STRONG CONTROLLABILITY AND OPTIMAL CONTROL OF THE HEAT EQUATION WITH A THERMAL SOURCE

Kamyad, A.V.,Borzabadi, A.H. 한국전산응용수학회 2000 The Korean journal of computational & applied math Vol.7 No.3

In this paper we consider an optimal control system described by n-dimensional heat equation with a thermal source. Thus problem is to find an optimal control which puts the system in a finite time T, into a stationary regime and to minimize a general objective function. Here we assume there is no constraints on control. This problem is reduced to a moment problem. We modify the moment problem into one consisting of the minimization of a positive linear functional over a set of Radon measures and we show that there is an optimal measure corresponding to the optimal control. The above optimal measure approximated by a finite combination of atomic measures. This construction gives rise to a finite dimensional linear programming problem, where its solution can be used to determine the optimal combination of atomic measures. Then by using the solution of the above linear programming problem we find a piecewise-constant optimal control function which is an approximate control for the original optimal control problem. Finally we obtain piecewise-constant optimal control for two examples of heat equations with a thermal source in one-dimensional.

A new approach for solving of linear time varying control systems

Ali Vahidian Kamyad,Mehran Mazandarani 장전수학회 2011 Advanced Studies in Contemporary Mathematics Vol.21 No.3

This paper is concerned with the solution of Linear Time Varying [LTV] control systems. The concept of a solution for LTV systems is defined on the basis of finding the fundamental matrix corresponding to LTV control systems. There are some numerical methods such as Euler method, Taylor method and Runge-Kutta method for obtaining approximate solution of LTV system [LTVs], each of them has some limitations. In the recent years, other kinds of constructive approaches for the solution of LTVs are presented limited to the particular cases of it. In this paper, we introduced a new approach that we call it AVK approach to obtain a global optimal approximation of the fundamental matrix of LTVs, by introducing a problem in calculus of variations corresponding to our LTVs problem. A global optimal approximate solution (general solution of LTV systems) by using Linear Programming [LP] is considered.

A NEW METHOD FOR SOLVING THE NONLINEAR SECOND-ORDER BOUNDARY VALUE DIFFERENTIAL EQUATIONS

Effati, S.,Kamyad, A.V.,Farahi, M.H. 한국전산응용수학회 2000 Journal of applied mathematics & informatics Vol.7 No.1

In this paper we use measure theory to solve a wide range of second-order boundary value ordinary differential equations. First, we transform the problem to a first order system of ordinary differential equations(ODE's)and then define an optimization problem related to it. The new problem in modified into one consisting of the minimization of a linear functional over a set of Radon measures; the optimal measure is then approximated by a finite combination of atomic measures and the problem converted approximatly to a finite-dimensional linear programming problem. The solution to this problem is used to construct the approximate solution of the original problem. Finally we get the error functional E(we define in this paper) for the approximate solution of the ODE's problem.

SOLVING OF SECOND ORDER NONLINEAR PDE PROBLEMS BY USING ARTIFICIAL CONTROLS WITH CONTROLLED ERROR

Gachpazan, M.,Kamyad, A.V. 한국전산응용수학회 2004 Journal of applied mathematics & informatics Vol.15 No.1

In this paper, we find the approximate solution of a second order nonlinear partial differential equation on a simple connected region in $R^2$. We transfer this problem to a new problem of second order nonlinear partial differential equation on a rectangle. Then, we transformed the later one to an equivalent optimization problem. Then we consider the optimization problem as a distributed parameter system with artificial controls. Finally, by using the theory of measure, we obtain the approximate solution of the original problem. In this paper also the global error in $L_1$ is controlled.

A NEW APPROACH TO SOLVING OPTIMAL INNER CONTROL OF LINEAR PARABOLIC PDES PROBLEM

Mahmoudi, M.,Kamyad, A.V.,Effati, S. The Korean Society for Computational and Applied M 2012 Journal of applied mathematics & informatics Vol.30 No.5

In this paper, we develop a numerical method to solving an optimal control problem, which is governed by a parabolic partial differential equations(PDEs). Our approach is to approximate the PDE problem to initial value problem(IVP) in $\mathbb{R}$. Then, the homogeneous part of IVP is solved using semigroup theory. In the next step, the convergence of this approach is verified by properties of one-parameter semigroup theory. In the rest of paper, the original optimal control problem is solved by utilizing the solution of homogeneous part. Finally one numerical example is given.

