ST.PETERSBURG, VIRTUAL AGE, PENDULUM AND CONTROL
Good afternoon, dear Colleagues!
I call you Colleagues, because you are now students of the
University and I am connected to the Universites for a long time:
more than 30 years. We are also Colleagues, because you entered
the Department related to Mechanical Engineering, and I work in the
Institute for Problems of Mechanical Engineering.
The title of my lecture is pretty long:
ST.PETERSBURG, VIRTUAL AGE, PENDULUM AND CONTROL
First of all I would like to tell you about St.Petersburg.
It is my native city, my favorite city, my beloved city, and I would like
to share with you my love. It is a very beautiful city and I would like you
to taste its beauty now.
Second, I would like to use this unique opportunity to talk
to such a large audience of young Japanese students. I am 50, and
you are less then 20. I would like to share with you some
important information which a person of my generation need to share with
people of your generation. That is why I would like to speak
about "Virtual Age" or "Information Age", as the next, XXI century
is sometimes called.
Third, since we are Colleagues, I would like to share with you
the most interesting and exciting information about our profession
and to tell you what I am interested in now. I would like to give
you an idea of new results in my field by simple example.
My field is Control Theory and currently I am interested in control
of oscillatory systems. I believe you agree that the pendulum is
perhaps the simplest oscillatory system of Mechanical Engineering
and you understand why the third part of my lecture is devoted to
pendulum and control.
Now let us start.
- Virtual Look
I arrived to Kumamoto from St.Petersburg. It is at the Far West
of Russia (far, if you are looking from Japan!)
And you will NOT find it
on the World Map, if you click here:
because World is too large. However you will find it
on the Map of Europe, if you click here:
It is even better seen on the map of Finland:
because it is
very close to the border of Finland - less than 200 kilometers or
half an hour of flight.
My flight took 20 hours, with two stops: in Copenhagen and in Tokyo.
Its last part (from Tokyo to Kumamoto) took only 1.5 hours.
Now I know that Kumamoto is near Tokyo!
St.Petersburg is second biggest city in Russia, after Moscow.
The population of St.Petersburg is
about 5 millions, while the population of Moscow is about
13 millions. Of course these are "small" cities compared with Tokyo
By the way, you may find more up-to-date information about these
and other big cities in the World on the WEB:
As you know, Moscow is Russian Capital city. But St. Petersburg
is former capital of Russia! It used to be the Russian capital for
200 years, almost from its foundation in 1703 till 1918 when the
capital was moved to Moscow because of the Wars (World War I and
Russian Civil War).
Thanks to being the capital in the past, St.Petersburg has many
attractions: beautiful buildings, churches and cathedrals, palaces
and museums. Perhaps it can be compared with Kyoto, which also is
former capital city.
St.Petersburg is almost 300 years old. It was founded in 1703 by
the Russian Emperor Peter the First (or Peter the Great) and called in
honor of St. Peter, guardian of the gates of Heaven.. Peter was
at war with Sweden and after a few victories his troops reached the
mouth of a big river (Neva) that falled into a gulf of the Baltic Sea.
And he began building the fortress on the small Zayachij island in the
mouth of Neva. You may see Zayachij island at
He has built the fortress and called it the Peter and Paul fortress:
and he invited the italian architect Domenico Tresini who built this
beautiful shrine - the Peter and Paul cathedral:
It reaches 122.5 meters height and there is an angel on the top of
Peter the Great himself is buried in this
cathedral and other Russian Tsars and Tsaritsas (Kings and Queens)
are also buried in this cathedral. In 1998 a very important event took
place here: the last Russian Emperor Nikolas II, who was killed
by communists in 1918 was finally buried in this cathedral too.
One of the first buildings in St.Petersburg was Admiralty -
the shipyard (verf) where different ships used to be built:
Another big and very beautiful building is St.Isaac's Cathedral
Its height is 101 meter. It was erected by french architect Monferrant.
You may also want to take its winter look:
Speaking about Tsar Peter, we should not miss the chance to take
a look at the monument dedicated to his memory:
It is a favorite attraction for many tourists. It was created by
French architect Falconet. It is beautiful even
in winter, under the snow, just look:
The horse of this monument is oppressing the snake which symbolizes the
victory of Russia over Sweden.
