Stabilization and Optimal Control of Dynamical Systems
MATH 403

Spring 2012
This examination has 5 questions and 3 pages.
The University of British Columbia
Final ExaminationsDecember 2006
Mathematics 403
Stabilization and Optimal Control of Dynamical Systems (Professor Loewen)
Time: 2 1 hours
2
Open book examination.
Any resourc
Stabilization and Optimal Control of Dynamical Systems
MATH 403

Spring 2012
This examination has 5 questions on 2 pages.
The University of British Columbia
Final ExaminationsApril 2003
Mathematics 403
Stabilization and Optimal Control of Dynamical Systems (Professor Loewen)
Time: 2 1 hours
2
Open book examination.
Any resources u
Stabilization and Optimal Control of Dynamical Systems
MATH 403

Spring 2012
Math 403 Problem Set 2
Due in class on Monday 24 September 2012
1. Evaluate eAt in each case:
(i)
1 2
2 1;
1 1
1
A = 1
2
1 0
A = 2 1
3 2
(ii)
0
2 .
1
2. Find a 2 2 matrixvalued function C() with these properties:
26
C(t) +
10
0
10
C(t) =
0
26
0
, t R;
0
Stabilization and Optimal Control of Dynamical Systems
MATH 403

Spring 2012
Math 403 Problem Set 1
Due in class on Friday 14 September 2012
This weeks problems are meant to be solved by hand calculation. But there is no penalty for using a
computer to check your work after the fact, or even to suggest the correct answer before yo
Stabilization and Optimal Control of Dynamical Systems
MATH 403

Spring 2012
This examination has 6 questions on 2 pages.
The University of British Columbia
Final ExaminationsDecember 2004
Mathematics 403
Stabilization and Optimal Control of Dynamical Systems (Professor Loewen)
Time: 2 1 hours
2
Open book examination.
Any resource
Stabilization and Optimal Control of Dynamical Systems
MATH 403

Spring 2012
This examination has 5 questions on 5 pages.
The University of British Columbia
Final ExaminationsApril 2010
Mathematics 403
Stabilization and Optimal Control of Dynamical Systems (Professor Loewen)
Time: 2 1 hours
2
Open book examination.
Any resources u
Stabilization and Optimal Control of Dynamical Systems
MATH 403

Spring 2012
This examination has 5 questions on 5 pages.
The University of British Columbia
Final ExaminationsDecember 2012
Mathematics 403
Stabilization and Optimal Control of Dynamical Systems (Professor Loewen)
Time: 2 1 hours
2
Standard UBC Examination Rules appl
Stabilization and Optimal Control of Dynamical Systems
MATH 403

Spring 2012
Math 403 Problem Set 3
Due in class on Monday 01 October 2012
1. The famous broombalancing problem involves a cart of mass M running along a straight horizontal
track under the inuence of a force u. Friction proportional to the carts velocity opposes its
Stabilization and Optimal Control of Dynamical Systems
MATH 403

Spring 2012
Math 403 Problem Set 4
Due in class on Monday 15 October 2012
1. Given a real matrix A Rnn , let t x(t; ) denote the unique solution of
x(t) = Ax(t), t > 0;
x(0) = .
Sometimes it is impossible to measure all n components of the state vector directly, and
Stabilization and Optimal Control of Dynamical Systems
MATH 403

Spring 2012
Math 403 Problem Set 9
Due in class on Friday 30 November 2012
1. Consider these two curves in the (t, x)plane:
S1 =
,x : x R .
S0 = (t, x) : 0 t , x = sin(2t) ,
2
2
Among all piecewise smooth functions x: [a, b] R for which (a, x(a) S0 and (b, x(b) S1 ,
Stabilization and Optimal Control of Dynamical Systems
MATH 403

Spring 2012
UBC Math 403(101)
19 October 2012
Midterm Test
Write your answers in the booklet provided. Start each solution on a separate page.
Any number of notes in your own handwriting are allowed; anything else is forbidden.
SHOW ALL YOUR WORK!
[15] 1. Consider th
Stabilization and Optimal Control of Dynamical Systems
MATH 403

Spring 2012
Math 403 Problem Set 8
Due in class on Wednesday 21 November 2012
1. The problem of stopping a controlled harmonic oscillator in prescribed time T with minimum energy
can be expressed succinctly as follows:
T
def
2
1
2 u(t)
minimize J[u] =
dt,
0
over al
Stabilization and Optimal Control of Dynamical Systems
MATH 403

Spring 2012
Math 403 Problem Set 7
Due in class on Wednesday 14 November 2012
1. Solve the following problem with a = 0, then repeat with a = 1:
2
minimize
subject to
a
(2x 3u u2 ) dt
2
0
x = x + u,
x(0) = 5,
0 u 2.
J=
(Suggestion: When a = 1, sketch the graph of w H
Stabilization and Optimal Control of Dynamical Systems
MATH 403

Spring 2012
Math 403 Problem Set 5
Due in class on Monday 29 October 2012
1. Consider this control system, in which U = cfw_v R : 1 v 1:
x1 = x1 u
x2 = 2x2 2u,
u [1, 1].
Find the
attainable set A(T ; 0, U ) for each T > 0. Show that every such set is contained in th
Stabilization and Optimal Control of Dynamical Systems
MATH 403

Spring 2012
Math 403 Problem Set 6
Due in class on Wednesday 7 November 2012
1. Consider the dynamical system
y (t) = u(t),
0 t 2;
y(0) = 0, y(0) = 0.
Call a piecewise continuous function u: [0, 2] R a control if it obeys
u(t) 1,
t [0, 2].
(a) Find the control that