Abstract
This paper deals with the numerical solution of the discrete classical optimal control problem (DCOCP) governing by linear hyperbolic boundary value problem (LHBVP). The method which is used here consists of: the GFEIM " the Galerkin finite element method in space variable with the implicit finite difference method in time variable" to find the solution of the discrete state equation (DSE) and the solution of its corresponding discrete adjoint equation, where a discrete classical control (DCC) is given. The gradient projection method with either the Armijo method (GPARM) or with the optimal method (GPOSM) is used to solve the minimization problem which is obtained from the necessary condition for optimality of the DCOCP to find the DCC.An algorithm is given and a computer program is coded using the above methods to find the numerical solution of the DCOCP with step length of space variable , and step length of time variable . Illustration examples are given to explain the efficiency of these methods. The results show the methods which are used here are better than those obtained when we used the Gradient method (GM) or Frank Wolfe method (FWM) with Armijo step search method to solve the minimization problem.
Highlights
Optimal control problems of partial differential equations PDEs have wide applications in many real live problems such as in economic, electromagnetic waves, biology, robotics, dynamical elasticity, medicine, air traffic and many others
Depending on the above initial control and its corresponding state, we get: (I) the GPARM, after 6 iterations is used to get the optimal control and its corresponding state, the results show with
(I) In the GPARM, after 9 iterations is used to get the optimal control and corresponding state, the results show with:
Summary
Optimal control problems of partial differential equations PDEs have wide applications in many real live problems such as in economic, electromagnetic waves, biology, robotics, dynamical elasticity, medicine, air traffic and many others. The our previous work push us to continue in studying the numerical solution for the DCOCP governed by LHBVP via GFEIM but instead of GM or FWM with the Armijo method to find the numerical DCOC, the GPM with both the ARM (GPARM) and the optimal step method (GPOSM) is used to find the numerical DCOP to solve the minimization problem which is obtained from the necessary condition for optimality of the DCOCP to find the DCC An algorithm is given and a computer program is coded in Mat lab software to solve the DCOCP, the results are drawing by figures and show that the GPARM is better than those methods which are used in [7], to solve the minimization problem
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