There are many beautiful churches and shrines in St.Petersburg. Among
the most beautiful ones is the Church of Resurrection:
It was erected just over the place where the Emperor Alexander II was
killed in 1881 by a terrorist. That is why the church is called:
Savoir on the Blood. Other magnificent churches are Cathedral of Smolny
and Kazan Cathedral:
You can enjoy the views of other St.Petersburg churches on the WEB:
Unfortunately, many beautiful churches were destroyed during the
communists rule in the 1930s. However - again thanks to the WEB - you
can take a look of some of them if you visit the site
"Lost St.Petersburg" with collection of views of nonexisting
For example, you will find there views of nice
or Savoir on the Water shrine
which was erected as a memorial to Russian seamen who died in the
Russian-Japanese war in 1904-1905. Both of them were destroyed
Now let us talk about museums.
There are more than 110 museums in St.Petersburg.
The list of them can be found at:
However the most famous of them and the largest one is The Hermitage.
The museum is located in 4 buildings. One of them is Winter Palace
- the former residence of Russian Emperors. You may enjoy its view from
or from the side of Palace Square:
By the way you can see here the column which is also very famous -
the Alexandrovsky column. It is amazing that the column is made from
single piece of the stone (granite). A very big stone for the monument was
found and mined in Finland and it was a big problem to transport it to
St.Petersburg. The column is 47.5 meters high, it weights 600 tons.
Look as big it is:
The Hermitage is the second largest museum in the world after the
Louvre in Paris.
Since it is located in the former Palace of the Emperors, you
can find reachly decorated interiors there.
There are some former gala and private Emperors' rooms on display:
The Hermitage houses huge collections of Western European Art,
Oriental Art, Applied Art, Ancient Roman, Greek and Egyptian Art.
However, the main fame of The Hermitage is due to its collections of
paintings and sculptures, especially those of Western European Art.
There are collections of Italian, Spanish, Flamish, Dutch,
French, English and German paintings dated from 13th to 20th centuries.
Just small celection already looks very impressive:
Madonna Litta by Leonardo da Vinci:
The Apostles Peter and Paul by El Greco:
One more Madonna - this one with flower, also by Leonardo da Vinci:
Danae by Titian:
Another Madonna with Joseph by Rafael:
"Venus and Cupid" by Lucas Cranach:
The famous "Return of the Prodigal Son" by Rembrandt:
The Hermitage has huge collection of Impressionists.
Vincent van Gogh:
Also Postimpressionists like Henri Matisse:
and Pablo Picasso:
You can know more about The Hermitage if you click at:
There are masterpieces of architecture not only in the center of the
city. As you can imadgine tsars needed palaces not only for winter
but also for summer. That is why we have a few brilliant architectural
ensembles in the suburbs. The most famous one is of course Petergof,
where the Summer Palace of Peter the Great was created:
Its style is similar to that of the Palace of Versailles,
the residence of the french Kings .
It is very rich and well decorated:
And the most famous attraction of the Petergof are its fountains
or, better here:
There are more than 200 fountains in the Petergof.
But the key figure among them is the famous Samson:
who is still fighting with the lion. It also symbolises the victory
of Peter the Great over Swedish King...
You can see even more fountains at
And you can find more fascinating views of St.Petersburg on the WEB,
for example here:
You can make a virtual tour of St.Petersburg here:
know more about its history and architecture here:
and even get a visitor's guide here:
You do not need to know Russian, just English.
However, you would be able to learn much more interesting if you would
So, why not to start virtual learning Russian language here:
Let us take a general view of St.Petersburg center:
In the left corner of this picture you can see the main building of the
St.Petersburg State University. It means that it is high time
now to speak about universities and students.
There are 43 state universities, 22 more military
universities and some number of private universities in St.Petersburg.
The most well known of them is St.Petersburg State Univerity
which is classical university with many different Faculties and
Departments. It was founded 275 years ago, in 1724 by special decision
of Tsar Peter the Great, together with Russian Academy of Sciences
and the special school: Academic Gimnasia. Therefore 1999 is
a kind of jubillee year and we have a lot of different special
events dedicated to this jubillee. Among them are special ceremony
of fountains opening in Petergof, and the chess match between
University alumni: grossmeister Viktor Korchnoy and World
ex-champion Boris Spassky. (By the way the match was won by Korchnoy
with the score 6:4).
You can know more about the St.Petersburg University at:
Among first professors of the St.Petersburg University were
great mathematicians: Leonhard Euler and Daniel Bernoulli.
By the way Euler worked for 30 years in St.Petersburg,
and his best papers were written there. Later great physicist and
chemist Mikhail Lomonosov was rektor of the University,
great chemist Dmitry Mendeleev, author of the periodic system
of chemical elements also was professor of our University.
