On the Discretized Algorithm for Optimal Problems Constrained by Differential Equation with Real Coefficients

A discretized scheme, Discretized Continuous Algorithm (DCA), for solving constrained quadratic optimal control problems was developed to ease the computational cumbersomeness inherent in some existing algorithms, particularly, the Function Space A lgorithm (FSA) by replacing the integral by a series of summation. In order to accomplish this numerical scheme, we resort to a finite approximation of it by discretizing its time interval and using finite difference method for its differential constraint. Using the penalty function method, an unconstrained formulation of the problem was obtained. With the bilinear form expression of the problem, an associated operator was constructed which aided the scheme for the solution of such class of problems. A sample problem was examined to test the effectiveness of the scheme as to convergence with relation to other existing schemes such as Extended Conjugate Gradient Method (ECGM), Multiplier Imbedding Extended Conjugate Gradient Method (MECGM) and Function Space Algorithm (FSA) for solving penalized functional of optimal control problem characterized by non-linear integral quadratic nature.


INTRODUCTION
The discretized scheme, DCA, with less computational rigour was proposed and compared to some existing algorithms, particularly the FSA which circumvented the use of operator, for solving a class of quadratic optimal control problems. ECGM and MECGM based on [1] on function minimization reviewed by [4] were ingredients to the development of the discretized scheme. Here, a generalized constrained formulation of the problem is given below for the discretization exercise of the scheme.

MATERIALS AND METHODS
The constrained problem can be turned into unconstrained problem via the penalty method [2] .
The problem may be put in the following equivalent form; x,U 0 2 Z,AZ Min ax t bu t x t cx t dux t }dt 0 is the penalty cons tan t. µ ≥ Discretization: By [3] , discretizing (2), subdivide [0,T] into n equal intervals at meshpoints x 0 <x 1 <x 2 < x 3 < …<x n-1 <x n with nx j = j j , j = 0,1,2,…,n, where j is the fixed length of each subinterval. x k =x t , u k =u t , k=0,1,2,...,n By finite difference method , We then have the discretized function in the form; Equivalent unconstrained and discretized formulation of the problem: (4) is turned into an unconstrained problem ( ) Simplifying (5), we have x t u t [ 2 d ] Construction of operator a: Now, Further simplifying (8) and using [6,7] , we have Setting µ K2 (T k ) = 0, in (11) and by [5] , we have 11 K 2 11 To obtain the component with the initial conditions V 11 (0) = p 0 and V 11 (0) = r 0 Solving (18) by Laplace method and letting ( )     the discretized algorithm is now applied to the following hypothetical problems P1 and P2 stated thus;

EXAMPLES
Example problem P1 The exact analytical solution is 1.0647 given by [7] . The numerical solution to this problem is obtained by assuming the following initial values and parameters; x0 = 1, u0 =.5 and.5 µ 2.5 ( ) The problem has been solved by other numerical methods such as Function Space Algorithm(FSA), Extended Conjugate Gradient Method (ECGM) and Multiplier Imbedding Extended Conjugate Gradient Method(MECGM)[ ] with results tabulated below. The concern here, in this paper, is solving the discretized problem numerically using penalty constant µ, where µ =.5(2.5).5. The stepsize = .2 is chosen arbitrarily constant. Also, the efficiency of each sheme is determined by the value of the objective function value closiest to the exact solution with a level of tolerance .3500 per iteration The solution to this problem is obtained by assuming the following initial values for the variables; The exact analytical solution is 0.7641 Applying the same algorithm to problem P2 and solving by qbasic programming language, we have the following table 1.2

COMMENTS AND CONCLUSION
In Table 1, for parameters, 0.5 µ 2.5, we see that the absolute value difference between the exact solution 1.0647 and the numerical solution per iteration in column 5 for DCA and MECGM all lie within the interval 0 x -x £ .3500, except for µ = 2.5 and µ = 0.5, respectively, Where, 0 x 1.0647, x is the iterate and .3500 is the tolerance level. FSA, for all parameters, is not comparable. Hence it performs worst of all the algorithms. Conclusively, DCA and MECGM are ranked qualitatively equivalent but comparably better than either FSA or ECGM.
In Table 2, step 2, the superiority of the DCA and ECGM has been exhibited for all values of the parameters in 1.0 µ 2.5 in column 5, except for µ = 0.5, where it ranks second to ECGM with minimum  at 0.79989. In fact, for µ = 1.00, the optimum 0.8303 is attained for DCA comparable to the exact solution 0.7641. Also, DCA's solution deviation from the exact solution has been within a tolerance level of ±0.1186 as seen in column 5 from its iterates, while other algorithms's iterates fall outside this tolerance level for some parameters in 1.0 µ 2.5. However, FSA did worst of all the algorithms.
Conclusively, DCA, with its less computational rigour, has performed better absolutely than either FSA or ECGM. So, it is an additional algorithm for the solution of such class of problems under consideration