the control, state, and the objective function are changed corresponding to the new time subdivision. The computational methods for these kinds of optimal control problems are constructed in Section 3.4. In this chapter, the control is also approximated by a step-function. Additional analysis of the control with different patterns leading to different minimums is discussed in Section 3.5. In Section 3.6, two sets of graphical results with different control patterns are presented.

In economics problems, the dynamic behavior may require a second order differential equation and more than one control. In Blatt [2, 1976], some generalizations of multi-state functions and controls were discussed. But Blatt did not give any explicit proof and explanation of these generalizations. The computational algorithms in this chapter effectively solve the second order differential equation. This technique is also applied to an optimal corporate financing model later in Chapter 4.

2. Controlling a Damped Oscillator in a Financial Model

We consider an aggregate dynamic financial system described by a second-order differential equation as follows:

This financial system second order differential equation can be transformed into an equivalent pair of first-order differential equations:

We now consider an oscillator problem with the bang-bang optimal control solution. A damped oscillator financial model for the above aggregate financial system with forcing function u(.), and parameter T, is presented as follows (see another version of this financial model in Section 5.3.1): /indexforcing function

x1(t) = Tx2{t) x2 = T{-x1(t) + u(t)-Tx2(t))

subject to:

where x\(t) = stock prices, u(t) = the proportion of investment in stocks compared to other form of financial investment.

Here ф{.) is a given target function showing the desired path to be achieved by the financial system. Suppose that u(.) is restricted by:

Since the dynamics are linear in u(.), if an optimum is reached, then bang-bang control may be expected, with possibly a singular arc for a time interval. However, singular arc control is not considered here.

The model (3.2) to (3.7) represents a financial decision making problem where the cost of changing control is added to the objective function, shown as follows:

Here, K is the cost of switching control, n is the number of control jumps. The modified model calculates the optimal allocation for stocks with an explicit consideration of changing allocation shares. The cost function becomes the new objective function to be computed.

The model (3.2) to (3.7) is similar to the model (2.11) to (2.15) and represent a financial decision making problem of choosing the optimal allocation of finance in an economy for stocks. The forcing function is an additional feature of this model compared to the model in (2.11) to (2.15). To make the model realistic, it is necessary to incorporate the cost of changing control in this model.

3. Oscillator Transformation of the Financial Model

In this section, a good transformation of time scale is introduced. The computer package nqq is also used for solving the differential equations in this chapter. However, the nqq package does a limited job, it only does one Runge-Kutta step for each subinterval given to it. So when smaller subinter-vals are needed for accuracy, they must be supplied by the small subintervals

construction. Although nqq package has this limitation, it has a great benefit for solving the differential equation with a non-smooth right-hand side of the differential equation which means there can be jumps at the end-points of intervals. Since a time-optimal control problem is not considered in this research, T is a constant. To make it easy, the time period of the optimal control problem described in last section is simplified to be [0,1] (the transformation of time scales when T is variable was given in Section 2.3).

First, the time horizon [0, 1] is divided into nb big subintervals , with end-points:

0 = pto < pt\ < ... < ptj < ptj+i < ... < ptnbi < ptnb = 1

Since the functions change quite rapidly within each big subinterval, further subdivisions into small intervals are needed to get sufficient precision in solving the differential equations. Then each subinterval \ptj,ptj+i] (1 < j < nb) is subdivided into ns small subintervals with end-points:

ptj,ptj+±,ptj+j ,... ,pi,-+nsi,pij+i

J ns J ns J ns

The whole subdivision of the time interval [0, 1] is shown as follows:

0 - to < t\ < £2 < < U < U+\ < < t(nb*ns)-\ < tnbms = 1

The relationship between and can be represented as follows:

j+ns

Vj e [0, nb] 2 -U= Ptj+l - ptj

The control only jumps from one big interval to the next one. It takes values as follows:

u(t) = Uo, 0,---, o, l, i,...,Ui,U2,U2,...,M2,tt3,---,

ипъ-2,ипъ-1,ипъ-\,..., unb-i (3.8)

A scaled time is constructed for the computer package, corresponding to the total number of subintervals nn = nb* ns, which takes values: