**Математическая
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**Электронный
математический и медико-биологический журнал. - Т. 14. -**

**Вып****. 4. - 2015. - URL:**

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УДК 517.977.5

**Equivalent substitution in the control theory**

**© 2015 г****. Proudnikov** **I. M.**

In this paper a system of the differential equations with a
control is considered. We study a problem of looking for an optimal control
that gives an infimum for an optimized functional. The system of differential
equations is replaced by two systems with the upper and lower envelopes of a
function on the right hand side of the initial system of the differential
equations. The optimized functional is replaced by its lower envelope. All
replacements are done in a region of attainability. The necessary conditions of
optimality are sufficient for the substituted system. The rules for evaluation
of the attainability set with the help of positively definite functions are
given in the second part of the paper.

**Key words.** Optimal control, optimal
trajectories, convex functions, lower and upper convex envelopes, attainability
set, convex analysis, linear and convex functions.

Consider
the following general problem of the control theory. Suppose we have a system
of the differential equations

_{} (1)

_{}, and an optimized functional has a form

_{} (2)

where _{}, _{} takes values in _{}, where _{} is a convex
compact set in _{}, _{}. We assume that the function _{} is continuous
and the function _{} is a Lipschitz
one in all arguments in totality, so that the system (1) satisfies the
conditions of uniqueness for a solution with the given initial values. We
consider the autonomous systems of the differential equations it does not
restrict the generality of consideration, as soon as if in the non-autonomous
case _{} is Lipschitz in
all arguments uniformly in _{}, then all argumentations, made below, will be true as
well.

We
have to find an optimal control _{} that is a
piecewise continuously differentiable vector-function from _{} with values in _{}. We will assume that the derivatives _{}, where they exist, are bounded in the norm, i.e.

_{}

uniformly
in _{}. Here _{} is a set of
points in _{}, where the derivatives _{} exist. In this
case pointwise convergence of a sequence _{} on _{} is equivalent to
uniform convergence of the functions _{} on a set of
continuity and, of course, is equivalent to convergence in the metrics _{} of the space _{}. The metrics of _{} is equal, by
definition, to the metrics of the space _{}, i.e.

_{}

The
function _{} is defined on
the segment _{}. We are looking for _{} for that a
solution to the system (1) gives an infimum for the functional _{}.

In
the beginning we replace the optimization problem (2) by the following
problem

_{} (3)

Let
us include into consideration all the functions _{} resulting from
pointwise convergence. It is obvious that all limit functions belong to a
closed, bounded set of functions defined on _{}, which we denote by _{}. The two functions from the set _{} are equivalent
(equal) if these functions are equal on a set of full measure. It is clear that
all measurable functions on _{} belong to _{}.

We
will solve the above formulated optimization problem (3) on the set _{}, i.e.

_{} (4)

The
problem is that an optimal control does not exist always. For this reason
generalized control (lower or upper semicontinuous) is considered.

As
an example, consider the following system of differential equations

_{}

and
the optimized functional is defined as

_{}

The
similar problem we can in [10] with replacement _{} on _{}. This problem does not have an optimal
control _{} in the set of
piecewise continuously differentiable functions on _{}, but it has an optimizing sequence of controls _{} that are
piecewise continuous functions with values _{}. It is easy to see that an optimizing sequence _{}, corresponding to the sequence of controls _{}, has the limit _{} on _{}. The control _{}, that corresponds to the solution _{}, can not be received as the pointwise limit of _{} and it’s not an
optimal control.

The
right-hand side of the equation (1) can be very complex, and the exact solution
of this equation can often be found approximately using numerical methods.
Optimization of the function _{} is also not easy
if it has a complex form. But optimization of the lower convex envelope (LCE)
of _{} is easier.
Moreover, a global optimum point does not disappear if we construct a lower
convex envelope of our optimized functional. In addition, the construction of
the lower convex envelope of _{}, that we denote by _{}, turns it into a lower weakly semicontinuous function.
It means that

_{}

for
any sequence _{} converging to _{} weakly. This
requirement is important for weak convergence of an optimizing sequence to a
solution of the problem (2).

