Astronauts in training are required to practice a dockingmaneuver under manual control. As part of this maneuver, they arerequired to bring an orbiting spacecraft to rest relative toanother orbiting craft. The hand controls provide for variableacceleration and deceleration, and there is a device on board thatmeasures the rate of closing between the two vehicles. Thefollowing strategy has been proposed for bringing the craft torest. First, look at the closing velocity. If it is zero, we aredone. Otherwise, remember the closing velocity and look at theacceleration control. Move the acceleration control so that it isopposite to the closing velocity, and proportional in magnitude.After this time, look at the closing velocity again and repeat theprocedure. Under what circumstances would this strategy beeffective?

In the example from the textbook, there were no assumptions madewith regards to any of the reaction times, but to make this a bitmore concrete, we are going to assume that the astronaut’s reactiontime is five seconds, that he or she waits 10 seconds before makingthe next observation of closing velocity, and that the constant ofproportionality between the closing velocity and manualacceleration is 0.02.

Reconsider the docking problem of Example 4.3, and now assumethat c=5 sec, w=10 sec, and k=0.02.

a. Assuming an initial closing velocity of 50 m/sec, calculatethe sequence of velocity observations v₀, v₁,v₂. . ., predicted by the model. Is the dockingprocedure successful?

b. An easier way to compute the solution in part (a) is to usethe iteration function G(x)=x+F(x), with the property thatx(n+1)=G(x(n)). Compute the iteration function for this problem,and use it to repeat the calculation in part (a).

c. Calculate the solution x(1), x(2), x(3), . . ., startingx(0)=(1,0). Repeat, starting at x(0)=(0,1). What happens as n→∞?What does this imply about the stability of the equilibrium (0,0)?[Hint: Every possible initial condition x(0)=(a,b) can be writtenas a linear combination of the vectors (1,0) and (0,1) and G(x) isa linear function of x].

d. Are there any states x for which G(x)=λx for some real λ? Ifso, what happens to the system if we start with this initialcondition?