Title: Impulsive Stabilization for Systems of Linear Differential and Difference Equations
1Impulsive Stabilization for Systems of Linear
Differential and Difference Equations
- Author Xiaochen Xu
- Supervisor Dr. Elena Braverman
- Date May-September, 2005
2Outline
- 1- Stochastic matrices and their applications
2- Historical notes
3- Important results of research project
3Stochastic Matrices and Their Applications
- A matrix is called row stochastic if its
entries are nonnegative and the sum of entries in
each row equals 1.
A matrix is called column stochastic if its
entries are nonnegative and the sum of entries in
each column equals 1.
4Example
population in the north at time n.
population in the south at time n.
a prob. North people will stay at time n1.
1-a prob. North people will move at time n1.
b prob. south people will move at time n1.
1-b prob. south people will stay at time n1.
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6Matrix form
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8Important property of Stochastic matrix
- All eigenvalues of a stochastic matrix
- satisfy
-
- 1 is always the largest eigenvalue of a
stochastic matrix.
9The importance of eigenvalues
- If all , then for any
vector v.
If all , then for any vector
v.
10Some Historical Notes
Andrei A. Markov
11Pafnuty Lvovich Chebyshev
12Some Biographical Data for A.A. Markov
- At the age 30, Markov became a professor at
St. Petersburg university.
He published more than 120 scientific papers in
number theory, continuous fraction theory,
differential equations, probability theory and
statistics.
His classical textbook, Calculus of
Probabilities, was published four times in
Russian and was translated into German.
Many of his papers were devoted to creating a new
field of research, Markov chains.
13Historical Reference About Markov Chains
In 1930, Kolmogorov extended the mathematical
theory to chains with infinite number of states.
During 1935 to 1945, Doeblin and Doob made
important contributions.
Chung summarized the present state of the theory
of Markov chains.
1950, Markov chain was well recognized as a
useful model for physical processes, and
mathematical theory applications.
Andrei Nikolaevich Kolmogorov
14Historical Reference About Markov Chains
- 1953, L.S. Shapley studied one of the
earliest sequential model in a Markov chain with
alternative transition probabilities.
Zachrisson and Shor examined similar game
formulation more recently.
Blackwell, Derman, Howard and others have
investigated a more general class of Markovian
decision models.
Howard and Jewell extended the Markovian decision
model to semi-Markov processes.
15Some important results of the research project
- Impulsive Stabilization for Systems of Linear
Differential Equations
(1)
(2)
(3)
16Theorem 1
- Let A be a symmetric stochastic matrix, and
, we can find a Matrix B, such that for
any Lgt0 there exists Tgt0 - for any tgtT, the solution of
- (1)
-
- (2)
- satisfies .
17Corollary 1
- Suppose A has at least k negative eigenvalues,
then there exists a matrix B, such that B has at
least k eigenvalues, - .
18Purpose of This Project
- we want to avoid overreaction in this project.
- When A has some eigenvalues that are greater than
0, then we will use B matrix to reduce the length
of the vector in that directions. - When A has some negative eigenvalues, then we can
even allow some growth in that directions.
19Impulsive Stabilization For Systems of Linear
Difference Equations
(4)
(5)
20Theorem 2
- Suppose A is a symmetric stochastic matrix,
- is an eigenvector of A. Then for any
solution of - (4)
-
- we have either or
as .
21Theorem (Poincare, Pituk)
- If X is the solution of
- (4)
- then either X(n)0 for all large n or
- ,
- is one of the eigenvalues of matrix A.
22Theorem 3
- Suppose A is a symmetric stochastic matrix. We
can find matrix B, such that for any Lgt0 there
exists for any , the solution of -
(4) -
(5) -
- satisfies .
23Corollary 2
- If A has at least k eigenvalues, such that
, then we can find a matrix B associated with
at least k eigenvalues that are greater than 1.
24 The end