Numerical Solution for Classical Optimal Control Problem Governing by Hyperbolic Partial Differential Equation via Galerkin Finite Element-Implicit method with Gradient Projection Method

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.


Introduction
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 numerical solution of the DCOCP is studied by many researches. The GPARM or GPOSM are used to find the numerical solution of the DCOCP governing either by systems of nonlinear elliptic PDEs as in [1,2], or by systems of semi linear parabolic PDEs as in [3,4], or by systems of nonlinear ordinary differential equations (ODEs) as in [5,6], or by systems of LHBVP so as our previous work [7]. Since the GFEM is one of the most an efficient and fast methods for

Numerical Solution for Classical Optimal Control Problem Governing by Hyperbolic Partial Differential Equation via Galerkin Finite Element-Implicit method with Gradient Projection Method
solving different types of differential equations in general [8,9], and DCOCP in particular. In fact, the GFEIM is used in [7]. To find the numerical solution for the state of the LHBVP and its adjoint equations while the Gradient method (GM)and Frank Wolfe method (FM) with the Armijo method is used there to solve the minimization problem which is obtained from the necessary condition for optimality of the DCOCP to find the DCC. 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. The cost functional is given by

Description of the CCOC and the DCOC Problems
where ⃗ and ⃗ are the desired state and the desired control respectively.
The CCOC problem is to minimize the cost functional (5) subject to . In this work, the inner product and the norm in it will be indicated by and respectively, while the norm in Sobolev space by , and the norm in by . Now, the weak form (WF) of the problem (1-4) is given by: Where is a bilinear form which is symmetric, and satisfies the following assumptions, , ̅ and for some positive scalars and .

Description of the DCCOCP
The discretization of the CCOC is obtained by using the GFEM. Assume that is polyhedron domain. For every integer (s), let be an admissible regular triangulation of ̅ , be a subdivision of the interval ̅ and be the space of continuous piecewise affine mapping in .
For each , the discrete state equations (DSEs) of (1-4) is written by (for The DCOC problem here is to find , such that ̅ ̅

The solution of the DCOCP
This section deals with some theorems and lemmas which are important in the next section they can be proved by using the same techniques which are used in [7].

Theorem 1: For any fixed j (
, and , the DSEs (9-12) has a unique solution for sufficiently small .

Theorem 2:
The operator is continuous.
(15) (16) where ,( The discrete directional derivative of G in equation (13) is given by: Where , for ( ), and is called the discrete Hamiltonian functional. Remark: To prove equation (17) is equivalent to the minimum principle block wise.
Let and , for all j except once(say ) i.e. , then , since is arbitrary then we have equation (18), and converse is very clear. [10]. The GPM is an iterative method used to find the point that minimizes the problem. The following algorithms describe the GP, GPARM and GPOSM; we will use the norm with respect to vector space Q. [10] Let Q be a Hilbert space, is a convex subset of an open set , , and let , be a sequence with , for each , and let be an initial classical control,  [10] Let Q be a Hilbert space, is a convex subset of an open set , , and let , , be a sequence with , for each . , and let be an initial control. Step1: Set , solve the system of the WF (9-12) (the system of the adjoint WF (14-16)) by GFEM to get ( , and then Calculate from equation (17) and from equation (13).

Algorithm
Step 2: Find a direction point , (i.e. a direction ) by applying the GPM as follows: Find the unique , such that Step 3: Solve the system of the WF (9-12) to find the new state corresponding to the new control .
Step 4: Choose using one of the following methods: ARM: Assume an initial value . Find the ( ) from equation (13). If satisfies the inequality ( ) we set ⁄ , and choose for , the last ) that satisfies the above inequality. If unsatisfied, set , and choose for the first that satisfies this inequality. OPSM: Find an , such that ( ) Step 5: Set , and we go to step 2. Practically the ARM is faster and is a finite procedure than OPSM. The following examples are solved by using the above algorithm; a computer program in mat lab software version 8.1.0.604 is coded. The solution and in step (1) are found using the GFEM with , ( ), the parameters in ARM are taken the value , and the parameter 0.5 in the GPM. with the initial control, ⃗ , ⃗ Algorithm (5.1) is used here to solve this problem. The following figures are represented the initial control and its corresponding state.  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 =5.9085e-08, 6.6150e-04, and =7.6800e-05 where and are the discrete maximum error of the state and the control respectively.

Numerical Examples
The following figures represent the optimal control and its corresponding state which are obtained by using GPARM.  (II) The GPOSM, after 2 iterations is used to obtain the optimal control and its corresponding state, with: =5.8424e-08, 6.6150e-04, and =7.1644e-05 The following figures are posted by using GPOSM and are represented the optimal control and its corresponding state.  Example 2: Consider the COCP, which was considered in example 1, but the control constraint is , and the desired control is given by ⃗ with the initial control,

⃗ {
The following figures represent the initial control and its corresponding state.  (I) In the GPARM, after 9 iterations is used to get the optimal control and corresponding state, the results show with: =5.9048e-08, 6.6150e-04, and =4.0813e-05 The following figures are represented the optimal control and its corresponding state.  (II) The GPOSM, after 2 iterations is used to get the optimal control and corresponding state, with =5.8570e-08, 6.6150e-04, and =7.4298e-05 The following figures are represented the optimal control and its corresponding state.

Conclusion
The following conclusions can be derived from the obtained results: (1) the GFEIM which is used here to find the DSE of LHBVP as well as the discrete adjoint equation for the state equation is fast and efficient than other method e.g., the finite difference methods, methods of lines, variational methods which clearly are needed more time and hence have more computations which are Grow up the error.
(2) By the results of the above examples, with (step length of space variable , and step length of time we conclude that: (I) the GPARM and the GPOSM which are used to solve the minimization problem (the cost functional), are suitable and efficient methods to find the DCOC governed by hyperbolic boundary value problem, with parameters , and . (II) Although the GPOSM is needed less iterations than GPARM, but the GPARM remain better since it is used for general minimization problems, on the contrary of the GPOSM which is used only for quadratic functional.