In Memory of Andrei I. Subbotin
Academician of Russian Academy of Sciences Andrei Ismailovich Subbotin died
on October 14, 1997.
A.I.Subbotin was an outstanding researcher utterly committed to
science, a personality of high morale, a devoted son of his mother country.
His
attitude to life was based on his belief in good. Andrei Ismailovich
Subbotin was characterized by tactful and attentive attitude to others. He
used to speak very modestly of his own scientific achievements. However,
his
ideas influenced the others significantly. It was especially visible at the
seminars that he conducted. During the last months of his life, Andrei
Ismailovich Subbotin worked actively to complete the research he started.
A.I.Subbotin was born on February 16, 1945 in a family of a military
officer. In 1962 he entered the faculty of mathematics and mechanics of
Ural
State University, where he began his research activity. At that time this
was
already an active national centre of research in stability theory, which
gradually shifted its interests towards an emerging topic, the mathematical
theory of control processes. The mentioned activity materialized in the
creation by one of the authors of this text (N.N.Krasovskii) of a new chair
of applied mathematics, where he invited a group of fairly young
enthusiastic
graduate local researchers. Stimulated by numerous applications and by the
knowledge of the activities of L.S.Pontryagin's group, by the information
from abroad (Bellman's 'Dynamic Programming'), the group was eager to
develop their own original vision of the upcoming problems.
Being a talented student, A.I.Subbotin attracted attention of
\`E.G.Al'brekht, who worked at the chair of applied mathematics.
He gladly accepted
the invitation to join this chair. He has found himself in
creative scientific atmosphere and was coming to know the research results
of the young but already well known scientific group. The gifted youth
joined this group naturally, and started to develop his talent quickly.
A.I.Subbotin quickly understood the sense of the problems being
investigated, started to attend special courses and seminars. He was
granted
an individual curriculum.
The main interests of the chair at that time were already focused on
new problems of conflict control and control under uncertainty, which are
formalized as differential games. In this insufficiently investigated
field,
A.I.Subbotin demonstrated his ability not only to grasp the essence of the
problems quickly, but also to find his own new ways of solving them.
In 1969 a part of staff of the chair including A.I.Subbotin was
transferred to Sverdlovsk branch of V.A.Steklov Mathematical Institute. The
same year A.I.Subbotin defended his PhD dissertation 'Problems of Encounter
and Evasion in Differential Games' and soon, in 1973, he defended his DSc
dissertation and received the Gold Medal of Academy of Sciences of the USSR
for Young Scientists.
In mid-1960s in investigations of Ural specialists in mathematics
and mechanics, the rule of extremal shift was formulated for a wide class
of
feedback game control problems. It gives a connection between problems of
program control and problems of synthesis in the minimax setting. To reduce
the latter ones to program constructions directly is not always correct.
Thus the finding of conditions that give the basis for this reduction
(regularity conditions) became one of the central problems. A.I.Subbotin
made an essential contribution to the solution of these questions.
A.I.Subbotin and N.N.Subbotina also established the following fundamental
fact. Implementing continuous positional strategies, the players use the
available information as a rule not in the best possible way, and the
corresponding result can be improved in the class of irregular strategies.
More than that, they showed later that the classical definition of motions
based on contingency is either not enough for a formalization that provides
the best possible result.
An essential element of the second (connected with the first) way of
the constructing of positional strategies is an extremal shift towards the
stable bridge, in whose vicinity the trajectory is to be kept. This way of
controlling provides the best guaranteed result and is not necessarily
described by a continuous strategy. This principal fact leads to the
theorem
of alternative. Thus, the existence is established of an equilibrium
solution to a differential game in the corresponding class of positional
strategies. This theorem asserts the following. For a pursuit-evasion
differential game for any choice of the initial position either the problem
of pursuit or the problem of evasion is solvable. The direction of
investigations that incorporates the extremal aiming and extremal shift
became known as the extremal approach.