In XIX-XX centuries the famous Russian mathematician P.L.Chebyshev
and his successors A.A.Markov, A.M.Lyapunov, V.A.Steklov created
tradition of St.Petersburg mathematical school.
And many famous Russian writers, poets, composers, musicians,
painters graduated from St.Petersburg University.
Among its alumni and professors are 7 Nobel prize winners,
namely, I.P.Pavlov (1904), I.I.Mechnikov (1908), N.N.Semenov (1956),
L.D.Landau (1962), A.M.Prokhorov (1964), V.V.Leontjev (1973),
Now there are 20 Faculties in the University with about
20000 students. Among them is the Faculty of Mathematics and
Mechanics which I proudly graduated from in 1971.
Now it is situated in the new campus of the University in Petergof,
not far from the famous Petergof Fountains. You may visit its
home page and take a look of its building:
A disadvantage of the Petergof campus is that it is rather far
from the center of the city: it takes 1.5 hours to get there by
Frankly speaking, the life of students in Russia is rather difficult
during recent years, especially after the financial crisis that happened
in Russia on August 17, 1998. The student's
scolarship is about US$10-US$15, while before August 17, 1998
it was about US$40-US$60. And professors' salary is about
US$40-US$50, while before August 17, 1998 it was some US$150-US$200.
Therefore both students and teachers need to find some extra jobs
in order to survive. And it is very difficult to maintain traditions
of Euler, Lomonosov and Mendeleev novadays. Nevertheless, we are
trying and we are happy that we have many talanted students.
I give you only one example. May be some of you know about the
International Collegiate Programming Contest. The teams of young
software programmers from different universities worldwide are
competing there in solving tasks from Computer Science.
This Contest is organized by ACM (Association of
Computing Machinery), sponsored by IBM Corporation and it is
very prestigeous. In 1998 the Contest took place in Atlanta, USA.
And in 1998 the team of young programmers from St.Petersburg
University was second in this prestigeous competition among
54 teams advanced to the Finals from 1250 teams.
You may take a look of the Final Standings of 1998:
In 1999 the Contest took place on April 11 in Eindhoven, The Netherlands.
This year the competition was still more strong:
62 teams from 1457 were selected for participation in the Finals.
This year our team was not so lucky: it was only 9th.
However, another team from St.Petersburg was the 3rd!
It was the team of St.Petersburg Institute of Fine Mechanics
So now two student teams from St.Petersburg are among top ten
in this Contest.
You may take a look of the Final Standings of 1999:
You may also look at the team of St.Petersburg University:
or at the Champions of Asia - the team of National Taiwan University:
You may know more about the ACM Contest at:
Why not to download the tasks set and try to solve them!
And may be next year the team of the Kumamoto University will
be the Champions of Asia!
After discussing such real things as St.Petersburg and St.Petersburg
University let us pass to the second part of the lecture
and talk about some virtual things.
- Paradoxes of the Virtual World
During the first part of my lecture you could learn some information
about St.Petersburg and university life and also could enjoy some
interesting and beautiful pictures. An important and impressive thing
for me was that all this information, pictures and much more about
St.Petersburg one can find in the Internet (terms "on the WEB"
or "in the Net" are also of use).
It means that there exists not only real city St.Petersburg, but also
a kind of "virtual St.Petersburg" which is, perhaps, not so large and
as the real one but is very interesting nevertheless. Same is valid for
other cities of the World. Therefore we can say that there exists a kind of
"VIRTUAL WORLD". It was created very recently, in fact in the late 90s.
It was generated by the real world, but now it has already its own independent
life. Not only the cities with their business, travel, shopping
information are objects
of this Virtual World. There exist virtual literature, virtual music,
virtual video, virtual games, entertainments, so on.
Just imagine: the Virtual World did not exist some 10 years ago. Some 5 years
ago it was in its very beginning. And now many people of the Real World
spend a few hours per day in the Virtual World!
So what can we expect for new century?
Let me tell you about one of my observations here in Kumamoto.
I saw big aquariums with fish in some public places like banks and
department stores in Kumamoto. And I was very surprised when learning
that these fishes were virtual ones! They were very nice small things:
they were floating and looking at me, but they were virtual. They were
floating in the virtual water, eating virtual sea plants and even making
And I began thinking that may be soon the Virtual World will become
as rich, interesting and important as the Real one...
That is why I decided to share with you my thoughts about some features
of this Virtual World in order to better understand its virtual life.
