Mixed Implicit Galerkin – Frank Wolf, Gradient and Gradient Projection Methods for Solving Classical Optimal Control Problem Governed by Variable Coefficients, Linear Hyperbolic, Boundary Value Problem

This paper deals with testing a numerical solution for the discrete classical optimal control problem governed by a linear hyperbolic boundary value problem with variable coefficients. When the discrete classical control is fixed, the proof of the existence and uniqueness theorem for the discrete solution of the discrete weak form is achieved. The existence theorem for the discrete classical optimal control and the necessary conditions for optimality of the problem are proved under suitable assumptions. The discrete classical optimal control problem (DCOCP) is solved by using the mixed Galerkin finite element method to find the solution of the discrete weak form (discrete state). Also, it is used to find the solution for the discrete adjoint weak form (discrete adjoint) with the Gradient Projection method (GPM) , the Gradient method (GM), or the Frank Wolfe method (FWM) to the DCOCP. Within each of these three methods, the Armijo step option (ARSO) or the optimal step option (OPSO) is used to improve (to accelerate the step) the solution of the discrete classical control problem. Finally, some illustrative numerical examples for the considered discrete control problem are provided. The results show that the GPM with ARSO method is better than GM or FWM with ARSO methods. On the other hand, the results show that the GPM and GM with OPSO methods are better than the FWM with the OPSO method.


Introduction
Optimal control problems (OCP) have various applications [1,2]. These problems are usually governed by partial differential equations (PDEs) or ordinary differential equations (ODEs). Many researchers have been interested to study the numerical solution of optimal control problems described by nonlinear elliptic PDEs [3,4], by semilinear parabolic PDEs [5][6][7], or by one dimensional linear hyperbolic PDEs with constant coefficients (LHPDES) [8]. The researchers also include two dimensional linear and nonlinear hyperbolic PDEs with constant coefficients [9][10][11][12], or by nonlinear ODEs [13]. These works attracted our attention to focus our interest on studying OCP described by LHPDES but with variable coefficients (LHBVPVC).
This paper investigates the numerical solution of the DCCOCP that is described by the LHBVPVC. Here, the continuous classical optimal control problem (CCOCP) is described, which is discretized by applying the Galerkin finite element method (GFEM). The GFEM is applied for variable space and the implicit finite difference scheme (IFDS) which is employed for the time variable to obtain the discrete CCOCP(DCCOCP). The existence and the uniqueness theorem for the discrete solution (DS) of the discrete weak form (DWF) is stated and proved. The DCCOCP is found numerically by using the mix of the GFEM with the IFDS (GFEM-IFDS) [10] to find the DS of the DWF, while the DCOC is obtained through solving the optimization problem (finding the minimum of the cost function) by separately using one of the following optimization methods: the Gradient method (GM) , the Gradient projection method (GPM), and the Frank Wolfe method (FWM) [14]. Within each one of these three methods the Armijo step option (ARSO) or the optimal step option (OPSO) is used to improve the direction of the optimal search [14]. Some illustrative examples for this considered problem are given to show the accuracy and the efficiency of each of the three methods.

The Statement of the CCOCP: Let
be a bounded open region, be the boundary of , and , -0<T< be a time space, . The CCOCP governed by the LHBVPVC is: with BC and ICs , a.e. in },with is a convex and compact set . The cost functional [9] is given by where, the desired state is symbolized by ( ⃗ ) ( ) . The CCOCP is to find which minimizes (5). Now, the weak form (WF) of the problem (1)(2)(3)(4) for where , ( ) and ( ) ( ̅ ( ⃗ ) ) ( ̿ ( ⃗ ) ) is a symmetric bilinear form. Assumptions A [9]: For each and the following inequality is satisfied where . Now, suppose , then (6)(7)(8) can be rewritten as 2. The DCCOCP [9]: The CCOCP is discretized by using the GFEM as follows: First, the region can be divided into subregions (a polyhedron) for every integer (s), be an admissible regular triangulation of ̅ i.e. ̅ ⋃ . Second, let , be a subdivision of the interval and for , where each interval has same lengths ( ). Let ( ) be the space of continuous piecewise affine mapping (CPAM) in .
The set of admissible discrete classical controls (DCC) is , and for , the DWF (9)-(12) can be given by . The discrete cost functional (DCF) ( ) is defined by Hence, the DCCOCP is to find a DCOC ̅ , such that ( ̅ ) ( )

