The main aim presented in this article is to provide an efficient transferred Legendre pseudospectral method for solving pantograph delay differential equations. At the first step, we transform the problem into a continuous-time optimization problem and then utilize a transferred Legendre pseudospectral method to discretize the problem. By solving this discrete problem, we can attain the pointwise and continuous estimated solutions for the major pantograph delay differential equation. The convergence of method has been considered. Also, numerical experiments are described to show the performance and precision of the presented technique. Moreover, the obtained results are compared with those from other techniques.

In this paper, we define a functional optimization problem corresponding to smooth functions which its optimal solution is first derivative of these functions in a domain. These functional optimization problems are applied for non-smooth functions which by solving these problems we obtain a kind of generalized first derivatives. For this purpose, a linear programming problem corresponding functional optimization problem is obtained which their optimal solutions give the approximate generalized first derivative. We show the efficiency of our approach by obtaining derivative and generalized derivative of some smooth and nonsmooth functions respectively in some illustrative examples.

In this paper, a new approach using linear combination property of intervals and discretization is proposed to solve a class of nonlinear optimal control problems, containing a nonlinear system and linear functional, in three phases. In the first phase, using linear combination property of intervals, changes nonlinear system to an equivalent linear system, in the second phase, using discretization method, the attained problem is converted to a linear programming problem, and in the third phase, the latter problem will be solved by linear programming methods. In addition, efficiency of our approach is confirmed by some numerical examples

Abstract-In this paper we present a new approach for linear Volterra integral equations that is based on optimal control theory. S ome optimal control problems corresponding Volterra integral equation be introduced which we solve these problems by discretization methods and linear programming approaches. Finally, some examples are given to show the efficiency of approach.Index Terms -Volterra integral equations, Optimal control, Linear programming. I. INTRO DUCTIO NVo lterra integral equations arise in many physical applications, e.g., potential theory and Dirichlet problems and electrostatics. Also, Volterra integral equations are applied in the biology, chemistry, engineering, mathematical problems of radiat ion equilibriu m, the particle transport problems of astrophysics and reactor theory, and radiation heat transfer problems [1,2,3,4,5]. There exist the some valid appro ximate and numerical methods for solving Vo lterra integral equation such as Adomian decomposition method [6], Walsh functions method and mult igrid approach [7,8] [35,36,37,38,39]. In this paper, we present a d ifferent approach fro m above methods for solving linear volterra integral equations which is based on the optimal control theory [40,41,42]. Consider the following linear Vo lterra integral equations of second kind: ( ) ( ) ( , ) ( ) , [ , ],   

In this paper, we propose a new approach to design the asymptotically stabilizing control and adaptive control for nonlinear systems. We first propose an infinite-horizon optimal control (OC) problem according to the nonlinear control system. We prove that its OC can be an asymptotically stabilizing control or adaptive control for the original nonlinear system. Then, we convert the infinite-horizon OC problem into an equivalent finite-horizon OC problem. Moreover, we utilize the Legendre pseudospectral method to solve the proposed problem. Finally, several examples are given to illustrate the efficiency of the approach.KEYWORDS adaptive control, brush DC motor, infinite-horizon optimal control, Legendre pseudospectral method, stabilizing control 1952