-
for the numerical construction of solutions to di erential games [
3
], [
6
], [
77
], [
139
],
[
141
], and developed the rst algorithms for calculating stable bridges in linear
di erential games [
12
], [76], [
82
], [
106
], [
146
], [
201
], [
203
], [
204
], [
209
], [
211
]{[
214
],
[
218
], [
219
].
At the age of 27, in 1983, Nikolai defended his PhD thesis \Numerical
solution of linear di erential games" [
1
], [
2
]. During the decade 1982{1992, the
methods developed by him were successfully applied [
7
], [
9
], [
13
], [
16
], [
78
]{[
81
],
[
107
], [
140
], [
142
]{[
145
], [
202
], [
205
], [
206
], [
208
], [
210
], [
215
], [
217
] to the
problems of aircraft control under wind disturbances in the framework of cooperation
with the Academy of Civil Aviation in Leningrad, Russia. A method of aircraft
landing control relying on switching lines was patented [
104
] in 1988. In the early
90s, based on an analysis of the asymptotic behavior of solutions to nonlinear
di erential games, he proposed a novel algorithm for computing the
discriminating kernel of di erential inclusion [
11
]. This algorithm was later on e ectively
implemented [
46
] in terms of level sets, which allowed to use it for solving many
applied problems.
In 1992, Nikolai received a scienti c grant from the Alexander von Humboldt
Foundation and worked for a year at the University of Wurzburg, and for the
next six months at the Technical University of Munich. During this period, he
kept working on di erential games [
14
], [
83
], [
147
], [
148
], [
150
], and was engaged
in the development of new e ective algorithms in the eld of convex optimization
and linear and quadratic programming [
15
], [
17
], [
84
], [
149
], [
151
], [
152
], [
220
],
[
221
], continuing this work in later years [
26
], [
31
]. In particular, a very e ective
numerical algorithm for nding Chebyshev center of a nite point set in the
Euclidean space Rn was created, which is often used by researchers in various
scienti c elds.
From August 1993 to January 1999 Nikolai Botkin worked at the Technical
University of Munich: until March 1994 as a Humboldt Fellow in the Division
of Higher and Numerical Mathematics, and since March 1994 as a researcher
in the Division of Applied Mathematics. In line with the scienti c focus of the
Division of Applied Mathematics, which was primarily on mathematical
models of complex physical processes, he completed the works on identi cation of
unknown parameters both of nonlinear heat conductivity equations [
18
], [
109
],
[
110
] and of models describing phase transitions in shape memory alloys [
108
],
[
155
], [
222
].
In the years 1997{2002, within the framework of the DFG-project \Adaptive
materials in real-time optimization" Nikolai was engaged in research of
problems related to oscillations excited by piezoelectrics in thin plates. Along with a
theoretical study of thin plate models excited by piezoelectric patches [
19
], [
20
],
[
111
], [
112
], [
153
]{[
156
], [
159
], [
223
], [
224
], including homogenized variants of the
models [
21
], [
23
], [
85
], [
86
], [
113
], [
157
], [
158
], [
160
], [
161
], as the number of
piezoelectric patches tends to in nity, he designed and implemented active real-time
controls which provide the desired performance of exible constructions
subjected to varying disturbances [
87
], [
225
]. Participation in this project continued
after moving in early 1999 to Bonn for work at the multidisciplinary research
ceniii
ter CAESAR (Center of Advanced European Studies and Research). For seven
and half years, he took part in numerous theoretical and applied projects. These
included:
- Nonlinear interactions in piezoelectric excited surface waves (2001{2004)
- Microbalance Array Mass Spectrometry for Proteomics (2001{2004)
- Tire Sensor (2000{2004)
- Flexible Microrobot for Scanning Electron Microscopy (2000{2004)
- Shape Memory Composites for Orthodontic Archwires (2002{2005)
- Distributed Simulation of Complex Technological Systems (2004{2006)
- Elasto-Optical Biosensor on the Nanostructural Base (2004{2006)
- Safe Production and Use of Nanomaterials (2006{2007).
The work on these projects motivated the creation and substantiation of
new mathematical models, the development of e cient algorithms and
numerical methods. In particular, not only was a mathematical model of an acoustic
biosensor, a multi-layered structure consisting of anisotropic piezoelectric and
isotropic layers, created [
24
], [
114
], [
162
], [
165
], [
166
], [
169
], [
171
], [
174
], but also
e ective numerical methods and corresponding software [
29
], [
118
], [
119
], [
123
],
[
173
], [
177
], [
181
], [
183
], [
188
] for the treatment of such multi-layered structures
were developed. To take into account complex interactions with uid of special
biostructures (aptamers) that selectively bind biomolecules, a novel
homogenization approach was elaborated [
25
], [
27
], [
116
], [
117
], [
164
], [
167
], [
172
], [
178
]. In
this method, the interaction between the bristles of the aptamer molecules and
the uid is accounted for through an averaged material whose properties are
derived using the passage to the limit in the model as the number of the bristles
goes to in nity whereas their thickness goes to zero. It should also be noted that
his contributions led to advances in the development of experimental devices
such as the elasto-optical biosensor [
45
], [
182
], [
187
], the micro uidic reactor for
producing nanoparticles [
184
], [
189
], the tire sensor [
170
], [
179
], and in the
modeling composite materials and materials with memory [
115
], [
120
], [
124
], [
163
],
[
176
], [
180
], [
186
], [
226
], as well as other important theoretical and experimental
work [
22
], [
28
], [
168
].