Theorem (existence and uniqueness of the DWF):
The DWF (13-16) and for any fixed , with fixed DCC , has a unique solution ( ), for sufficiently small . Proof: To find the solution ( ) for any fixed j ( ), let ( ( ⃗) , ( ) are CPAM in with ( ⃗) on ) being a span of . Then for any and , , , , equations (13-16) can be formed as: Now by using the GFEM, we obtain , , where ( ) and ( ) are unknown constants, and to be determined.
(34) Substituting equation (34) into equation (33) gives Now, by choosing ̿ , the second and the fourth terms in the LHS of equation (35) are positive.

Theorem [10]:
Suppose that is convex and compact, and if ( ) is coercive, there exists a DCOC. Proof: By using the same technique used in (Theorem 4 in [10]).

The Necessary conditions for DCCOCP
The following theorem deals with the state and proof for the necessary conditions of the DCCOCP 4.1 Theorem: Assume that DCF of equation (17) where , for ( ) and is called the Hamiltonian. Proof: By using equation (31) and set , then summing over (for to ), we get Now, for any given values of ( ) in a vector space, the following functions are defined almost everywhere on E: On the other hand, since the FD of the DCF exists, then ( ) ( ) where ( ) and as . By substituting equation (48) into equation (49), one can have where ( ) and as . Finally, the FD of the DCF is ) .

Optimization methods:
The following algorithm shows the numerical calculation for the DCOC by using the mixed GFEM-IFDS with each of the methods of GM, FWM, or GPM (with ARSO and Step 7: Set ( ), and we go to step 2.

Numerical results for solving the DCOCP
This section contains some illustrative examples to show the activity of the methods which are given in algorithm (5.1). Mat lab software is used to achieve the solution of the methods. The GFEM is used in step (2) to find the DS ( ), with , and . In the GM, GPM and FWM, the parameters are taken the values of and . 6.1 Application 1: Consider the following CCOCP governed by the LHBVPVC: with the initial control ( ⃗ ) ( ( )) , ( ⃗ ) . First, depending on the above initial control and its corresponding state, the following results are obtained according to the optimization methods with ARSO: (I) In the GM: the optimal control and corresponding state are obtained after iterations, and the results are: ( )=1.4362e-06, 4.2e-03, and =2.5438e-04 where and are the discrete maximum errors for the state and control, respectively.
, in The control constraint is , and the cost function (5) with is with the initial control ( ⃗ ) , ( ⃗ ) . First, depending on the above initial control and its corresponding state, the following results are obtained according to the optimization methods with ARSO: (I) In the GM: the optimal control and corresponding state are obtained after iterations, and the results are: ( )=8.3096e-06, 8.7e-03, and =3.6968e-04 The optimal control and its corresponding state at are shown by Figures-(11 and 12).    8.7e-03, and =5.7906e-04 Figures-(19 and 20) show the optimal control and its corresponding state at .

Conclusions
In this paper, the proof of the existence and uniqueness theorem for the DS of the DWF for the LHBVPVC is achieved. The existence theorem for the DCOC and the necessary conditions for optimality of the problem are proved under suitable assumptions. On the other hand, the DCOCP was solved numerically by using the mixed GFEM-IFDS to find the DS, the DWF and its adjoint of the DAWF, with step length of space variable and step length of time . While the DCOC is obtained by finding the minimum of the cost function by using each one of the optimization methods of GPM, GM and FWM with either ARSO or ORSO step options with parameters ( , and ). From the numerical solutions we concluded that; the GFEM was a suitable and fast method to solve the DWF and DAWF, beside this we saw from the results obtained using the GPM with ARSO method were better than those obtained using the GM or FWM with ARSO methods, on the other hand the results obtained using the GPM and GM with OPSO methods were better than those obtained using the FWM with the OPSO method. The OPSO method needed less or equal number of iterations than the ARSO method. This comparison happened when we had a quadratic cost function. Finally, when we had a more general function, the OPSM was not easy to be applicable, while the ARSO method can be considered as a general method to improve the direction search.