In this paper, we will study the  Optimal control problem of a system containing a differential integral (D-I) operator. We will deduce the necessary Optimality conditions and apply it first to the problem of minimum energy to find the lowest energy for an electrical circuit containing a resistor, a coil and a capacitor (RLC circuit), and second to the problem of the minimum time to transfer electrical current in  RLC circuit from one state to another in the shortest possible time.

The Optimal control of problem is about finding a control law for a given system such that a certain Optimality criterion is achieved. Methods of solving the Optimal control problems are divided into direct methods and mediated methods (through other equations). In this paper, the PSO- SVM indirect method is used to solve a class of Optimal control problems. In this paper, we try to determine the appropriate algorithm to improve our answers to problems.

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.

In this paper, one numerical method is presented for numerical approximation of linear constrained Optimal control problems with quadratic performance index. The method with variable coefficients is based on Hermite polynomials. The properties of Hermite polynomials with the operational matrices of derivative are used to reduce Optimal control problems to the solution of linear algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

In this paper we consider the Time Optimal control Problem with Bounded state (TOCPB). By means of a process of embedding and using measure theory, this problem is replaced by another, in which we seek to minimize a linear form over a subset of a measure space defined by linear equalities. The theory allows us to convert the new problem to an infinite-dimensional linear programming problem. Afterwards, the infinite-dimensional linear programming problem is approximated by a finite dimensional one. Then by the solution of the final linear programming problem one can find an approximate value of the trajectory function x (0), control function u (0) and Optimal time T as well. AMS classification (49A).