=Paper=
{{Paper
|id=Vol-2098/paper31
|storemode=property
|title=Properties of Admissible Set of an Optimal
Non-destructive System Exploitation Problem in
Some General Formalization
|pdfUrl=https://ceur-ws.org/Vol-2098/paper31.pdf
|volume=Vol-2098
|authors=Vladivir D. Mazurov,Alexander I. Smirnov
}}
==Properties of Admissible Set of an Optimal
Non-destructive System Exploitation Problem in
Some General Formalization==
https://ceur-ws.org/Vol-2098/paper31.pdf
Properties of Admissible Set of an Optimal
Non-destructive System Exploitation Problem in
Some General Formalization
Vladivir D. Mazurov and Alexander I. Smirnov
Krasovskii Institute of Mathematics and Mechanics UB RAS, Ekaterinburg, Russia
asmi@imm.uran.ru
Abstract. We study the properties of admissible strategies for the prob-
lem of using of a renewable resource system that does not lead to its
destruction, in some general formulation. It is assumed that the sys-
tem evolution is described by some iterative process. The conditions for
the stable existence of this system, originally formulated in terms of the
asymptotic properties of this iterative process, reduce to the existence
of admissible controls for some mathematical programming problem. We
analyzed the properties of its admissible set in terms of the dominant
eigenvalues of some positively homogeneous maps. We obtained the con-
ditions for the existence of optimal controls under the influence of which
the system can stably exist for an indefinitely long time. Introduced by
author’s generalization of the classical concept of the irreducibility of
a non-linear map, the concept of local irreducibility, is essentially used.
The problem under consideration has important applications in the tasks
of rational exploitation of renewable natural resources, in particular, in
the tasks of managing ecological populations.
Keywords: Concave programming · Irreducible map · Positive equilib-
rium · Rational exploitation of natural resources
1 Introduction
One of a vital problems that require the use of mathematical methods is a prob-
lem of a rational use of renewable resources. Particularly acute in practice is
a problem of rational non-destructive ecosystem exploitation. This problem is
characterized by an abundance of specific approaches for specific natural sys-
tems (see the review in [4]). In the matrix formulation, the approach to optimal
exploitation of a population in a stationary state has prevailed for a long time.
This approach uses as a structure of the population a stable distribution —
the structure of an eigenvector corresponding to the dominant eigenvalue of the
Copyright c by the paper’s authors. Copying permitted for private and academic purposes.
In: S. Belim et al. (eds.): OPTA-SCL 2018, Omsk, Russia, published at http://ceur-ws.org
360 V.D. Mazurov, A.I. Smirnov
projection matrix (that must be greater than unity). If at some point in time
the structure of the population coincides with a stable distribution, then it is
possible to remove the surplus without damage to the population, returning the
population to the previous state. This corresponds to the withdrawal of the same
proportion of all its constituent groups.
One of the first theoretical works in this direction was [2], where the concept
of sustainable yield was defined, and the solvability of the problem of finding the
maximum level of permissible exploitation was proved. In work [1] some concepts
natural for the exploitation of ecosystems were formalized and the corresponding
statements were rigorously proved. The first successful attempts to generalize
these results to the case of a nonlinear density dependence are related to papers
[11, 3] that were considered the dependence on density only for the first age class.
In [5] was considered the model with dependence on density for all age classes.
We are developing here a general approach to this problem that considers
the ecosystem as a discrete system, with its representation in the form of an
iterative process
xt+1 = F (xt ), t = 0, 1, 2, . . . (1)
where xt means the state of the system at time t = 0, 1, 2, . . ., the step operator
F transforms the system from the previous state to the following. With this
approach, one can naturally formalize the inadmissibility of excessive load on
the dynamic system condition.
