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
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)
T
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 thes
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 th
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
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)
T
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
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.
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 meas
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 functi
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 els
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 fo
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
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
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 Loewe