Optimal Control and Applied mathematics

1 Problem I

The trajectory of a vertical sounding rocket vehicle traveling over a

at earth

in the absence of an atmosphere could be computed from the following equa-tions

_

h=v (1a)

_ v= g+a(t) (1b)

wheregis the acceleration due to gravity and a(t) is the acceleration provided

by the rocket motor, which in this case represents the control input. The total

impulse of the motor is limited, so

I=

Z

T

0

a(t)dt= constant (2)

We would like to choosea(t) so that the rocket vehicle’s peak attitude is

maximized, that is

max J=h(T) (3)

A) Write the necessary conditions for optimality, assuming that there is

no constraint ona(t) and comment on the optimal solution.

B) Assume thatja(t)ja

max. Does this make a dierence? Explain.

1

2 Problem II (optional)

Suppose that the quality of a rental house is characterized by a single variable

x. The quality is governed by the equation

x(k+ 1) =x(k) +u(k)

u(k)

2

x x(k)

; (4)

where 0< <1,u(k) is the maintenance expenditure in periodk, and x >0

corresponds to `perfect’ conditions. The rent is proportional to the quality.

A landlord wishes to determine the maintanance policy that maximizes his

discounted net prot up to periodN, at which point he plans to sell the

house. In particular, he wishes to maximize

J=

N

cx(N) +

N 1 X

k=0

[px(k) u(k)]

k

; (5)

wherep >0, 0< <1. The quantitycx(N) is the sale price at time N.

A) Using the variational approach, write the necessary conditions and

nd an equation for(k) that is independent ofx(k) andu(k).

B) Show thatu(k) can be expressed in feedback form.

C) Find the optimal return function (and show that is linear inx).

D) What is a fair pricec? That is, what value of c would make the

landlord indierent to selling or retaining the house?

2

1 Problem I

The trajectory of a vertical sounding rocket vehicle traveling over a

at earth

in the absence of an atmosphere could be computed from the following equa-tions

_

h=v (1a)

_ v= g+a(t) (1b)

wheregis the acceleration due to gravity and a(t) is the acceleration provided

by the rocket motor, which in this case represents the control input. The total

impulse of the motor is limited, so

I=

Z

T

0

a(t)dt= constant (2)

We would like to choosea(t) so that the rocket vehicle’s peak attitude is

maximized, that is

max J=h(T) (3)

A) Write the necessary conditions for optimality, assuming that there is

no constraint ona(t) and comment on the optimal solution.

B) Assume thatja(t)ja

max. Does this make a dierence? Explain.

1

2 Problem II (optional)

Suppose that the quality of a rental house is characterized by a single variable

x. The quality is governed by the equation

x(k+ 1) =x(k) +u(k)

u(k)

2

x x(k)

; (4)

where 0< <1,u(k) is the maintenance expenditure in periodk, and x >0

corresponds to `perfect’ conditions. The rent is proportional to the quality.

A landlord wishes to determine the maintanance policy that maximizes his

discounted net prot up to periodN, at which point he plans to sell the

house. In particular, he wishes to maximize

J=

N

cx(N) +

N 1 X

k=0

[px(k) u(k)]

k

; (5)

wherep >0, 0< <1. The quantitycx(N) is the sale price at time N.

A) Using the variational approach, write the necessary conditions and

nd an equation for(k) that is independent ofx(k) andu(k).

B) Show thatu(k) can be expressed in feedback form.

C) Find the optimal return function (and show that is linear inx).

D) What is a fair pricec? That is, what value of c would make the

landlord indierent to selling or retaining the house?

2