Numerical Solutions for the Optimal Control Governing by Variable Coefficients Nonlinear Hyperbolic Boundary Value Problem Using the Gradient Projection, Gradient and Frank Wolfe Methods

This paper is concerned with studying the numerical solution for the discrete classical optimal control problem (NSDCOCP) governed by a variable coefficients nonlinear hyperbolic boundary value problem (VCNLHBVP). The DSCOCP is solved by using the Galerkin finite element method (GFEM) for the space variable and implicit finite difference scheme (GFEM-IFDS) for the time variable to get the NS for the discrete weak form (DWF) and for the discrete adjoint weak form (DSAWF) While, the gradient projection method (GRPM), also called the gradient method (GRM), or the Frank Wolfe method (FRM) are used to minimize the discrete cost function (DCF) to find the DSCOC. Within these three methods, the Armijo step option (ARMSO) or the optimal step option (OPSO) are used to improve the discrete classical control (DSCC). Finally, some illustrative examples for the problem are given to show the accuracy and efficiency of the methods.


Introduction
Optimization problems have wide applications in medicine, sciences and many other fields [1,2]. These applications are usually governed by partial differential equations (PDEs) or ordinary differential equations (ODEs).
Many researchers investigated the numerical solution of optimal control problems (NSOCPs) governed by nonlinear elliptic PDEs [3], semilinear parabolic PDEs [4], one dimensional linear hyperbolic PDEs with constant coefficients(LHPDES) [5], two dimensional linear and nonlinear hyperbolic PDEs with constant coefficients [6][7][8][9], two dimensional linear hyperbolic PDEs but with variable coefficients [10], or by one dimensional nonlinear ODEs [11]. The outcomes of these works have driven us to focus our interest on investigating the NSDCOC governed by the VCNLHBVP.
In this paper, the continuous classical optimal control problem (COCOCP) described by the VCNLHBVP is discretized by applying the Galerkin finite element method (GFEM) for the space variable and implicit finite difference scheme (GFEM-IFDS) for the time variable to get the DSCOCP ((the discrete weak form (DWF) for the VCNLHBVP and the discrete cost function (DGF)). To find such solutions, we should discuss the existence and the uniqueness theorem for the NS for the DWF. The proof of the existence theorem for the discrete classical optimal control (DCOC) and the necessary conditions of the problem are studied in a previous article [9] and they are all needed here. On the other hand, the DSCOCP is found numerically by using the GFEM-IFDS to find the NS of the DWF and then the DCOC by solving the optimization problem (the minimum of DCF) by using, separately, each one of the optimization methods; the GM, the GPM and the FWM. Within these three methods, the ARSO or the OPSO are used ,separately, to get better direction of the optimal search. Some illustrative examples for this problem are given to show the performance of each of these methods.

Statement of the COCOCP [6]
Let be a bounded open region, with boundary , and let , -0<T< be a time space. The COCOCP governed by the VCNLHBVP, with control ( ⃗ ) ( ) and state , a.e. in }, with is a convex and compact set. The cost functional is defined by where ( ⃗ ) ( ) is the desired state. The CCOCP is to obtain which minimizes equation (5). Now, the weak form (WF) of the problems (1-4) for , ( ) and the bilinear form is obtained by Assumption A: For each and , the following inequality is satisfied

Statement of the DSCOCP [6]:
The COCOCP is discretized by applying the GFEM as follows: First, consider ( ) is dependent of t, the region can be divided into subregions (a polyhedron) for every integer (s), and be an admissible regular triangulation of ̅ i.e. ̅ ⋃ . Second, let , be a subdivision of the interval and for , where each interval has a same length ( ). Let be the space of continuous piecewise affine mapping (CPAM) in . The set of admissible discrete classical controls (DCC) is , and for , the DWF of (9-12) can be obtained by The discrete cost functional (DCF) ( ) is given by Hence, the DSCOCP is to find ̅ , such that ( ̅ ) ( )

Assumption (B):
(I) Suppose that the function is defined on continuous with respect to and satisfies the following: denotes the Lipschitz constant for any .

Existence of the DSCOCP:
The following assumptions are useful to study the existence of the DCC. Assumption C: The cost functional is of Caratheodary type, and satisfies : where ( ⃗) ( ⃗ ) ( ) and , The proofs of the following theorem and lemmas are shown in a previous article [9] where , for ( ) and is called the Hamiltonian.

Main results (Solution methods):
This section is devoted to present our method which is used to solve the DCCOC governed by the VCNLHBVP, the DWF (and the DAWF) are solved by using the mixed GFEM-IFDS, while the minimum values for the DCF and the DCOC are found by using ,separately, each one of the GM, FWM, or GPM. Within each of these three methods, the ARSO and the OPSO are used, separately, to improve the value of the DCOC. The following algorithm shows the steps of this method in details.
Step 4: Solve the DWF (13-16) to find the state solution corresponding to the new . If , then stop the process . (where is the norm-2 with respect to vector space ). Step 6: Choose by using one of the following methods: ARSO: Assume an initial value , )(or , -). If satisfies the inequality ( ) ( ( )) ( ) we set ⁄ , and choose the last ( ) for , that satisfies the above inequality. If not satisfied, we denote , and choose for the first ( ) (or (in GM) that satisfies the above inequality.

Numerical examples
This section contains some illustrative examples which show the activity of the methods which are given in algorithm (6.1). MATLAB software is used to achieve the above algorithm. The GFEM-IFDS is used in step (2) to find the DS ( ), with ( ) , ( ). In the GM, GPM and FWM, the parameters take the values of and . 7.1 Example: Consider the following COCOCP governed by the VCNLHBVP: , and The control constraint is , and the cost function (5)      , and      (II) The results obtained by applying the GPM with ARSO are better than those obtained by using the GM On the other hand, the results obtained by these two methods are better and faster than those obtained by applying the FWM with ARSO. (III) The results obtained by the GPM and GM with OPSO method are better and faster than those obtained by using the FWM with the OPSO method. (IV) For the OPSO and the ARSO, which were used inside the GPM, although the GM method was faster than the FWM one, i it cannot be used in general, since it is only suitable for a quadratic cost function.