We
propose here a method of equivalent substitution with the help of which we can
overcome these difficulties. Namely, we suggest a replacement of the right-hand
side of the equation (1) by another function with a simpler structure. The
search for solutions of the system (1) (numerical or not) becomes simpler. The
principle of equivalent replacement claims that although we have another
function with a simpler structure, but the function _{} attains the same
infimum on the set of piecewise continuously differentiable functions and on
its closure. At the same time the new optimized functional _{} becomes lower
semicontinuous.

This
idea is different from the idea of relaxation [1] because we take the convex
(concave) envelopes of functions on the right hand side of our system of the
differential equations for all points from the region of attainability.

Taking
into consideration the information about the replacements of the functions _{} and _{}, we can conclude that searching for an optimal control
and optimal trajectory becomes easier and the new optimization problem is
equivalent to the initial one in the sense of finding an optimal control. In
this case the conditions of optimality become necessary and sufficient.

For
the first time the author used the idea about replacing a function with its
lower convex approximation for finding its optimal points in [2] -[3].

Let
us consider the same system of differential equations (1), and the optimized
functional (4). We rewrite the system (1) and (4) in the following form

_{} (5)

with the initial conditions _{}. The optimization problem (4) is replaced by another
optimization problem

_{} (6)

Any solution of (5) is a solution of the
integral equations

_{} (7)

where _{} is a piecewise
continuously differentiable control.

Unite
all solutions of (7) in one set _{} for _{}

_{}

_{} (8)

which is called the set of attainability for
the systems (1) and (4) at time _{}.

It
is easy to see that the optimization problem (6) is equivalent to the following
optimization problem

_{} (9)

The
function _{} is linear in the
coordinates _{}. (The function _{} depends only on
the coordinate _{}). It is well known that any linear function reaches
its maximum or minimum on boundary of any compact set on which maximum or
minimum are looked for.

Since
the set of solutions of (1) in accordance with the assumptions is bounded on _{}, the set of the vector-valued functions _{} is closed and
bounded in _{}. Then _{} is closed and
bounded for any _{} in the metrics _{} of the space _{}. Indeed, if _{}, then, as it was mentioned above, there is uniform
convergence of _{} to _{} on _{}, where _{} is a set of any
small measure. Then
convergence in measure [4] holds. The convergences

_{}

and

_{}

follow
from continuity of the functions _{}, _{} in all variables.

Uniform convergence of the integrals

_{}

and

_{}

in
_{} and _{} follows from
Egorov’s theorem [5].

Indeed,
otherwise the sequences _{} and _{} exist, for that
and for some _{}, the inequalities

_{}

and

_{}

hold.
The integrals can be considered as functions of _{}.

According
to Egorov’s theorem for any small _{} there is a set _{} with measure _{} , that the
integrals, as the functions of _{}, will converge uniformly in _{} on a set _{}. As soon as the integrals are absolutely continuous in
measure, the integrals over the set _{} with measure _{} will be
arbitrarily small if _{} is also
arbitrarily small. As a result, we come to the contradiction with existence of _{}, for which the inequalities, written above, are true.
Thus we have proved the following theorem.

**Theorem
2.1 ***The set *_{}* is closed and bounded in *_{}* for any *_{}* in the metrics *_{}* of the space *_{}*. *

Consider
a sequence of the functions defined on _{},

_{} (10)

The sequence _{} converges on _{} uniformly in _{}, if the sequence _{} converges in the
metrics _{} to the function _{} a.e. on _{}. Prove this fact.

Indeed,
we know from the said above that the functions _{} converge to _{} uniformly on _{}. We replace the control _{} by the control _{} in (10). The
difference between the original value of the integral (10) and the new value of
the same integral can be evaluated in the following way. According to the inequality

_{}

where
_{} is a Lipschitz
constant of the function _{}, the mentioned above difference is arbitrarily small
for large _{} as well. Indeed, we have

_{}

and
the right hand side of this inequality is arbitrary small for large _{}.

We
will use the following result.

**Lemma
2.1 ***[6], [7]. The sequence *_{}* converges uniformly for *_{}* and *_{}* to a solution of (1). *

Summing
up everything mentioned above, we can conclude about uniform convergence on _{} of the solutions
_{} of (1) for _{} to a solution _{} of the same
system (1) with the control _{} as _{}.

