Dinkelbach Approach to Solving a Class of Fractional Optimal Control Problems

We consider optimal control problems with functional given by the ratio of two integrals (fractional optimal control problems). In particular, we focus on a special case with affine integrands and linear dynamics with respect to state and control. Since the standard optimal control theory cannot be used directly to solve a problem of this kind, we apply Dinkelbach’s approach to linearize it. Indeed, the fractional optimal control problem can be transformed into an equivalent monoparametric family {P_q} of linear optimal control problems. The special structure of the class of problems considered allows solving the fractional problem either explicitly or requiring straightforward classical numerical techniques to solve a single equation. An application to advertising efficiency maximization is presented. This work was partially supported by the Università Ca’ Foscari, Venezia, Italy, the MIUR (PRIN cofinancing 2005), the Council for Grants (under RF President) and State Aid to Fundamental Science Schools (Grant NSh-4113.2008.6). We thank Angelo Miele, Panos Pardalos and the anonymous referees for comments and suggestions.     

A Combination of Variational and Penalty Methods for Solving a Class of Fractional Optimal Control Problems

This paper develops an approximate method, based on the combination of epsilon penalty and variational methods, for solving a class of multidimensional fractional optimal control problems. The fractional derivative is in the Caputo sense. In the presented method, utilizing the epsilon method, the given optimal control problem transforms into an unconstrained optimization problem; then, the equivalent variational equality is derived for the given unconstrained problem. The variational equality is approximately solved by applying a spectral method.     

A Simple Accurate Method for Solving Fractional Variational and Optimal Control Problems

We develop a simple and accurate method to solve fractional variational and fractional optimal control problems with dependence on Caputo and Riemann–Liouville operators. Using known formulas for computing fractional derivatives of polynomials, we rewrite the fractional functional dynamical optimization problem as a classical static optimization problem. The method for classical optimal control problems is called Ritz’s method. Examples show that the proposed approach is more accurate than recent methods available in the literature.     

An Approximation Scheme for Uncertain Minimax Optimal Control Problems

In this work, we address an uncertain minimax optimal control problem with linear dynamics where the objective functional is the expected value of the supremum of the running cost over a time interval. By taking an independently drawn random sample, the expected value function is approximated by the corresponding sample average function. We study the epi-convergence of the approximated objective functionals as well as the convergence of their global minimizers. Then we define an Euler discretization in time of the sample average problem and prove that the value of the discrete time problem converges to the value of the sample average approximation. In addition, we show that there exists a sequence of discrete problems such that the accumulation points of their minimizers are optimal solutions of the original problem. Finally, we propose a convergent descent method to solve the discrete time problem, and show some preliminary numerical results for two simple examples.     

Epsilon-Ritz Method for Solving a Class of Fractional Constrained Optimization Problems

In this paper, epsilon and Ritz methods are applied for solving a general class of fractional constrained optimization problems. The goal is to minimize a functional subject to a number of constraints. The functional and constraints can have multiple dependent variables, multiorder fractional derivatives, and a group of initial and boundary conditions. The fractional derivative in the problem is in the Caputo sense. The constrained optimization problems include isoperimetric fractional variational problems (IFVPs) and fractional optimal control problems (FOCPs). In the presented approach, first by implementing epsilon method, we transform the given constrained optimization problem into an unconstrained problem, then by applying Ritz method and polynomial basis functions, we reduce the optimization problem to the problem of optimizing a real value function. The choice of polynomial basis functions provides the method with such a flexibility that initial and boundary conditions can be easily imposed. The convergence of the method is analytically studied and some illustrative examples including IFVPs and FOCPs are presented to demonstrate validity and applicability of the new technique.     

Meshfree Approximation for Stochastic Optimal Control Problems

In this work,we study the gradient projection method for solving a class of stochastic control problems by using a mesh free approximation ap-proach to implement spatial dimension approximation.Our main contribu-tion is to extend the existing gradient projection method to moderate high-dimensional space.The moving least square method and the general radial basis function interpolation method are introduced as showcase methods to demonstrate our computational framework,and rigorous numerical analysis is provided to prove the convergence of our meshfree approximation approach.We also present several numerical experiments to validate the theoretical re-sults of our approach and demonstrate the performance meshfree approxima-tion in solving stochastic optimal control problems.     

A Generalization of Ritz-Variational Method for Solving a Class of Fractional Optimization Problems

This paper presents an approximate method for solving a class of fractional optimization problems with multiple dependent variables with multi-order fractional derivatives and a group of boundary conditions. The fractional derivatives are in the Caputo sense. In the presented method, first, the given optimization problem is transformed into an equivalent variational equality; then, by applying a special form of polynomial basis functions and approximations, the variational equality is reduced to a simple linear system of algebraic equations. It is demonstrated that the derived linear system has a unique solution. We get an approximate solution for the initial optimization problem by solving the final linear system of equations. The choice of polynomial basis functions provides a method with such flexibility that all initial and boundary conditions of the problem can be easily imposed. We extensively discuss the convergence of the method and, finally, present illustrative test examples to demonstrate the validity and applicability of the new technique.     

