ciently rapid, the result of step-function approximations converges to the exact optimum while n - oo. It is necessary to have some assumption of this qualitative kind in order to ensure that the chosen finite dimensional approximations will permit a good approximation to the exact optimum. The RIOTS 95 package can run faster than SCOM, maybe because of its implementation in the C programming language. The efficiency of the STC method has been demonstrated by comparisons with results through MISER3 optimal control software (a general-purpose optimal control software package incorporating sophisticated numerical algorithms). MISER3 did not get results as fast as the STC method did, perhaps because the general-purpose might be hampering its agility to certain extent because of some default settings regarding the tolerances for the accuracy of the ODE solver and optimization routine in the software.

This research is only concerned with pure bang-bang control problems within a fixed-time period. All the algorithms and transformations are made for this purpose. The control function is also approximated by a step-function. However, because the control does not always jump at the grid-points of the subdivisions of the time intervals which are usually equally divided in other works, it is necessary to calculate the optimal divisions of the time horizon. This research is mainly computing the optimal ranges of the subdivisions in time period as well as calculating the minimum of the objective function. Situations when a cost of changing control is involved in the cost function are discussed as well as how this cost can effectively work on the whole system. Although the STC method is also concerned with the calculation of the optimal switching times, it does not include the cost of each switching control.

The limitations of the above computational approaches are summarized in Chen and Craven [10, 2002]. From the above survey it will also appear that each of the above computational methods has characteristics which are computationally efficient for computing optimal control financial models with switching times. A new approach which can adapt various convenient components of the above computational approaches is developed in the next section. Although the present algorithm has similarity with CPET, the details of the two algorithms are different. A new computer package called CSTVA is also developed here which can suitably implement the proposed algorithm. The present computational method consisting the STV algorithm and the CSTVA computer programs does, therefore, provide a new computational approach for modeling optimal corporate financing. The computational approach can be suitably applied to any other disciplines as well.

This approach (STV) consists of several computational methods (described in Chapter 2):

1 The STV method where the switching time is made a control variable optimal value of which is to be determined by the model.

2 A piecewise-linear (or non-linear) transformation of time.

3 The step function approach to approximate the control variable.

4 Finite difference method for estimating gradients if gradients are not provided.

5 An optimization program based on the sequential quadratic programming (SQP) (as in MATLABs constr program similar to the Newton Method for constrained optimization).

6 A second order differential equation to represent the dynamic model.

8. Conclusion

This book is mainly concerned with using the computational algorithms to solve certain classes of optimal control problems. The next chapter will introduce the computational approach named Switching Time Variable (STV) algorithm developed in this book that can solve a class of financial optimal control problems when the control is approximated by a step-function. The piecewise-linear transformation that is constructed for the computer software package is described in Section 2.2. Some non-linear transformations, which were first introduced by Craven [14, 1995] , are also discussed in Section 2.3. These transformations are used to solve the large time period optimal control problem in Chapter 4. A computer software package that was developed by Craven and Islam [18, 2001] and Islam and Craven [37, 2001] is presented in Section 2.4. The nqq function for solving differential equations is quoted as a part of the computer software package in this research. The thrust of this book involves a general computer software for certain optimal control problems. The principal algorithms behind it are introduced in Section 2.6. All the computing results of an example problem for optimal investment planning for the economy are shown in graphs and tables in Section 2.7. A cost of changing control is also discussed in Section 2.8.

Chapter 2

THE STV APPROACH TO

FINANCIAL OPTIMAL CONTROL MODELS

1. Introduction

In this chapter, a particular case of the optimal financial control problems, which has one state function and one control function that is approximated by a step-function, is discussed. Before the problem is defined, it is necessary to cover some concepts and transformations in Section 2.2 and Section 2.3, to explain the problems and algorithms that will be introduced in later sections and chapters. As part of the computer software package SCOM (that was constructed by Craven and Islam), nqq is used as a DE solver in the algorithms in this research. This program is described in Section 2.4. Then a simplified control problem is introduced in Section 2.5. The computational algorithms , which are used to solve this simplified control problem, are indicated in Section 2.6. Some problems with different fitting functions will be discussed later in Chapter 5. Lastly, graphs and tables in Section 2.7 represent all the computing results of this problem. The analysis of the results is discussed in Section 2.8

2. Piecewise-linear Transformation

The idea of piecewise-linear transformation of the time variable was first introduced by Teo [40, 1991], but the time intervals were mapped into + 1) instead of (jh, (j + h = l/n, where n is the number of time intervals. In Lee, Teo, Rehbock and Jennings [51, 1997], the time transformation is described by where is a piece-wise constant transformation. In this book, a similar idea is used, but the implementation is a little simpler, not requiring another differential equation. A non-linear transformation of the time scale given by Craven [15, 1998], is introduced in Section 2.3. The transfor-