We will use the following notation: Rq+ denotes the non-negative orthant of
R ; x ≤ y means y − x ∈ Rq+ ; x < y means y − x ∈ int Rq+ ; int M — the
q
interior of the set M ; x y means x ≤ y and x 6= y; Z — the set of integers;
m, n = {i ∈ Z | m ≤ i ≤ n}. Vector x = (x1 , x2 , . . . , xq ) can be written in
the short-hand form as x = (xi ). The iteratives of the map F are denoted by
F t (x) (t = 1, 2, . . .), F 0 (x) ≡ x.
A controlled system is modelled by iterative process
xt+1 = Fu (xt ), t = 0, 1, 2, . . . , (2)
where x = (xi ), u = (ui ), Fu (x) = (F (x) − u))+ , a+ = max{a, 0}, x+ = (x+ i ).
We will denote by {xt (x0 , u)}+∞ i
t=0 , xt (x0 , u) = (xt (x0 , u)) the realization of this
process starting from the initial vector x0 . In [7] we formulated an optimization
problem for dynamical system (1): determine
c̃ = max{c(u) | u ∈ U }, (3)
where c(u) — the total effect of using of system elements in the quantities
u1 , u2 , . . . , uq , the set U is the closure of the set
U = {u ∈ Rq+ | X0 (u) 6= ∅}
of controls. We will be more interested in the properties of the admissible set U ,
so for simplicity we shall assume that c(u) is linear nonnegative on Rq+ function.
Admissible Set of Optimal Non-destructive System Exploitation Problem 361
The control vector u is considered as admissible if there is at least one initial
state x0 , for which all units of the system stably exist for an indefinitely long
time:
X0 (u) = {x0 ∈ Rq+ | inf xit (x0 , u) > 0 (∀i ∈ 1, q)}
t
Thus, the size of each unit of the system should not decrease with time to
zero. As can be seen from this definition, the set U formalizes the requirement
of the indestructability of the managed object.
A necessary condition for U 6= ∅ is the existence of a positive fixed point of
the map F , i. e., condition NF+ 6= ∅. In this case the set U contains at least the
trivial control: 0 ∈ U .
Unfortunately, the set U is not always closed; thus, in problem (3), its closure
is considered as the admissible set.
Denote by ũ and U e the optimal vector and optimal set of problem (3); and
+
by Nu and Nu the sets of nonzero fixed points and positive fixed points of Fu ,
respectively. It is assumed that the original model without control (1) has the
trivial equilibrium state: F (0) = 0.
We are interested in the positive realizations of the iterative process (2) only,
when xt (x0 , u) > 0 (∀t = 1, 2, . . .). In this case Fu (x) = F (x) − u > 0 for all
x = xt (x0 , u). Note that if x ∈ Nu+ , then Fu (x) > 0 and F (x) = x + u also must
be fulfilled.
The main requirement imposed on the class of maps under consideration is
a concavity or its weakening, a subhomogeneity [6]. The definitions of subhomo-
geneous and superhomogeneous maps, as well as other standard concepts used
below, are given, for example, in [8]. The subhomogeneity of a map make it pos-
sible to depict adequately a so-called saturation effect observed in many natural
systems with limited resources.
In what follows, the concept of irreducibility (indecomposability) of a map
is essentially used. Along with the classical concept of irreducibility [10], we will
also use its local analogues [12], such as a notion of irreducibility of a map at
zero. We give a classical definition of the irreducibility of a map in a somewhat
different way, using the following sets:
I + (x, y) = {j ∈ 1, q | xj > yj }, I 0 (x, y) = {j ∈ 1, q | xj = yj },
I + (x) = {i ∈ 1, q | xi > 0}, I 0 (x) = {i ∈ 1, q | xi = 0}.
A map F is said to be reducible at a point y ∈ Rq+ , if
∃x ∈ Rq+ : x y, I 0 (x, y) 6= ∅, I 0 (x, y) ⊆ I 0 (F (x), F (y)). (4)
A map F is said to be irreducible at point y, if it is not reducible at point y.
A map reducible (or irreducible) at every point of set M is said to be reducible
(or irreducible, respectively) on the set M .