Generalized Lagrange Function and Generalized Weak Saddle Points for a Class of Multiobjective Fractional Optimal Control Problems

By constructing a kind of generalized Lagrange function for a class of multiobjective fractional optimal control problems, sufficient and necessary conditions for existence of generalized weak saddle points are established. In addition, the relationship between weak efficiency and generalized weak saddle points is discussed.     

A Pseudospectral Method for Fractional Optimal Control Problems

In this article, a direct pseudospectral method based on Lagrange interpolating functions with fractional power terms is used to solve the fractional optimal control problem. As most applied fractional problems have solutions in terms of the fractional power, using appropriate characteristic nodal-based functions with suitable power leads to a more accurate pseudospectral approximation of the solution. The Lagrange interpolating functions and their fractional derivatives belong to the Müntz space; such functions are employed to show that a relationship exists between the Karush–Kukn–Tucker conditions associated with nonlinear programming and the first optimal necessary conditions. Furthermore, the convergence of the method is investigated. The obtained numerical results are an indication of the behavior of the algorithm.     

An Efficient Monte Carlo Method for Optimal Control Problems with Uncertainty

A general framework is proposed for what we call the sensitivity derivative Monte Carlo (SDMC) solution of optimal control problems with a stochastic parameter. This method employs the residual in the first-order Taylor series expansion of the cost functional in terms of the stochastic parameter rather than the cost functional itself. A rigorous estimate is derived for the variance of the residual, and it is verified by numerical experiments involving the generalized steady-state Burgers equation with a stochastic coefficient of viscosity. Specifically, the numerical results show that for a given number of samples, the present method yields an order of magnitude higher accuracy than a conventional Monte Carlo method. In other words, the proposed variance reduction method based on sensitivity derivatives is shown to accelerate convergence of the Monte Carlo method. As the sensitivity derivatives are computed only at the mean values of the relevant parameters, the related extra cost of the proposed method is a fraction of the total time of the Monte Carlo method.     

Error Estimates for a Class of Elliptic Optimal Control Problems

In this article, functional type a posteriori error estimates are presented for a certain class of optimal control problems with elliptic partial differential equation constraints. It is assumed that in the cost functional the state is measured in terms of the energy norm generated by the state equation. The functional a posteriori error estimates developed by Repin in the late 1990s are applied to estimate the cost function value from both sides without requiring the exact solution of the state equation. Moreover, a lower bound for the minimal cost functional value is derived. A meaningful error quantity coinciding with the gap between the cost functional values of an arbitrary admissible control and the optimal control is introduced. This error quantity can be estimated from both sides using the estimates for the cost functional value. The theoretical results are confirmed by numerical tests.     

Optimal Control Problems for a Class of Semilinear Multisolution Elliptic Equations

We study the optimal control problem for a class of elliptic problems that may possess multiple solutions. We obtain necessary conditions for optimal control by constructing a related parabolic problem and using known results for the parabolic problem.     

多元Kantorovich算子的最优逼近类

本文研究定义在单纯形上的多元Kantorovich算子逼近的正逆不等式与饱和定理,给出该算子在Lp(1≤p≤∞)空间的最优逼近类,即利用K-泛函的特征刻画分别满足‖Knf-f‖p=O(n-1) 与‖Knf-f‖p=o(n-1)的函数类．     

多元Kantorovich算子的最优逼近类

本文研究定义在单纯形上的多元Kantorovich算子逼近的正逆不等式与饱和定理,给出该算子在Lp(1≤p≤∞)空间的最优逼近类,即利用K-泛函的特征刻画分别满足‖Knf-f‖p=O(n-1)与‖Knf-f‖p=o(n-1)的函数类.     

Finite Element Approximation of Elliptic Dirichlet Optimal Control Problems

We develop a priori error analysis for the finite element Galerkin discretization of elliptic Dirichlet optimal control problems. The state equation is given by an elliptic partial differential equation and the finite dimensional control variable enters the Dirichlet boundary conditions. We prove the optimal order of convergence and present a numerical example confirming our results.     

一类非线性奇异最优控制问题的离散解法

本文研究非线性奇异最优控制问题的离散解法。利用在每个“小”时间区间上的积分形式来刻画奇异最优控制的特征,并构造了求解问题的差分方程,同时建立预估、校正格式给出了最优控制近似解的计算方法。     

On a Class of Optimal Control Problems with State Jumps

In this paper, we consider a class of optimal control problems in which the dynamical system involves a finite number of switching times together with a state jump at each of these switching times. The locations of these switching times and a parameter vector representing the state jumps are taken as decision variables. We show that this class of optimal control problems is equivalent to a special class of optimal parameter selection problems. Gradient formulas for the cost functional and the constraint functional are derived. On this basis, a computational algorithm is proposed. For illustration, a numerical example is included.     

Error Estimates for the Approximation of a Class of Optimal Control Systems Governed by Linear PDEs

This paper deals with a class of optimal control problems in which the system is governed by a linear partial differential equation and the control is distributed and with constraints. The problem is posed in the framework of the theory of optimal control of systems. A numerical method is proposed to approximate the optimal control. In this method, the state space as well as the convex set of admissible controls are discretized. An abstract error estimate for the optimal control problem is obtained that depends on both the approximation of the state equation and the space of controls. This theoretical result is illustrated by some numerical examples from the literature.