Important thing is that the Virtual World deals with neither substance
or matter, nor with energy. It deals with information and the laws in it
are quite different from the laws of the physical world. I think that
some of these laws are stil to be discovered.
In XXI century people will make new efforts to understand the laws of the
Virtual World. May be some of you will discover these laws in future.
And now I would like to draw you attention to some paradoxes which may
give rise to new discoveries.
EXAMPLE 1. Suppose I give some money to one of you. In this case I will
have less money, won't I? Similarly, if the moving ball strikes another
ball and transfers its energy to another ball it will have less energy.
In the contrary, if I have some information and give it to one of you
or, even to all of you I will not have less information! Instead,
you will have more information and the total amount of information
will increase. Moreover, the value of my information for myself will also
grow, because more people know about my information. And if you pass
this information to other people, saying "It is information from
Mr. Fradkov from St.Petersburg, Russia", the value of my information
will grow further and I will be pleased still more! It is surprising, but
it is true. So, please, pass the information from my lecture to
everybody and do not forget to mention that "It is information from
Professor Fradkov from St.Petersburg, Russia"
Sometimes people do not want information to grow and to be distributed
freely. They introduce restrictions of access, passwords and so on,
because they want to get money for their information. But information has
its own laws: it may go through even very small holes. You can find a lot
of free information on the WEB: free books, free software, free news, etc.
You can hardly imagine how much can be found!
EXAMPLE 2. There are no distances on the WEB.
Internet has some geography,
but it is topic-oriented rather than space-oriented. During my staying in
Kumamoto I discussed many things with my friends and colleagues. One of them
is in USA now, another one is in Holland, the third one is in Norway, and
of course many of them are in St.Petersburg. Looks like far from each other.
However I never noticed this spatial difference just because we had same
language and culture and the things we
were discussing were important for all of us. We were close in the
cyberspace because we understood each other very well.
Summarizing: Cyberspace has no distances.
The difference between points is
determined by language and culture rather than by geographical location.
EXAMPLE 3. Information and ideas in the Virtual World can be created and
freely distributed as texts, images and sounds independently of
the Real World.
The things (texts, images and sounds) which do not exist or are not allowed
in the Real World are possible in the Virtual World. For example, look at
How many teeth does it have?
Another puzzle: How many legs does this elephant have?
Many images which are impossible in the Real World were created by the
famous Dutch graphic artist Maurits Cornelis Escher. For example,
look at one of his so called "impossible buildings"
(lithograph "Ascending and Descending", 1960):
Or, look at this lithograph ("Drawing hands", 1948):
You can learn more about M.C.Escher and enjoy other his images and
other optical illusions at:
M.C.Escher was born in 1898 and died in 1970. It is amazing that he
created his images before the Computer Age and long before creation
of the Virtual World. In 1998 the 100th anniversary of his birth
was celebrated and the big exhibition of his works was organized in
Rotterdam, in his native country Holland. I was lucky to visit this
exhibition. You can learn more about this celebration at:
Perhaps, Maurits Escher was the first WEB-designer in the history.
I believe, profession of WEB-designer will be very popular in the XXI
- Virtual Watches Real
Although the Virtual World looks exactly like a free world, it is not the case.
In fact it is driven by the Real World. It means that the Virtual World
is watching and copying the images of the Real World.
An interesting thing is that live copying is possible. Perhaps you know
that there are live cameras in many places of the Real World that are
connected to Internet. Images are renewed every few minutes or even
seconds. You can check what is going on in New York, Paris or Tokyo
RIGHT NOW. There are live cameras in St.Petersburg too. One of them
watches St.Isaac Cathedral which you are already familiar with:
The view is beautiful and it is listed among Top Ten live cameras
of the World. You can look at the other of Top Ten and also to browse
other live cameras at:
By the way, there is Japanese site in Moscow with live view of Kremlin:
One nice thing about live cameras is that you can make pictures
of your friends or relatives. For example from Kumamoto I have made the
picture of my wife Olga in St.Petersburg with the help of another live
camera located near beautiful small river Mojka:
It was not so difficult. I just asked my wife to come near this
camera and copied the image when I saw her in the picture.
Then I sent it to my wife by E-mail. She was very happy to
receive her picture "Made in Japan".
There are also other projects related to watching the Real World.
I would like to tell you a few words about space flights.
Perhaps you know about the project of International Space Station (ISS),
carried out by 16 nations, including Russia, USA and Japan.
