# Integrating factor in optimal control theory

• Jun 4th 2010, 03:27 PM
AmberLamps
Integrating factor in optimal control theory
Hey guys,

This is an optimal control problem for an econ class but it is the maths behind it I am struggling with - so have left the context out. I have derived two differential equations in my state and costate variables:

costate:$\displaystyle \frac {dy} {dt} = Ae^{(r+\beta)t}$

state: $\displaystyle \frac {dk} {dt} = \beta k - (Ae^{(r+\beta)t})^2$

where $\displaystyle A$ is some arbitrary constant.

Now i need to solve $\displaystyle \frac {dk} {dt}$ for $\displaystyle k$

Using the integrating factor I get:

$\displaystyle K = Ce^{\beta t} - \frac {A^2e^{(4r+4\beta)t}} {4r+3\beta}$

Where $\displaystyle C$ is another arbitrary constant. Is this right?

I then have to apply the transversality condition:

$\displaystyle e^{-rt} k(t) y(t) = 0$

Which using my solutions for $\displaystyle k$ and $\displaystyle y$ is:

$\displaystyle ACe^{2\beta t} - \frac {A^3e^{(4r+5\beta)t}} {4r+3\beta} = 0$

I won't go any further with the problem here because I am convinced I have gone wrong somewhere since the equation isn't very pretty :( can anyone check my value for $\displaystyle k$ please?
• Jun 4th 2010, 04:23 PM
ANDS!
This is what I get:

$\displaystyle \frac{dk}{dt}=\beta k-A^{2}e^{(2r+2\beta)t} \Rightarrow \frac{dk}{dt}-\beta k= -A^{2}e^{(2r+2\beta)t}$

$\displaystyle \mu = e^{-\int \beta dt} \Rightarrow \mu = e^{-\beta t}$

$\displaystyle \left(K \cdot e^{-\beta t} \right)' = -A^{2}e^{(2r+2\beta)t}\cdot e^{-\beta t} \Rightarrow \left(K \cdot e^{-\beta t} \right)' = -A^{2}e^{(2r+\beta)t}$

$\displaystyle \int \left(K \cdot e^{-\beta t} \right)' = -A^{2}\int e^{(2r+\beta)t}$

$\displaystyle K \cdot e^{-\beta t} = -A^{2}\frac{e^{(2r+\beta)t}}{2r+\beta}+C$

$\displaystyle K = -A^{2}\frac{e^{(2r+2\beta)t}}{2r+\beta}+C_1e^{\beta t}$

When you integrate y(t), I think you are leaving off a constant:

$\displaystyle Y = \frac{Ae^{(r+\beta)t}}{r+\beta}+C_2$
• Jun 5th 2010, 02:39 PM
AmberLamps
thanks - have gone through it myself and got the same - my dy/dy should have just read y(t) but thanks anyway!