The Maximum Principle of Pontryagin
in control and in optimal control
Andrew D. Lewis1
16/05/2006
Last updated: 23/05/2006
1
Professor, Department of Mathematics and Statistics, Queens University,
Kingston, ON K7L 3N6, Canada
Email: [email protected]
Set Identities.
Identity
Name
AU =A
A=A
Identity laws
AU =U
A=
Domination laws
AA=A
AA=A
Idempotent laws
A=A
AB =BA
AB =BA
Complementation law
Commutative laws
A (B C ) = (A B ) C
A (B C ) = (A B ) C
Associative laws
A (B C ) = (A B ) (A C )
A (B C ) = (A
Proofs
A theorem is a mathematical statement that can be shown to be
true.
An axiom or postulate is an assumption accepted without proof.
A proof is a sequence of statements forming an argument that shows
that a theorem is true. The premises of the arg
Examples of proofs (by type)
(Theorems to be proved in class, and required denitions)
Direct proof
Denition 1 An integer n is called odd if n = 2k + 1 for some integer k , and is called
even if n = 2m for some integer m.
Theorem 2 If n is an odd integer,
Examples of fallacies
Fallacy of arming the conclusion:
Example 1.
H1 : If I eat spicy food, I have bad dreams.
H2 : I had a bad dream.
C : I ate spicy food.
Wrong. I watched a horror movie.
Example 2.
H1 : If there is a trac jam, I will be late for scho
Arguments
An argument is a set of propositions in which one (called the conclusion) is
claimed to follow from the others (called the hypotheses or premises). In other
words, an argument is a proposition of the form
(P1 P2 . . . Pk ) C,
where P1 , P2 , .
Truth Trees
Like truth tables, truth trees are used to determine all truth assignments of the
propositional variables (atomic propositions) that make a compound proposition
true. Unlike a truth table, whose size grows exponentially with the number of
pro
Disjunctive Normal Form
An atomic proposition is a proposition containing no logical connectives.
A literal is either an atomic proposition or a negation of an atomic proposition.
A conjunctive clause is a proposition that contains only literals and (p
Examples of DeMorgans Law
Negate the following statements:
A: It is cold and snowy.
p: It is cold.
q : It is snowy.
A p q.
A p q
A: It is not cold or not snowy.
B: I will go to Mexico or visit family.
p: I will go to Mexico.
q : I will visit family
MAT 1348 A - Winter 2013
Examples
January 10, 2013
Add. Ex. 6a) Rewrite the following argument using the propositional variables:
If the dog is barking, then there is someone at the door only if it is snowing.
The dog is playing outside unless there is
A Method of Solving Knights-And-Knaves Questions
There is an island far o in the Pacic, called the Island of Knights and Knaves. On
this island, there are people called knights, who always tell the truth, and people called
knaves, who always lie. The two
Welcome to MAT 1348 A
Discrete Mathematics for Computing
MAT 1348A (uOttawa)
Intro
January 7, 2013.
1/7
Welcome to MAT 1348 A
Discrete Mathematics for Computing
. study of discrete (= not connected, countable (nite)
mathematical objects
MAT 1348A (uOttawa
Logical connectives (operators)
Name
Compound
proposition
Negation
p
Conjunction
pq
Disjunction
pq
Exclusive or
pq
Biconditional
pq
Corresponding
true if and only if
both p and q are true
false if and only if
both p and q are false
true if and only if
exa
MAT 1348 Second Homework Assignment
Due Wednesday, Feb. 6 at 10:00am in DGD
Instructions:
Write MAT 1348 (A or B) and DGD session (1 or 2) in the top left corner of the rst
page. Also write your TAs name.
Put your rst and last name and your student numb
MAT 1348 First Homework Assignment
Due Wednesday, Jan. 23 at 10:00am in DGD
Instructions:
Write MAT 1348 (A or B) and DGD session (1 or 2) in the top left corner of the rst
page. Also write your TAs name.
Put your rst and last name and your student numb
SAMPLE TEST QUESTIONS, PART 1
FOR MAT 1348
Instructions- These questions have been taken from my old exams from the course MAT
1361, which no longer exists but was similar to MAT 1348. They are an EXCELLENT
indicator of the sort of questions that will app
Discrete Mathematics for Computing MAT134A
Midterm Examination Front Page
14 February 2013
Instructor: Nevena Franceti
c
Instructions:
This is an 80-minute closed-book exam; no notes are allowed. Calculators (without graphing or
programming function) are