**Lemma
2.2 ***The sequence *_{}* defined by (10), converges uniformly on *_{}* in *_{}* to a solution *_{}* of (1).
*

**Remark
2.1 ***Lemma (2.2) is also valid for the case when *_{}* in the metrics *_{}* of the space *_{}*, i.e. *

_{}

The
problem (9) has a solution if the functional (2) is lower semicontinuous. It
will be shown how to make it lower semicontinuous.

If
there is a solution to the problem (9) on the set of piecewise continuously
differentiable functions _{}, then we have a solution to the problem

_{} (11)

where _{} is a symbol of
taking convex hull.

We
introduce a set of attainability (or an attainability set) for the time _{}, which, by definition, is

_{} (12)

where _{} means closed
convex hull. It is easy to see that for an arbitrary

_{}

such as

_{}

and

_{}

the inclusion

_{}

will
be true. Therefore, the closure in (12) can be removed and definition of the
set _{} can be given as
the following

_{} (13)

Moreover, the problems (9) and (11) are
equivalent that means: if one of them has a solution, then the other one has a
solution as well and these solutions are equal to each other. In addition,
since projections of the set _{} on the axes _{} are closed and
bounded, and, hence, compact in the corresponding finite-dimensional spaces and
_{} is continuous in
_{} as the
set-valued mapping, then _{} in (11) can be
replaced by _{} and the problem
(11) can be rewritten in the following way

_{} (14)

But
a global optimal point of the problem (14) will not change if we replace the
functions _{} by their upper
concave and lower convex envelopes and _{} by its lower
convex envelope constructed on a set of attainability for the time _{}. Further we will understand under taking the upper
concave and lower convex envelopes of _{} the similar
operations for all coordinates of _{}.

Indeed,
take two arbitrary points _{} and _{} _{} from the set _{}. Consider a combination with nonnegative coefficients _{}, _{}, _{}. Then, the point _{} will belong to
the set _{}, if we replace the functions _{} and _{} by the
following:

_{}

and

_{}

But
this construction, performed for all points of the regions _{} it just means
that we construct the lower and upper convex envelopes of the function _{} and the lower
convex envelope of the functions _{} in the
attainability set for the time _{}, i.e. in _{}. Indeed, it follows from the above formula (13) for _{} that the
function _{} reaches its
minimum (14) on some _{}. Consequently, we can construct the lower convex or
upper concave envelopes of the functions _{} and _{} in all region _{}.

Denoted
by _{} a new
optimization function obtained after the replacement of the function _{} by _{} in _{}

_{}

It
is clear that _{} takes the same
optimal value in the attainability set _{}, that the functional _{} _{} takes for the
system (1).

Replace
the system _{} by the
system

_{} (15)

and the optimized functional by the functional

_{} (16)

It
is easy to see that the minimum or the maximum of the functional _{} did not change.
Hence, the problems _{}, _{} and _{}, _{} are replaceable.
So "convexification" _{} of the function _{}, in contrast to the procedure of
"convexification" of the function _{} should be as
following:

1.
Construction of the lower convex envelope (LCE) of the function _{} in the variables
_{} for each _{} from the
attainability set for the time _{}, i.e. _{}, which we denote by _{}. LCE of _{} is the biggest
convex function that does not exceed _{} in _{}.

2.
Construction of the upper concave envelope (UCE) of the function _{} (or,
equivalently, we construct the lower convex envelope for the function _{} and after that
take minus of this function) in the variables _{} for each _{} from the
attainability set for the time _{}, i.e. _{}, which we denote by _{}. UCE of _{} is the smallest
concave function that is not less _{} in _{}.

3.
Let us replace the system _{}, _{} by two systems
of the equations:

_{} (17)

with the optimization function _{} and

_{} (18)

with the same optimization function _{};

4.
Let us find among the solutions of _{} and _{} such that gives
the smallest value of the functional _{} in _{}.

We obtain the following result.

**Theorem
2.2 ***There are the solutions among the solutions of *_{}* and *_{}* such that deliver a minimum (maximum) in *_{}* and *_{}* for the functional *

_{}

that
coincides with an infimum (supremum) of the functional _{} (see (4)).
Moreover, necessary conditions for the minimum (maximum) are also sufficient
conditions.

