Промышленный лизинг Промышленный лизинг  Методички 

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maximum principle should be replaced by the indifference principle , and optimal control will also exist. The existence of an optimal solution can be proved by Theorem 2 in Blatts paper.

This research originated from Blatts work. The goal of this book involves some novel computational algorithms for solving (1.28-1.32) based on the theorems in Blatts paper. The work on the cost analysis, difference of optimal control policy sequences and division of the time intervals are extended from Blatts original work. The methods, which could avoid the solution staying at local minimum without reaching the global minimum, are also discussed. This book is more concerned with using the computer software package to solve the problems rather than solutions analysis.

There are some computing optimal control methods (see next section) which have been successfully applied in many fields. They involve subdividing the interval [0, T] into many (usually equal) subintervals. However, the accuracy will be lost if the switching times do not lie on the end-points of the equal subintervals. Hence it is essential to compute the optimal switching times.

7. Different Approaches to Optimal Control Problems

With the advances of modern computers and rapid development of software engineering, more and more people are concerned with computational algorithms, which could shorten computing time and provide more accurate results for complex optimal control problems that are difficult to solve analytically. During last thirty years, many efficient approaches have been developed and successfully applied in many models in a wide range of fields. Several numerical methods are available in the references ([24, 1981]; [56, 1975]; [57, 1986]; [58, 1986]; [76, 1981]; [75, 1980]; [84, 1991]; [85, 1991]), while some typical computing optimal control problems and efficient computational algorithms relevant to the present study will be discussed in the next few sections, the general computational approaches and algorithms for optimal control may be classified as the following (Islam [36, 2001]).

There is a wide range of algorithms which can be used for computing optimal control models in finance and can be classified under the algorithms for continuous and discrete optimal control models (Islam [36, 2001]). Algorithms for continuous optimal control models in finance include: (i) gradient search methods; (ii) algorithms based two value boundary problems; (iii) dynamic programming, approximate solution methods (steady-state solution, numerical methods based on approximation and perturbation, contraction mapping based algorithm and simulation); and (iv) control discretization approach based on step function, spline etc. Algorithms for discrete optimal control models in finance may be classified as follows: (i) algorithms based on linear and nonlinear programming solution methods; (ii) algorithms based on the difference equations of the Pontryagin maximum principle and solved as a two value-



boundary problem; (iii) gradient search method; (iv) approximation methods; and (v) dynamic programming.

For computing optimal control finance models, some recently developed computer packages such as SCOM, MATLAB, MATHEMATICA, DUAL, RIOTS, MISER and OCIM can be used.

7.1 OCIM

In reference [15, Craven 1998], a FORTRAN computer software package OCIM (Optimal Control in Melbourne) was discussed for solving a class of fixed-time optimal control problems. This computational method is based on the Augmented Lagrangian algorithm which was discussed in Section 6.3.2 in Craven [14, 1995]. Powell and Hestenes first made the Augmented Lagrangian for a problem with equality constraints. Rockafellar extended it to inequality constraints in paper [74, 1974]. OCIM can be run on a Macintosh as well as

UNIX.

The basic idea of this method is to divide the time interval [0, 7] into n subintervals. The control u{.) is approximated by a step-function as in MISER [33, 1987], with u(t) = щ (constant) on subinterval [jT/n, (j + l)T/n}. In MISER [33, 1987], Goh and Teo had obtained good numerical results using the apparently crude approximation of by a step-function, which is called control parameterization technique . In Craven [14, 1995], the theory in Section 7.6 and 7.8 proves that this occurs when the control system acts as a suitable low-pass filter; also the smoothing effect of integrating the state differential equation will often have this result. Increasing the number n of subdivisions will lead to a greater accuracy. Note in the calculation of the co-state equation for A(.), the interpolation of the values of state x(.) is required. It is done by linear interpolation of the value of x(.) at subdivision points The linear interpolation is also used for calculating the gra-

dient in this research.

A brief description of the computing method is discussed here. First, given the approximated control to solve the state equation, obtain the state values taken at the grid-points of the subintervals; second solve the co-state equation with respect to x{.)\ thirdly, calculate the augmented Lagrangian; and lastly, calculate the gradients of the cost function respect to This algorithm had been programmed in FORTRAN. Note that jumps in co-states are not yet implemented. Unconstrained minimization is done using the CONMIN package [70], which uses the BFGS quasi-Newton algorithm [28, 1980].

Using the augmented Lagrangian allows gradients to be calculated by a single adjoint differential equation, and instead of having to code a separated formula for each constraint. In this algorithm, a time interval [0, T] is divided into equal subintervals. A diversity of control problems can be solved by this method, but the equality constraint like presents some



problems with OCIM. Several famous models were successfully solved by this method. They are a damped oscillator (the approaches developed in this research also solved an example of an oscillator problem), the Dodgem problem [29, 1975], and the Fish problem [12, 1976]. In Cravens paper, there was a useful non-linear transformation of the time scale that can effectively give a good subdivision in big time ranges, without requiring a large number of subdivisions, and also avoid computing problems. The details of the transformation will be described in Section 2.3.

7.2 RIOTS 95

Relating to another computing optimal control software package SCOM, which will be introduced in Section 1.7.5, the RIOTS (Recursive Integration Optimal Trajectory Solver 95) package [77, 1997] also runs on MATLAB . It solves optimal control problems by solving (in various ways) the differential equations that lead to function values and gradients, and then applying a mini-mizer package. The computing scheme is similar to the one that was described by Craven [14, 1995] in Section 6.4.5. This package obtains good accuracy of the switching times by using a sufficiently small subdivision of the time interval.

7.3 A computational approach for the cost of changing control

A computational method based on the control parameterization technique [84, 1991] was introduced in reference [82, 1992] for solving a class of optimal control problems where the cost function is the sum of integral cost, terminal cost, and full variation of a control. The full variation of a control is used to measure changes on the control action.

This method involves three stages of approximation. In the first stage, the problem is approximated by a sequence of approximation problems based on the control parameterization technique. Every approximation problem involving full variation of control is an optimal parameter selection problem. They become non-smooth functions in the form ofi - norm. In the second stage, the non-smooth functions are smoothen by a smoothing technique [83, 1988]. Another smoothing technique [42, 1991] is used to approximate the continuous state inequality constraints by a sequence of conventional canonical constraints in stage three. All these approximation problems are standard optimal parameter selection problems which can be solved by the general optimal control software package MISER3 [40, 1991]. The method is supported by the convergence properties. A calculation of a fishery harvesting model (which was computed in reference [41, 1990] with penalty с = 0), was involved to illustrate the method, and the chosen value penalty term was explained in this



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