We consider separately the case of irreducibility at the point y = 0. A map
F ∈ {Rq+ 7→ Rq+ } is said to be reducible at point y = 0 (reducible at zero), if
∃x ∈ Rq+ : x 0, I 0 (x) 6= ∅, I 0 (x) ⊆ I 0 (F (x)). (5)
362 V.D. Mazurov, A.I. Smirnov
We note that the concept of reducibility at zero makes sense only for maps
satisfying condition F (0) 6> 0.
The global irreducibility of a map means its irreducibility at all points Rq+
and, in particular, irreducibility at zero.
In a research of asymptotic properties of iterative processes a primitivity
property of a map [10] is usually used. Although irreducibility at zero is a weaker
property in comparison with primitivity at zero, it also guarantees the positiv-
ity of nonzero points of a map. Indeed, suppose, by way of contradiction, that
I 0 (x̄) 6= ∅. Then from fi (x̄) = x̄i (∀i ∈ I 0 (x̄)) we get I 0 (x̄) ⊆ I 0 (F (x̄)), which
conradicts, by (5), the irreducibility of F at zero. Therefore I 0 (x̄) = ∅ and
x̄ > 0.
The local irreducibility properties for monotone increasing subhomogeneous
maps are considered in [8, 9].
We refine our assumptions about properties of dynamical system (1). The
step operator F is assumed to be nonnegative and concave on Rq+ (and hence it
is monotone increasing on Rq+ ). It is also natural to assume that the condition
for the absence of identically zero coordinates of F is satisfied:
∀ i ∈ 1, q ∃ x ∈ Rq+ : fi (x) > 0. (6)
It is important that this property for monotone increasing subhomogeneous map
F implies that F (int Rq+ ) ⊆ int Rq+ . A sufficient condition for (6) is the irre-
ducibility of F at zero. Since irreducibility at zero also guarantees the positive-
ness of non-zero fixed points, it is convenient to assume from now on that the
map F is irreducible at zero.
2 Some Preliminary Results
For the subhomogeneous map F we can define the following positively homoge-
neous maps:
F0 (x) = lim α−1 F (αx), F∞ (x) = lim α−1 F (αx), (7)
α→+0 α→+∞
It is easy to see that
F∞ (x) ≤ F (x) ≤ F0 (x) (∀x ∈ Rq+ ). (8)
Define the following sets:
PF+ = {x > 0 | F (x) > x}, Q+
F = {x > 0 | F (x) < x}. (9)
It is shown in [13] that for a subhomogeneous, monotone increasing and irre-
ducible at zero map F ∈ {Rq+ 7→ Rq+ } the following properties are valid:
λ(F0 ) > 1 ⇔ PF+ 6= ∅; λ(F∞ ) < 1 ⇔ Q+
F 6= ∅. (10)
Admissible Set of Optimal Non-destructive System Exploitation Problem 363
Likewise, for the superhomogeneous, monotone increasing and irreducible at zero
map F the following properties are valid:
λ(F0 ) < 1 ⇔ Q+
F 6= ∅; λ(F∞ ) > 1 ⇔ PF+ 6= ∅. (11)
We introduce the set
MF (x) = {α > 0 | F (αx = αF (x)}, (12)
and quantities
αF (x) = inf MF (x), βF (x) = sup MF (x), (13)
where x is an arbitrary nonzero vector. Note that the set MF (x) always contains
unity as an element.
Existence conditions of positive fixed points for subhomogeneous maps on
cone Rq+ were studied in [13]. It follows from (8) due to [10, Theorem 10.3] that
the condition λ(F∞ ) ≤ 1 ≤ λ(F0 ) is necessary for existence of a positive fixed
point for monotone increasing, subhomogeneous and irreducible at zero map. A
similar condition with strict inequalities:
λ(F∞ ) < 1 < λ(F0 ) (14)
is sufficient for existence of a positive fixed point. Sufficient conditions with
λ(F∞ ) = 1 or λ(F0 ) = 1 require the use of the global irreducibility property.