**Remark
2.2 ***The set *_{}* is not necessarily compact, although its
projections on the axis *_{}* are compact . **That’s why we are able to go to the
problem *

_{}

if
the problem (9) has a solution. The last one coincides with the formulation of
Mazur’s theorem. It asserts that in any weakly convergent sequence _{}, _{} a subsequence
can be chosen for each _{} convex hull of
which is almost everywhere on _{} converges as _{} to some _{}. In our case, there exists a sequence _{}, the convex hull of which will converge to an optimal
control _{}. The sequence of the solutions _{}, corresponding to the controls _{}, will converge to an optimal solution _{}, corresponding to the control _{}, provided that the solutions have been calculated to
the problems with the modified right-hand side.

**Remark
2.3 ***The rules for construction of LCE and UCE are given in
Appendix.*

**Remark
2.4 ***In many cases we have to construct only LCE or UCE for
the function *_{}*.*

Return
back to the initial problem (2) with the fixed time _{}. Consider a set

_{}

_{}

that
is called the set of attainability of the system (1),(2) at time _{}.

Let
us introduce a set of attainability for the time _{} for the system
(1),(2) that is by definition

_{} (19)

As
above it is possible to prove that we can remove the closure in (19) and write

_{}

The
optimization problem can be reformulated in the form

_{} (20)

The
problems (1), (2) and (20) are equivalent which means if one has a solution,
then another one has a solution and these solutions are the same. Moreover, as
soon as the projections of the sets _{} on the axes _{} are closed,
bounded and continuous as a set valued mappings, then we can write instead of _{} _{} if a solution of
(20) exists.

We
come to the following result.

**Theorem
2.3 ***There are some solutions among the solutions of *_{}* and *_{}* such that deliver a minimum (maximum) in *_{}* and *_{}* to the functional *

_{}

that
coincides with an infimum (supremum) of the functional _{} (see (2)) where _{}.is LCE of the function _{} Moreover, the
necessary conditions for minimum (maximum) are also the sufficient conditions.

Consider
some examples. It is clear that an equivalent replacement of one system by
another can be applied to a differential system without control _{}.

Example 1. Consider the differential
equation

_{}

with
the initial condition _{}. The optimized functional is given by

_{}

The
general solution of the differential equations for _{} has the form

_{}

which
tends to _{} as _{}. The general solution of the differential equation for
_{} is given by

_{}

which
tends to _{} as _{}. In order to meet the initial condition we have to put
_{}. The projection of the attainability set _{} on _{} axis is the
interval _{}

It
is clear that the function _{} takes its
minimum at _{}. But we get the same solution if instead of the
function _{} we take its
lower convex envelope, namely, the function

_{}

Example 2. The same example, but

_{}

which
we minimize for _{}. The equation of the solution is

_{}

Here
the replacement of the function _{} by the function _{} on the whole
line is not correct, since the projection of the attainability set _{} on _{} axis is the
interval _{}.

Example
3. Let us give the differential equation

_{}

with
the initial condition _{}. The general solution has the form

_{}

a
solution, satisfying the initial condition, is

_{}

The
optimized functional has the form

_{}

for
_{}.

It
is easy to compute its optimal value

_{}

In
this case, the projection of the attainability set _{} on _{} axis is the set _{}. It is easy to see that the lower convex envelope of
the functional _{} on _{}, which we denote by _{}, takes the same infimum value. It is also true for the functional

_{}

Example
4. Let us consider the following problem

_{}

The optimized functional is

_{}

We
will get the following system after construction of the lower and upper
envelopes

_{}

and

_{}

The
optimal solution exists among their solutions _{}.

Example
5. Let us consider the differential equation

_{}

with
the initial condition _{}. We are considering piecewise continuously
differentiable functions _{} on the segment _{} for such that
delivers minimum to the functional

_{}

The
solution when _{} is constant on
the segment _{} is given by the
form

_{}

Here
the constant _{} is defined by
the initial conditions. We can see from here that if _{} is not constant
on _{}, then _{} for any initial
conditions. It means that a curve _{} will be in a set
bounded by the lines _{} on the plane _{}, where _{}. The set of attainability _{} will belong to
the same set. UCE of the function _{} in _{} is a function
the graph of which goes through the point _{}. Therefore, if we solve the differential equations
with the right sides _{} and _{}, then among the solutions there are such that deliver
the minimum _{} to the
functional _{} i.e. the
formulated theorem is true.

