OPTIMIZATION OF QUADRATIC CONSTRAINED OPTIMAL CONTROL PROBLEMS USING AUGMENTED LAGRANGIAN METHOD

Abstract

In this research, the analytical and numerical solutions of continuous optimal control problems governed by both ordinary differential equation and linear damping evolution with delay and real coefficients are presented. The necessary conditions for optimality are obtained from the knowledge of calculus of variation. These necessary conditions, obtained for optimal control problem constrained by delay differential equation are a linear two-point boundary value problem involving both delay and advance terms. Clearly, this coupling that exists between the state variable and the control variable is not amenable to analytical solution, hence a direct numerical approach is adopted. We propose an augmented discretized continuous algorithm via quadratic programming, which is capable of handling optimal control problems constrained by both ordinary and delay differential equations. The discretization of the problem using trapezoidal rule (a one step second order numerical scheme) and Crank-Nicholson with quadratic formulation amenable to quadratic programming technique for solution of the optimal control problems are considered. A control operator (penalized matrix), through the augmented Lagragian method, is constructed. Important properties of the operator as regards sequential quadratic programming techniques for determining the optimal point are shown. Numerical examples to illustrate the efficiency and robustness of this new technique are presented. Results show that the new technique converges faster and gives a better solution than the existing algorithms.