This project evolved from program of International flights on the
Russian space station "Mir". You can see the photo of real
Station "Mir" works already for 13 years in the space, much more than was
initially planned. It is an absolute World record. Soon it will go down
and finish its life. And the ISS station is now under development. Its
first two parts were launched in December, 1998 by Russians and Americans
and connected in the space as follows:
You can see the Russian made module "Zarya" and US made module "Unity".
The next module is to be launched and connected in May, 1999. The first
mission will start in 2000, while the whole station will be ready
in 2004. So how is this story related to the
Virtual World? Let me explain. You can watch current location of
already existing part of ISS if you visit the WEB-site with
The picture is changing every 5 seconds. And when you find out that ISS is
near Kumamoto NOW, why not to go out and to check it in the sky?!
Similarly, you can watch current location of Russian station "Mir" at:
And you can learn more about space flights, if you visit
WEB sites of US space agency NASA:
Russian Space Agency:
or National Space Development Agency of Japan:
You can also visit the page of Russian Space Agency at NASA
but information is different there.
Among different historical photo you will find there
Russian shuttle "Buran" flying on the huge Russian-Ukrainian
Or, you will see "Buran" prototype staying in the Park in the
center of Moscow and open for visitors: everybody can
touch the spacecfaft, enter it and walk in it:
And this is photo of Mir-Shuttle joint station (from NASA):
More pictures and stories are available from
Russian Aerospace Guide:
I would like to say that it is a very good example of international
cooperation. You know that US and Soviet Union were competing in
the space flights for many years. E.g. SU launched the first
artifucial satellite ("sputnik" in Russian) in 1957, then SU first
launched a man (his name was Yuri Gagarin) into the space in 1961.
US first performed the manned moon landing (Apollo-11)in 1969.
And now you can see US and Russian modules together in the sky.
Isn't it better than bombing each other?!
Also it is very good that Japan participates in the project. May be
some of you will take part in this project in future and watch the Real
World from the sky!.
- What is Control?
During the previous part of my lecture we were discussing about
the Virtual World and how it can watch the Real World. But the
Virtual World can not only watch, it can influence the Real
World and even control it. Therefore let us talk about Control.
Such a back influence is called "feedback" It is studied by
Control Theory and in a more general context, by Cybernetics
(this term was introduced in 1948 by American mathematician Norbert
Wiener for the science studying communication and control in the animal
and machine, but now its meaning is even more wide).
The Control Theory is very much related to your future field:
Mechanical Engineering and Material Science. And it is also
my field where I am doing research for about 30 years.
During recent years I am interested in Control of Oscillations
and Chaos. Now I would like to take a simple example of real
mechanical oscillatory system and demonstrate how you can change
its physical properties by means of Cybernetics.
I believe you agree that pendulum is the simplest mechanical oscillatory
system. Let me create an experimental pendulum device right now.
I take simple wooden ruler which I brought from Russia, having a small hole
near its end.
Then I take a metal pin borrowed in the University of Kumamoto and put it
into the hole -
it will be the rotation axis of our pendulum. The device is ready. Isn't it
a real "Russian-Japanese made" device!
Now I should explain what is
control system. In general control system consists of 4 parts:
controlled plant (in our case it is the pendulum itself);
actuator (in our case it is just my hand), sensor (in our case it is my
eye which can measure the angle of the pendulum and, possibly, its
angular velocity) and, finally, controller acting according to some rule
or algorithm (in our case the controller is is in
my mind, while its rule or algorithm, I suppose, belongs to the Virtual
World). Thus, controlling the pendulum requires some influence from
the Virtual World.
Look at the two possible schemes of the control system. The simplest case is
when controller does not use the measurements (so called open loop systems,
see Fig.1) while more advanced controllers use online measurements and
that is why they may achieve better quality (such systems are called
"feedback" or "closed loop" control systems, see Fig.2).
In our case moving the pin horisontally back
and forth will be input or control action. Angle of the pendulum will
be output variable. Now let us perform some experiments.
- Feedback Resonance
First let the pendulum be affected by small periodic control
action without any feedback. Then it will perform some periodic
oscillatory motions - swings - with the same frequency. If the
input frequency is changing, the amplitude of the swings will also be
changing and reach its maximum at the so called resonance frequency
that is just natural frequency of small oscillations of the pendulum.
This phenomenon is called resonance and it is well known in physics
since Galileo Galilei first mentioned it in 1638. Resonance has many
applications in science and engineering, e.g. it provides base for
spectroscopic investigations, telecommunications, etc.
