. Results of the first problem are extended to problems containing multiple fractional derivatives and unknown functions. For the second problem, we also present a Lagrange type multiplier rule. For both problems, we develop the Euler-Lagrange type necessary conditions which must be satisfied for the given functional to be extremum. Two problems are considered to demonstrate the application of the formulation. The formulation presented and the resulting equations are very similar to those that appear in the field of classical calculus of variations.     

Optimal harvesting of renewable resources

The aim of the paper is to investigate different types of calculus of variations solutions maximizing the present value of a renewable resource. It is found that there are two types of optimal stationary harvesting policies. For certain cost functions, bang-bang harvesting may be more profitable than either of these.     

Another Jacobi sufficiency criterion for optimal control with smooth constraints

In order to tighten the gap between necessary and sufficient conditions, new second-order sufficient conditions are developed for optimal control problems, where the control set is given by smooth functions. When the control set is polyhedral, our criterion generalizes prior results of the same kind, namely, the Jacobi criterion in Hamiltonian form and that in Lagrangian form (Refs. 1–3).The research of V. Zeidan was supported by NSERC Grant A-8570, which is gratefully acknowledged.     

Natural boundary conditions in the calculus of variations

We prove necessary optimality conditions for problems of the calculus of variations on time scales with a Lagrangian depending on the free end‐point. Copyright © 2010 John Wiley & Sons, Ltd.     

Optimal trajectory to overflow in a queue fed by a large number of sources

We analyse the deviant behavior of a queue fed by a large number of traffic streams. In particular, we explicitly give the most likely trajectory (or optimal path) to buffer overflow, by applying large deviations techniques. This is done for a broad class of sources, consisting of Markov fluid sources and periodic sources. Apart from a number of ramifications of this result, we present guidelines for the numerical evaluation of the optimal path.     

Performance Optimization of a Class of Discrete Event Dynamic Systems Using Calculus of Variations Techniques

We explore an approach involving the use of calculus of variations techniques for discrete event dynamic system (DEDS) performance optimization problems. The approach is motivated by the observation that such problems can be described by separable cost functions and recursive dynamics of the same form as that used to describe conventional discrete-time continuous-variable optimal control problems. Three important difficulties are that DEDS are generally stochastic, their dynamics typically involve max and min operations, which are not everywhere differentiable, and the state variables are often discrete. We demonstrate how to overcome these difficulties by applying the approach to a transportation problem, modeled as a polling system, where we are able to derive an explicit and intuitive analytic expression for an optimal control policy.     

The Optimal Control of a Train

We consider the problem of determining an optimal driving strategy in a train control problem with a generalised equation of motion. We assume that the journey must be completed within a given time and seek a strategy that minimises fuel consumption. On the one hand we consider the case where continuous control can be used and on the other hand we consider the case where only discrete control is available. We pay particular attention to a unified development of the two cases. For the continuous control problem we use the Pontryagin principle to find necessary conditions on an optimal strategy and show that these conditions yield key equations that determine the optimal switching points. In the discrete control problem, which is the typical situation with diesel-electric locomotives, we show that for each fixed control sequence the cost of fuel can be minimised by finding the optimal switching times. The corresponding strategies are called strategies of optimal type and in this case we use the Kuhn–Tucker equations to find key equations that determine the optimal switching times. We note that the strategies of optimal type can be used to approximate as closely as we please the optimal strategy obtained using continuous control and we present two new derivations of the key equations. We illustrate our general remarks by reference to a typical train control problem.     

Some Extensions to a Direct Optimization Method

A direct method for the global extremization of a class of integrals, introduced in Refs. 1–3, is generalized to allow for constraints in the form of differential conditions and by considering the so-called infinite-horizon case.     

Homogenization in L

Homogenization of deterministic control problems with L^∞ running cost is studied by viscosity solutions techniques. It is proved that the value function of an L^∞ problem in a medium with a periodic micro-structure converges uniformly on the compact sets to the value function of the homogenized problem as the period shrinks to 0. Our main convergence result extends that of Ishii (Stochastic Analysis, control, optimization and applications, pp. 305-324, Birkhäuser Boston, Boston, MA, 1999.) to the case of a discontinuous Hamiltonian. The cell problem is solved, but, as non-uniqueness occurs, the effective Hamiltonian must be selected in a careful way. The paper also provides a representation formula for the effective Hamiltonian and gives illustrations to calculus of variations, averaging and one-dimensional problems.     

基于Panel Data的地区财政支出结构优化模型研究

本文用平面数据(PanelData)的协整理论和误差修正模型、多目标规划等方法,利用1994—2001年中国各地区财政支出结构的数据,构建了地区财政支出结构优化模型,并对河北省2000年和2001年的财政支出结构进行了实证分析,提出了优化方案。结论显示,在财政支出总量不变的前提下,模型拟合取得了满意的效果。     

Optimal impulsive transfers between real planetary orbits

This study presents the derivation of the equations necessary to establish a practical method for computing optimal transfers in the real case. It makes it possible to compute and use actual optimal transfers for advanced planning of preliminary mission analysis in place of the standard Hohmann transfers. This realizable solution includes the effects of the inclination and eccentricity of the orbits of the planets. It applies directly to cases in which a two-impulse transfer is the optimal solution and presents the results as functions of the corresponding idealized Hohmann transfers.     

Optimal design of structures with unspecified loading distribution

The problem of the optimal distribution of loading on a structure that corresponds to the minimum of the elastic compliance or the maximum of the safety factor for plastic collapse is considered. Optimality criteria are derived, and their applicability is illustrated in the case of beams. Besides the optimally varying cross section, also the support positions and the load distribution are determined from the optimal solution.