Since August 2006, Nikolai Botkin was with the Division of Mathematical
Modeling in the Mathematical Department at the Technical University of
Munich. From 2006 to 2009 he was a leading researcher in the DFG project
\Mathematical modeling and control in cryopreservation of living cells and tissues".
He developed and investigated models describing the processes of freezing and
thawing of living cells and tissues and ingeniously applied the optimal control
theory to the design of optimal cooling protocols that minimize the damaging
e ects caused by the release of the latent heat and by delayed freezing of
intracellular uid [
32
], [
90
], [
185
], [
190
], [
230
], [
233
]. N. D. Botkin was the co-author
of the patent [
105
] for a device for freezing dental tissues. These and subsequent
years, he also continued [
35
], [
39
], [
129
] his research on the numerical methods for
solving di erential games, developed grid algorithms for problems with nonlinear
dynamics and state constraints [
37
], [
40
], [
89
], [
193
], [
234
]. He was among the
rst who applied numerical algorithms for solving di erential games to
biotechnological problems [
237
], [
242
], in particular, to the problem of propagation of
surface acoustic waves in mathematical modeling of an acoustic biosensor [
92
],
[
121
], [
122
], [
162
], [
175
], [
227
], [
236
], as well as for nding optimal protocols of
freezing and thawing of living cells and tissues [
34
], [
38
], [
88
], [
93
], [
194
], [
228
],
[
231
], [
232
].
In 2009{2012 Nikolai worked on a joint project \Modeling of CO2
sequestration, including parameter identi cation and numerical simulation" of the
Technical University of Munich and King Abdullah University of Science and
Technology, Saudi Arabia. He obtained new results in the eld of mathematical
modeling of multiphase ows in porous medium, with accounting for the e ects
of hysteresis [
41
], [
42
], [
126
], [
128
], [
195
], [
197
].
His active work with graduate and postgraduate students of the Division of
Mathematical Modeling is re ected in publications [
30
], [
33
], [
44
], [
55
], [
125
],
[
198
], [
229
], where mathematical models for hydrodynamics, phase- eld
problems, transport in porous media, and economic problems have been developed.
Nikolai Botkin also paid great attention to research collaboration with the
Poznan University of Technology, Poland. The joint results on inverse heat
transfer problems were published in [
36
], [
52
], [
191
], [
192
]. Since 2012, he has been
actively involved in a scienti c cooperation with the Institute for Applied
Mathematics, Far Eastern Branch of Russian Academy of Sciences (Vladivostok,
Russia), on theoretical and numerical analysis of boundary-value problems of
complex heat transfer [
43
], [
47
], [
49
], [
53
], [
62
], [
68
], [
132
], [
196
], [
235
], [
239
] and
inverse and optimal control problems of complex heat transfer [
48
], [
56
]{[
58
],
[
63
], [
96
], [
131
], [
238
].
Since 2013, Nikolay Botkin with great enthusiasm participated in the joint
project \Mathematical modeling of cerebral blood circulation in premature
infants with accounting for germinal matrix" of the University Clinic \rechts der
Isar" and the Division of Mathematical Modeling. He applied the methods of
differential game theory to the problems of modeling cerebral autoregulation [
50
],
[
67
], [
98
], [
130
], [
240
], [
241
]. New models of cerebral blood circulation [
60
], [
64
],
[
65
], [
75
], including models of oxygen transport [
54
], [
69
], [
71
], [
136
], [
137
], [
243
]
in the brain were also developed and investigated with his direct involvement,
and some peculiarities of cerebral blood circulation in premature infants were
explored [
70
], [
72
], [
73
].
In 2015, he began work on the DFG project \Robust dynamic programming
approach for solution of ight dynamics control problems" (joint with the
Institute of Flight System Dynamics, Faculty of Mechanical Engineering, Technical
University of Munich), as a leading project researcher. The algorithms for
solving di erential games, especially for the numerical construction of viability sets
in nonlinear problems of high dimension were further developed [
46
], [
51
], [
99
],
using novel parallelization technologies. This made it possible to solve problems
with a phase vector dimension of 6{7, which was previously unattainable in
differential games, and successfully apply the developed algorithms to problems
of aircraft control at the stages of runway acceleration [
94
], take-o [
91
], [
100
],
[
102
], [
127
], [
133
], [
134
], cruise ight [
59
], [
61
], [
97
], [
135
], and landing [
95
]
unv
der wind conditions, as well as for creating algorithms for generating dangerous
wind disturbances [
66
], [
101
], [
103
]. Moreover, new robust control methods were
designed [
74
], [
100
], [
102
], [
138
].
Nikolai Botkin continued to work actively until his sudden death. Many of his
243 publications are frequently requested and cited by other researchers.
Numerical and analytical methods he developed for linear and quadratic programming,
optimal control and di erential game problems are used by theorists and
practitioners. Nikolai Botkin's friends, colleagues, and students will remember him
with tenderness and gratitude.