With some work it can be shown that if F is monotone increasing, subho-
mogeneous, and condition (14) is satisfied, then NF+ is bounded, convex and
contains a largest element x̄F . Moreover, a concave globally irreducible map
satisfying condition (14) has a unique positive fixed point [13, Corollary 2.2.1].
If Nu+ = ∅ for all u > 0, then it follows from condition NF+ 6= ∅ that
U = {0}. This case is uninteresting to us, therefore, in addition to condition
NF+ 6= ∅, it is necessary to assume that Nu+ 6= ∅ for some u > 0. In this case
F (x) = x + u > x (∀x ∈ Nu+ ), and by (10), we have λ(F0 ) > 1. For this reason,
the condition (14) is assumed everywhere below. Then the set U is also convex,
contains some positive vector and with each positive vector u also contains the
segment [0, u]; those U is so-called downward set. Besides the following equalities
hold:
U = {u ∈ Rq+ | Nu+ 6= ∅}, U = {u ∈ Rq+ | Nu 6= ∅}, (15)
Finally, it was shown in [7] that under above assumptions the problem of
mathematical programming
max{c(u) | x = F (x) − u, x ≥ 0, u ≥ 0} (16)
is solvable in the sense that c̃ < +∞ and this value is attained on certain
admissible vectors ũ, x̃. Moreover, c̃, ũ is a solution of problem (3) if and only if
c̃, ũ, x̃ is a solution of problem (16).
We now turn to properties of the admissible set for this problem. Proceeding
from its basic interpretation (10), we will be interested first of all in properties
and conditions for existence of the positive equilibria x̃ for the control ũ.
364 V.D. Mazurov, A.I. Smirnov
3 Some Properties of the Admissible Set of Problem (3)
Note that the set Nu for u ∈ U contains the largest element x̄u and the map
x̄(u) : u → x̄u is monotone decreasing on U . It can also be shown that x̄(u)
inherits the concavity of the map F .
We introduce the set
D = {u ∈ Rq+ | Nu 6= ∅, Nv = ∅ (∀v > u)} (17)
that forms part of a boundary of U . The set D is of interest in connection
with the fact that it contains all potentially optimal vectors of the problem (3):
e ⊂ D. We divide this set into two disjoint parts by the criterion of the presence
U
or absence of common points with U :
D0 = D \ D00 , D00 = U \ U. (18)
From equalities (15) we obtain the following representation of D00 :
D00 = {u ∈ Rq+ | Nu 6= ∅, Nu+ = ∅}. (19)
We note that, by definition of D00 , for elements of Nu with u ∈ D00 the following
property is true:
/ int Rq+ (∀u ∈ D00 ).
x̄u ∈ (20)
Directly from (17)–(19) the following representation follows:
D0 = {u | Nu+ 6= ∅, Nv = ∅ (∀v > u)}. (21)
The representations (19), (21) can be clarified. Indeed, it follows from (21), in
view of (19), that D0 ⊂ U . Since D0 ⊆ D, we get D0 ⊆ D ∩ U . On the other
hand, D ∩ U ⊆ D0 , because u ∈ U together with the second equality in (18)
means u ∈/ D00 . Hence, taking into account u ∈ D and the first equality in (18),
we obtain u ∈ D0 . Therefore, D0 = D ∩ U .
Further, from the second equality in (18) it follows that D \ U ⊆ D00 . On the
other hand, D00 ⊆ D \ U , since D00 ⊆ D and D00 ∩ U = ∅. Therefore D00 = D \ U .
Thus, we have the following equalities:
D0 = D ∩ U, D00 = D \ U. (22)
These equalities show that the positive optimal vector x̃ for the optimal control
ũ of the problem (3) exist if and only if ũ ∈ D0 . Using (22), we can refine that
U \ D = U \ D0 .
The set U \ D does not a fortiori contain the optimal vectors of the prob-
lem (3). We give without proof the following assertion characterizing this set.
Lemma 1. If F is concave on Rq+ , irreducible at zero and the condition (14)
is satisfied, then u ∈ U \ D if and only if there exists v > u belonging to the
same set: v ∈ U \ D; or equivalently, if and only if there exists x > 0 such that
Fu (x) > x.
