In this paper, a numerical technique is proposed to solve Optimal Control problems (OPCs) of Volterra integral equations (VIEs). We apply the linear B-spline polynomials to solve OPCs by VIEs. The B-spline function divides the interval into sub-intervals and then built a different approximating polynomial on each sub-interval. In this method, Optimal trajectory and Control functions are expanded in terms of B-spline functions. The linear B-spline operational matrix of integration and multiplication are utilized in the proposed method. The main characteristic this method is that by using the suggested numerical technique and the related operational matrices, Optimal Control problem governed by Volterra integral equations is converted to a system of equations. Suffice it to say that this scheme simplifies the main problems and also makes to obtain a good approximate solution for them. In the end, there are two illustrative examples which numerical results show the validity and applicability of our method.

The non-convex behavior presented by nonlinear systems limits the application of classical optimization techniques to solve Optimal Control problems for these kinds of systems. This paper proposes a hybrid algorithm, namely BA-SD, by combining Bee algorithm (BA) with steepest descent (SD) method for numerically solving nonlinear Optimal Control (NOC) problems. The proposed algorithm includes the merits of BA and SD simultaneously. The motivation of presenting the proposed algorithm includes that BA is showed to converge to the region that global optimum is settled, rapidly during the initial stages of its search. However, around global optimum, the search process will become slowly. In contrast, SD method has low ability to convergence to local optimum, but it can achieve faster convergent speed around global optimum and the convergent accuracy can be higher. In the proposed algorithm, at the beginning step of search procedure, BA is utilized to find a near optimum solution. In this case, the hybrid algorithm is used to enhance global search ability. When the change in fitness value is smaller than a predefined value, the searching procedure is switched to SD to accelerate the search procedure and find an accurate solution. In this way, the algorithm finds an optimum solution more accurately. Simulations demonstrate the feasibility of the proposed algorithm.

In this paper the Chebyshev finite difference method is em-ployed for finding the approximate solution of time varying Constrained Optimal Control problems. This approach consists of reducing the op-timal Control problem to a nonlinear mathematical programming prob-lem. To this end, the collocation points (Chebyshev Gauss-Lobatto nodes) are introduced then the state and Control variables are approx-imated using special Chebyshev series with unknown parameters. The performance index is parameterized and the system dynamics and con-straints are then replaced with a set of algebraic equations. Numerical examples are included to demonstrate the validity and applicability of the technique.

Multi-Parametric Programming (mpp) is a widely acknowledged mathematical method for its ability to handle parameter depended optimization problems explicitly. Using this powerful tool, the traditional model predictive Control (MPC), which is highly time consuming, can be solved efficiently, so-called explicit MPC (EMPC). Its main advantage is that it obtains the Control actions as explicit function of the plant measurements. This allows for the Control actions to be obtained online via simple function evaluations instead of solving repetitively a computationally demanding online optimization. This significant characteristic allows the MPC strategy to be applicable for the fast dynamics, which may be prohibitive for traditional MPC that relies on optimization methods.This paper aims to study this method from theoretical and practical point of view. The complete unified framework for developing EMPC and its mathematical foundations are studied. Also, in this paper a simple closed form for some matrices well established to achieve low computational complexity in design procedure. Finally the problem of servo system Control has been dealt with and thereby the performance and effectiveness of the presented method verified.

In this paper, generalized model predictive spread Control methodis developed to consider the intermediate constraints on system states and system inputs. Because of using the orthogonal basis functions and thus reducing the computational burden, this new method which is named Constrained generalized model predictive spread Control can be used in online implementation of a finite-time Constrained Optimal Control problem. For demonstrating the performance of the proposed technique, in this paper an interceptor midcourse guidance problem is formulated to reach a desired point in space. Several constraints are considered such as: hard and soft intermediate and terminal constraints on system states and different constraints on input acceleration command to the interceptor in different time intervals of the trajectory. It is shown that in all the above situations, the proposed method could produce the Optimal guidance commands such that all the interceptor trajectory constraints are satisfied.

This paper presents two numerical methods for solving the nonlinear Constrained Optimal Control problems including quadratic performance index. The methods are based upon linear B-spline functions. The properties of B-spline functions are presented. Two operational matrices of integration are introduced for related procedures. These matrices are then utilized to reduce the solution of the nonlinear Constrained Optimal Control to a non-linear programming one to which existing well-developed algorithms may be applied. Illustrative examples are included to demonstrate the validity and applicability of the presented techniques.

We develop a method to obtain an Optimal solution for a Constrained distribution system with several items and multi-retailers. The objective is to determine the procurement frequency as well as the joint shipment interval for each retailer in order to minimize the total costs. The proposed method is applicable to both nested and non-nested policies and ends up with an Optimal solution. To solve this large nonlinear and integer problem, a two-level algorithm is proposed. In the first level, the functional constraints are relaxed and a solution is obtained by taking advantage of its special structure. Then, we apply separable programming technique for finding the Optimal solution of the original problem. To decrease the size of the problem, some appropriate bounds on variables are introduced. We will show that under some conditions, the Optimal solution of the original problem is proportional with the solution of its unConstrained problem.

This paper presents a numerical fuzzy indirect method based on the fuzzy basis functions technique to solve an Optimal Control problem governed by Poisson’, s differential equation. The considered problem may or may not be accompanied by a Control box constraint. The first-order necessary Optimality conditions have been derived, which may contain a variational inequality in function space. In the presented method, the obtained Optimality conditions have been discretized using fuzzy basis functions and a system of equations introduced as the discretized Optimality conditions. The derived system mostly contains some nonsmooth equations and conventional system solvers fail to solve them. A fuzzy system-based semi-smooth Newton method has also been introduced to deal with the obtained system. Solving Optimality systems by the presented method gets us unknown fuzzy quantities on the state and Control fuzzy expansions. Finally, some test problems have been studied to demonstrate the efficiency and accuracy of the presented fuzzy numerical technique.

Structural optimization, when approached by conventional (gradient based) minimization algorithms presents several difficulties, mainly related to computational aspects for the huge number of nonlinear analyses required, that regard both Objective Functions (OFs) and Constraints. Moreover, from the early ' 80s to today' s, Evolutionary Algorithms have been successfully developed and applied as a computational alternative to many optimization problems, such as structural ones. In this study the effectiveness of a relatively new Evolutionary Algorithm, namely Differential Evolutionary, is investigated for Constrained optimization. This presents many interesting advantages and so that it is a candidate to be widely used in many real structural optimization problems. The algorithm version here used has been developed by hybridizing some recent versions of Differential Evolutionary algorithms proposed in literature, and uses a specific way for dealing with constraints which, always, concern real structural optimization problems. The effectiveness of proposed approach has been demonstrated by developing two cases of study, which regard simple but very significant structural problems for steel structures, one of which is a standard benchmark in structural optimization. The analyses show the simplicity and effectiveness of the proposed approach, so that it can be suitably ready for practical uses out of academic contest.