Internet Control Course

Exercises

1.1

Design a control with T > 0, which steers the cart (1.1) from to where are given and

1.2

Show that in order to drive the cart from to with either or one can use having only one point of switching, i.e., for all except one point of switching.

1.3

Show that in order to reach any point from a given point one can use either

or

Exercise 1.9 may be of some help for calculating of the values

1.4

Let be the set of oriented curves, each is contained in D , which are integral curves of the cart (1.1) for some Prove that if and only if for some parametrization we have

for all and is continuous for any and the functions

are continuous on [0,1] .

1.5

Show that a curve if and only if where N is a natural number ( N may depend on ) and there exist parametrizations such that for all

1.6

Let where is a ball of radius with center at x . Prove that if and joins then, for all there exists such that and joins and

1.7

Prove that the digraphs and are the same.

1.8

Prove that the cart is controllable on D by iff it is controllable on D by C .

1.9

Show that if and is its parametrization, then the control defining has the form

where is a solution of the equation

1.10

Prove that the transient time i.e., the time for the cart to move along the curve is expressed by an integral of the second kind,

1.11

Let and let the cart be controllable by both on the domain Q and on the domain D . Prove that the cart is controllable by on as well.