A.I.Subbotin was developing the extremal approach to nonlinear game
dynamics control problems. This approach on the one hand gives the general
structure of solutions, and on the other hand provides computer realization
of optimal strategies. The basic results of the first initial period of
these investigations are given in the monograph by N.N.Krasovskii and
A.I.Subbotin 'Positional Differential Games'.
Later A.I.Subbotin came to a natural idea to replace the
non-infinitesimal extremal shift by infinitesimal constructions for the
value function of differential game. By that time A.I.Subbotin was ready to
employ in his research the new advances of nonsmooth analysis. From
mid-1970s he was developing a new approach to defining a generalized
solution of equations of Hamilton-Jacobi type, which play an important part
in theory of differential games. (In this theory the name Isaacs-Bellman
equation has taken root.) A.I.Subbotin together with N.N.Subbotina have
investigated the class of game dynamics problems whose value is a piecewise
smooth function. For these problems, a representation of solution in the
form of a pair of differential inequalities was obtained. These
inequalities
give an infinitesimal form of the optimality principle, which consists of
two conditions, namely, of the properties of {\it u}-stability and {\it v}%
-stability of the value function of differential game. When studying
differential games with nonsmooth value function, A.I.Subbotin realized
from
the beginning the necessity to employ the new constructions of nonsmooth
and
convex analysis and in his turn contributed to the development of these
constructions. These constructions were later used by him for investigating
differential games and first-order partial differential equations.
Let us mention in this connection that in 1950-70s the problems that
involve nonsmooth solutions to first-order partial differential equations
were considered by many mathematicians.
In 1980s A.I.Subbotin extended his original approach concerned with
defining a generalized solution to a significantly more general form of
first-order partial differential equations of Hamilton-Jacobi type. He
introduced a fundamental notion of minimax solution, gave apparently
different but equivalent in essence forms of this definition. He also
established a highly nontrivial fact of equivalence of the minimax solution
and the viscosity solution introduced by M.G.Crandall, P.-L.Lions and
L.C.Evans.
The results of investigations of A.I.Subbotin in the theory of
minimax solutions are summarized in his monographs 'Minimax Inequalities
and
Hamilton-Jacobi Equations' (1991) and 'Generalized Solutions of First Order
PDEs. The Dynamical Optimization Perspective' (1995).
A.I.Subbotin was a broad-minded scientist and had a deep
understanding of the logic of development of science. Overcoming many years
of his grave illness, he was engaged in intensive scientific investigation,
taught students, supervised the work of young researchers. A.I.Subbotin
worked in close contact with scientists from many countries. Some leading
scientists gave talks at his seminar.
The results of intensive research work of A.I.Subbotin are
summarized in more than 100 articles and 4 books authored by him and
devoted
to the theory of differential games, Hamilton-Jacobi equations, optimal
control. His scientific achievements are highly evaluated by colleagues in
different countries, he has won everybody's respect. He was an invited
speaker at the International Congress of Mathematicians (Montreal, 1974),
Lenin prise laureate (1976), decorated with Order of the Red Banner of
Labour (1976), presidium member of the section 'Control theory' at the
International Congress of Mathematicians (Warsaw, 1981), corresponding
member of Russian Academy of Sciences (1991), academician of Russian
Academy
of Sciences (1997).
His wife, Nina Nikolaevna Subbotina (n\'ee Barabanova), is a
mathematician, specialist in control theory. His son, Ismail Andreevich
Subbotin, is a designer. A.I.Subbotin has two grandchildren.
Andrei Ismailovich Subbotin, a kind, tactful person, an outstanding
scientist, will be remembered by his colleagues, pupils, and all those who
knew him.
`E.G.Al'brekht,
A.G.Chentsov,
A.F.Kleimenov,
N.N.Krasovskii,
A.B.Kurzhanski,
Yu.S.Osipov,
V.S.Patsko,
V.E.Tret'yakov,
V.N.Ushakov
Last modified: Mon Jun 8 12:02:40 MET DST 1998