AN APPROACH FOR SOLVING NONLINEAR PROGRAMMING PROBLEMS

Basirzadeh, H.,Kamyad, A.V.,Effati, S. 한국전산응용수학회 2002 The Korean journal of computational & applied math Vol.9 No.2

In this paper we use measure theory to solve a wide range of the nonlinear programming problems. First, we transform a nonlinear programming problem to a classical optimal control problem with no restriction on states and controls. The new problem is modified into one consisting of the minimization of a special linear functional over a set of Radon measures; then we obtain an optimal measure corresponding to functional problem which is then approximated by a finite combination of atomic measures and the problem converted approximately to a finite-dimensional linear programming. Then by the solution of the linear programming problem we obtain the approximate optimal control and then, by the solution of the latter problem we obtain an approximate solution for the original problem. Furthermore, we obtain the path from the initial point to the admissible solution.

AN APPROACH FOR SOLVING OF A MOVING BOUNDARY PROBLEM

Basirzadeh, H.,Kamyad, A.V. 한국전산응용수학회 2004 Journal of applied mathematics & informatics Vol.14 No.1

In this paper we shall study moving boundary problems, and we introduce an approach for solving a wide range of them by using calculus of variations and optimization. First, we transform the problem equivalently into an optimal control problem by defining an objective function and artificial control functions. By using measure theory, the new problem is modified into one consisting of the minimization of a linear functional over a set of Radon measures; then we obtain an optimal measure which is then approximated by a finite combination of atomic measures and the problem converted to an infinite-dimensional linear programming. We approximate the infinite linear programming to a finite-dimensional linear programming. Then by using the solution of the latter problem we obtain an approximate solution for moving boundary function on specific time. Furthermore, we show the path of moving boundary from initial state to final state.

A NEW METHOD FOR SOLVING NONLINEAR SECOND ORDER PARTIAL DIFFERENTIAL EQUATIONS

Gachpazan, M.,erayechian, A.,Kamyad, A.V. 한국전산응용수학회 2000 Journal of applied mathematics & informatics Vol.7 No.2

In this paper, a new method for finding the approximate solution of a second order nonlinear partial differential equation is introduced. In this method the problem is transformed to an equivalent optimization problem. them , by considering it as a distributed parameter control system the theory of measure is used for obtaining the approximate solution of the original problem.

An approach for solving of a moving boundary problem

H. Basirzadeh,A. V. Kamyad 한국전산응용수학회 2004 Journal of applied mathematics & informatics Vol.14 No.-

In this paper we shall study moving boundary problems, and we introduce an approach for solving a wide range of them by using calculus of variations and optimization. First, we transform the problem equivalently into an optimal control problem by defining an objective function and artificial control functions. By using measure theory, the new problem is modified into one consisting of the minimization of a linear functional over a set of Radon measures; then we obtain an optimal measure which is then approximated by a finite combination of atomic measures and the problem converted to an infinite-dimensional linear programming. We approximate the infinite linear programming to a finite-dimensional linear programming. Then by using the solution of the latter problem we obtain an approximate solution for moving boundary function on specific time. Furthermore, we show the path of moving boundary from initial state to final state.

OPTIMAL CONTROL OF THE HEAT EQUATION IN AN INHOMOGENEOUS BODY

Borzabadi, A.H.,Kamyad, A.V.,Farahi, M.H. 한국전산응용수학회 2004 Journal of applied mathematics & informatics Vol.15 No.1

In this paper we consider a heat flow in an inhomogeneous. body without internal source. There exists special initial and boundary conditions in this system and we intend to find a convenient coefficient of heat conduction for this body so that body cool off as much as possible after definite time. We consider this problem in a general form as an optimal control problem which coefficient of heat conduction is optimal function. Then we replace this problem by another in which we seek to minimize a linear form over a subset of the product of two measures space defined by linear equalities. Then we construct an approximately optimal control.