Then let us increase input amplitude. The motion of the pendulum
is becoming more complicated and even nonperiodic. This phenomenon
is caused by nonlinear dynamics of the large oscillations of the
pendulum. It is called nonlinear resonance. It is well
studied by physicists during last two decades.
And now we come to the culmination, key point of the lecture.
Is it possible to swing the pendulum more than the physical
intuition predicts? Look: I am moving the pin slightly, but
following the swings of the angle, And I can make the swings
larger! Why it is possible? Because I was using the feedback!
Our control system is the closed loop system. It can provide
a physical system with new properties, and one of them we were
observing right now. This property can be called "FEEDBACK RESONANCE".
The concept of "Feedback Resonance" is a new one. The paper describing
it was published in the scientific journal "Physica D" in April,1999
(see references below).
So, you have got acquainted with very recent scientific result.
I believe, it will have applications in many fields, including
Mechanical Engineering and Material Science.
The question arises naturally: how to design the algorithm,
which provides the system with new properties? To answer it we need
to jump into the Virtual World and to write down the mathematical
model of the pendulum. In fact it is just the Newton's law applied to
the mechanical system:
m y''+ ry'+mg sin(y)=(cos y)u(t),
where u(t) is input acceleration, y=y(t) is angle of the pendulum,
y' is angular velocity, y'' is angular acceleration, m is mass,
r is friction coefficient. After applying the theory that was
developed in our Lab (see references) different control algorithms
can be obtain. For example, the following one:
u(t)= - K[E(t)-Ed](cos y)y',
where E(t) is the current energy of the pendulum, Ed is the desired
value of energy, K is proportionality coefficient called "feedback gain".
Our algorithms confirm existence of the Feedback Resonance
and allow to reach the desired energy level from almost all initial
If you get interested and want to play with the controlled pendulum
yourself - why not to play with the Virtual Controlled
Pendulum which you can find on the WEB
here (requires Internet Explorer)
The pendulum system is very simple and nicely demonstrates the feedback
resonance phenomenon. If you get more interested and would like to know
how to solve more complicated problems you may find the theory and examples
in our recent books and papers:
1. Fradkov A.L., Pogromsky A.Yu.,
``Introduction to control of oscillations and chaos",
World Scientific Pub. Co, Singapore, (1998).
2. Fradkov A.L., Miroshnik I.V., Nikiforov V.O.
``Nonlinear and adaptive control of complex systems''
Kluwer , Dordrecht, (1999).
3.Fradkov A.L., ``Exploring nonlinearity by feedback'',
Physica D, vol. 128/2-4 pp. 159-168, (1999).
4.Fradkov A.L., ``Investigation of physical systems by means of feedback",
Automation and Remote Control, vol. 60(3), (1999).
5.Fradkov A.L. ``Feedback resonance of nonlinear oscillators''
Proc. 5th European Control Conference, Karlsruhe, Aug.31-Sept.3, (1999).
6.Andrievsky B.R., Fradkov A.L. ``Feedback resonance in single and
coupled 1-DOF oscillators'', International J. of Bifurcations and Chaos,
vol 9(10),(1999) (to appear).
7.Fradkov A.L., ``Speed-gradient scheme in adaptive control'',
Automation and Remote Control, vol. 40(9), 1333--1342, (1979).
8.Fradkov A.L., ``Swinging control of nonlinear oscillations'',
International J. of Control, vol. 64(6), 1189--1202, (1996).
Also I invite you to visit the home page of our Lab:
where you can find more information and publications, including the
ongoing conferences on control and nonlinear dynamics.
you may get interested in the 2nd International Conference
"Control of Oscillations and Chaos" (COC'2000), see its WEB-site
It will take place in St.Petersburg, on July 5-7, 2000.
This time of the year is especially pleasant: it is so called "White Nights"
period. During this period the dark time of the night is very short and
light. It makes the sights of St.Petersburg still more fascinating
and attracts still more tourists from all over the World during this season.
Why not to take chance to visit our beautiful city during the
"White Nights" and to visit this Conference?!
I would like to thank Japan Society for Promotion of Science (JSPS) for
support of my staying in Japan which made this lecture possible.
I am also grateful to Prof.Z.Iwai for his kindest invitation and for
arrangements which made my visit to Kumamoto not only possible but
also very pleasant. I would like to thank other professors, students
and staff of the Department of Mechanical Engineering and Material
Science for their kind assistance during my staying in Kumamoto.
Finally I am grateful to member of our Lab Dmitry Tomchin for