Abstract
This paper presents an Algorithm for the numerical solution of the Optimal Control model constrained by Partial Differential Equation using the Alternating Direction Method of Multipliers (ADMM) accelerated with a parameter factor in the sense of Nesterov. The ADMM tool wasapplied to a partial differential equation-governed optimization problem of the one-dimensional heat equation type. The constraint and objective functions of the optimal control model were discretized using the Crank-Nicolson and Composite Simpson’s Methods respectively into a derived discrete convex optimization form amenable to the ADMM. The primal-dual residuals were derived to ascertain the rate of convergence of themodel for increasing iterates. An existing example was used to test the efficiency and degree of accuracy of the algorithm and the results were favorable when compared the existing method.
Highlights
This paper presents an Algorithm for the numerical solution of the Optimal Control model constrained by Partial Differential Equation using the Alternating Direction Method of Multipliers (ADMM) accelerated with a parameter factor in the sense of Nesterov
Ghobadi et al in [2] exential equations (PDE) are derivations arising from solid me- tended this approach to the one-dimensional heat equation chanics, fluid dynamics, engineering, Economics and sciences and enumerated a lot of advantages when compared with the amongst others
Sci. 3 (2021) 105–115 back of conditional stability will be circumvented in this research with the use of the implicit method of higher order in the discretization and the ADMM extended in obtaining the optimal solutions with better accuracy of results and faster rate of convergence
Summary
It is imperative to have the temperature profile f (x, t ) at each point of the bar that does not go below a given specific temperature function given by g (x, t )( known as a lower bound function), during the time interval. The objective of this modeling is to choose an appropriate temperature profile such that the energy consumed (objective function) is at minimum by reducing overheating, that is, to minimize the cost of energy consumed while keeping the temperature of the bar above the lower-bound profile. The appropriate choice of the objective function can be compared to the temperature profile of the bar with the lowerbound profile as closely as possible expressed mathematically as f ((x, t ), y) − g (x, t ). Assuming the temperature profile is the approximation of the temperature of the bar provided the heat equation and initial condition are satisfied. While u1(t ) and u2(t ) are the temperatures at the two boundary points, l1 and l2 respectively, of the bar whose optimal values will be obtained
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
