Abstract
This study presents a computational technique developed for solving linearly constraint optimal control problems using the Gradient Flow Method. This proposed method, called the Modified Gradient Flow Method (MGFM), is based on the continuous gradient flow reformulation of constrained optimization problem with three-level implicit time discretization scheme. The three-level splitting parameters for the discretization of the gradient flow equations are such that the sum of the parameters equal to one (\theta1 + \theta2 +\theta3=1). The Linear and quadratic convergence of the scheme were analyzed and were shown to have first order scheme when each parameter exist in the domain [0, 1] and second order when the third parameter equal to one. Numerical experiments were carried out and the results showed that the approach is very effective for handling this class of constrained optimal control problems. It also compared favorably with the analytical solutions and performed better than the existing schemes in terms of convergence and accuracy
Highlights
This study presents a computational technique developed for solving linearly constraint optimal control problems using the Gradient Flow Method
The gradient flow method was first applied to nonlinear programming problems by Evtushenko [5] and later improved by Evtushenko and Zhadan [6]; while the application to unconstrained problems was by Behrman [3] and the unified approach to nonlinear optimization problem was by Wang et al [15]
The numerical examples and respective results are presented below based on the construction of modified gradient flow method (MGFM) for discretized optimal control problems
Summary
This paper considered both the theoretical and numerical analyses of the problem in the development of the algorithm for solving eqns. (1) to (3). This paper considered both the theoretical and numerical analyses of the problem in the development of the algorithm for solving eqns. The discrete form of the quadratic objective functional and the linear differential constraint are constructed using the Trapezoidal and Euler method respectively. The recurrence relation for both the objective functional and constraint were developed to help construct their respective matrix operators.
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