Let
us have a system of differential equations

_{} (21)

with the initial condition _{}, where _{} is Lipschitz in
the variables _{}, _{} is a convex
compact set in _{}. The problem is to estimate the attainability set. By
definition, the area of attainability for the time _{} is the set

_{}

where

_{}

The
choice of the initial position and the initial time of zero is not a loss of
generality.

Take
an arbitrary positively definite function _{} (see [8]),
satisfying the condition

_{}

Let

_{}

and
_{} is a piecewise
continuous vector-function.

Consider
the systems of differential equations

_{} (22)

and

_{} (23)

We
denote by

_{} (24)

the
attainability sets for the systems (22), (23), where

_{}

_{} (25)

Let
the estimates of the attainability sets for the time _{} be given
respectively by the inequalities

_{}

We
get the estimation for the attainability set of the system (21) for the time _{}. We show that the attainability set _{} for this system
satisfies the inclusion

_{}

Indeed,
by definition, the set _{} will consist of
the points on the curves the tangents to which are the sum of the tangents to
the curves consisting of the points of the sets _{} and _{} for all _{}. It is clear, that for some vector-function _{} the resulting
set will include _{}, which consists of the points on the curves the
tangents to which are the sum of the tangents to the curves consisting of the
points of the sets _{} and _{} for all _{}.

As
a result, the following theorem is proved.

**Theorem
3.1 ***For the attainability set *_{}* of (21) the inclusion *

_{}

is
true for some vector-function _{}, where _{}, _{} are given by
(24), (25) .

The
following lemma follows from here.

**Lemma
3.1 ***The function *_{}* satisfies the inequality *

_{}

in
the attainability set _{} of the system
(21).

Now
consider two differential systems with the right sides _{}, _{} and zero initial
conditions at the time equaled to zero.

Let
us suppose that the inequality

_{} (26)

is true for all _{}. We assume that we know the attainability set _{} for the time _{} of the system
with the right hand side _{}. The problem is to obtain some estimates of the
attainability set of the system (21). The arguments will be carried out as
previously, considering the trajectories of the corresponding systems.

Any
vector in the set _{} for some _{} and _{} is

_{}

Consequently,

_{}

_{}

Since
the previous inclusion holds for any _{}, it follows that

_{}

The
following theorem is proved.

**Theorem
3.2 ***For the systems (21) and (23) with the attainability
sets *_{}* and *_{}* respectively, for which the inequality (28)
holds, the inclusion *

_{}

is
true.

From here we can easily obtain the following
conclusion.

**Lemma
3.2 ***In the attainability set *_{}* of the system (21) the function *_{}* satisfies the inequality *

_{}

where
the constant _{} limits the top
value of the function _{} in the
attainability set _{} of the system
(23).

Let
us give a general method for evaluation of _{} of the system
(21). This method does not require any additional information for the system
(21).

As
is known, a convex set can be given by its extreme points. There are no
problems if there is a finite number of such points. But very often these
points are unknown or their number is infinite. We can reconstruct a convex set
if we know its projections on different directions. If we project any
trajectory, then we project not only the points but the tangents constructed at
these points. It means that we have to consider the following system for any
direction _{}

_{}

As
a result, we have

_{} (27)

where _{} is the scalar
production _{}.

We
can do an orthogonal transformation that the direction of the vector _{} was the first
coordinate axis _{}. Then the projection of the velocity vector _{} on the line _{} will be equal to
_{}. It means that we have to substitute _{} into the first
equation and to solve the first order differential equation for different
values of the constants _{} that are
corresponding coordinates of the start point.

We
can make the following conclusion: * convex hull of the attainability sets of the
equation (27) for different *_{}* will include the attainability set of the
equation (21)*. It is possible to do, because calculation methods are
developed very well for the first order differential equations.

The
obtained results allow us to pass from local to global optimization problem. To
implement this it is required to construct the lower convex and upper concave
envelops of the function written on the right hand side of the differential
system (1). We also construct the lower convex envelops for the optimized
functional. All constructions are done in the attainability set for the time _{}.

A
method for estimation of the attainability set with the help of the positively
definite functions (Lyapunov functions) is suggested. The proposed method is
based on the decomposition of the function, stayed on the right hand side of
the system of the differential equations, into the components the sets of
attainability of which are already known. It makes it different from the paper
[9], where the linear systems are considered.

It
is suggested to find projections of _{} onto any
direction _{}. For this reason we have to find projections of the
trajectories of the differential system and the tangents to them to the
direction _{}. We come across the problem of definition of a set of
attainability for the differential equation of the first order.

The
proposed transformation method of the systems is especially useful when it is
difficult to get a solution of differential equations in an explicit form, but
while using approximate methods only. In addition, the sufficient conditions of
optimality for an optimal control are obtained according to the proposed
method.

** APPENDIX**

We
will prove a theorem giving a rule for construction of LCE and UCE.

Let
_{} be continuous
function on a convex compact set _{}. It is required to construct LCE and UCE in _{}. Consider a function _{}

_{}

where
_{} is a
distribution function satisfying the following equalities

_{} (28)

We
will consider the functions _{} for different
distributions _{}.

**Theorem 4.1 ***The functions *

_{}

are
UCE and LCE of _{} on _{} correspondingly.

** Proof.** Without loss of generality we will
consider that _{} for all _{}. Divide _{} into subsets _{} _{}, _{}. We can approximate the function _{} with any
precision by the integral sums

_{} (29)

where

_{}

It
follows from (28) that

_{} (30)

The sign _{} means that the
values on the left hand side from this sign can be close to the values on the
right hand side with any precision depending on _{}. The expression (29) means that we take a convex hull
of _{} vectors _{} with
coefficients _{}, i.e. we calculate a vector

_{}

and
define a value of the function _{} at this point
equaled to

_{}

Changing
the points _{} and the
coefficients _{} satisfying (30),
we define in such way the functions _{} with different
values at _{}.

Let
us prove that the function

_{}

is
LCE. As soon as _{} is taken for all
distributions _{}, then the inequality _{} is true for all _{}. The function _{} can be
approached by the sums (29) for any distribution _{} under conditions
on the coefficients (30). It follows from here that _{} can not be
smaller than LCE of _{}. The operation _{} keeps this
quality. Consequently, _{} is LCE of _{}. We can prove in the same way that _{} is UCE of _{}. The Theorem is proved. ∆

The
construction of LCE can be done using Fenchel-Morrey’s theorem [10]. According
to it LCE is equal to the second conjugate function _{}. Construction of _{} is not easy. To
find a value of LCE at one point we have to solve two difficult optimization
problems, namely,

_{}

and

_{}

**References**

[1]
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[4]
* Kolmogorov A.N., Fomin S.V.*
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[5]
* Vulich B.Z.* A short course in the
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[6]
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[7]
* Krasnov M.L.* Integral Equations.
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[8] Zubov V.I., Lectures in control theory. M:
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[9]
* Kostousova E.K.* External and
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[10] Alekseev V.M., Tichomirov V.M., Fomin S.V.
Optimal control. M.:
Nauka, 1979. 224 p.

**Эквивалентная замена в
задачах оптимального управления**

**Прудников
И. М.**

В работе
изучается задача поиска оптимального управления, заданная в виде системы
дифференциальных уравнений и оптимизируемого функционала. Определяется правило
эквивалентной замены одной задачи оптимального управления на другую с тем же
значением оптимального решения. При этом эквивалентная замена определяется таким
образом, чтобы условия оптимальности были бы также достаточными условиями. Во
второй части статьи даются методы оценки области достижимости, в которой как
раз и происходит эквивалентная замена систем.

**Ключевые слова**:
Оптимизационные задачи, оптимальное управление, оптимальные траектории,
выпуклые функции, нижние выпуклые огибающие (НВО), верхние выпуклые огибающие
(ВВО), оценка области достижимости.

*Прудников Игорь Михайлович,
к.ф.-м.н., доцент.*

*pim
10@hotmail.com*, ph.num. +79203011393

ООО «Научно-исследовательский центр
перспективных фундаментальных биотехнологий»

Поступила в редакцию 26.11.2015.