Admissible Set of Optimal Non-destructive System Exploitation Problem 365
Let us define the maps Φy ∈ {Rq+ 7→ Rq+ }, Ψy ∈ {[0, y] 7→ Rq+ }, where y ∈ Nu ,
Φy (x) = (φiy (x)), Ψy (x) = (ψyi (x)), with the following equalities:
Φy (x) = F (x + y) − F (x), Ψy (x) = F (y) − F (y − x). (23)
Obviously, x̄u ≤ x̄F ; therefore we can assume that y, which plays the role of a
parameter here, varies in [0, x̄F ].
It is easy to see that the maps Φy , Ψy inherit the monotonicity of the map F ,
as well as the presence of a trivial fixed point x = 0. Furthermore, the map Φy
is also concave on Rq+ , and the map Ψy is convex on [0, y]. Therefore, for these
maps there exist positively homogeneous (of the first degree) maps (Φy )0 , (Ψy )0
defined for F in (7). For their components the following equalities hold:
(φiy )0 (x) = fi0 (y, x), (ψyi )0 (y, x) = −fi0 (y, −x) (∀i ∈ 1, q),
where fi0 (y, x) is one-sided directional derivative of the function fi (x) at the
point y in the direction x.
The maps Φy , Ψy (and, hence, their dominant eigenvalues) are related by the
followings inequalities:
Φy (x) ≤ Ψy (x) (∀x ∈ [0, y]), (24)
λ((Φy )0 ) ≤ λ((Ψy )0 ). (25)
If F is differentiable at point y and F 0 is its derivative, then the following equa-
tions are valid:
(Φy )0 (x) = (Ψy )0 (x) = F 0 (y)x,
λ((Φy )0 ) = λ((Ψy )0 ) = λ(F 0 (y)).
The following relations between the sets of fixed points for the maps Fu ,
Φy , Ψy and the sets (9) for these maps follow directly from their definitions.
Lemma 2. Suppose that F is concave on Rq+ , irreducible at zero and the con-
dition (14) is satisfied. Then the following properties are valid:
(1) x ∈ PΦ+y , y ∈ Nu ⇒ x + y ∈ PF+u ;
(2) x ∈ Q+ +
Ψy , y ∈ Nu , x < y ⇒ y − x ∈ PFu ;
(3) x ∈ NΦy , y ∈ Nu ⇒ x + y ∈ Nu ;
(4) x ∈ NΨy , y ∈ Nu , x ≤ y ⇒ y − x ∈ Nu ;
(5) x ∈ Nu , y ∈ Nu , y ≤ x ⇒ x − y ∈ NΦy .
(6) x ∈ Nu , y ∈ Nu , y ≤ x ⇒ x − y ∈ NΨ+x .
We show that Φy and Ψy inherit also the global irreducibility of the map F .
Lemma 3. Suppose that F is concave on Rq+ and the condition (14) is satis-
fied. If the map F is globally irreducible, then the map Φy (y ∈ Rq+ ) is globally
irreducible, and the map Ψy (y ∈ int Rq+ ) is irreducible on [0, y).
366 V.D. Mazurov, A.I. Smirnov
Proof. Let F be globally irreducible and suppose for contradiction that Φy is
not globally irreducible for some y ∈ Rq+ . By (4) this means that there exist
x, x0 such that x0 x, I 0 (x, x0 ) 6= ∅ and φi (x, y) = φi (x0 , y) for i ∈ I 0 (x, x0 ).
Then for z = x + y, z 0 = x0 + y we have: z 0 z, I 0 (z, z 0 ) = I 0 (x, x0 ) 6= ∅ and
fi (z) = fi (z 0 ) for all i ∈ I 0 (z, z 0 ). Hence F is not globally irreducible, which
contradicts our assumptions. Thus, Φy is globally irreducible. The proof of the
irreducibility for Ψy is carried out analogously. The proof is complete.
We proceed to describe the set D by means of the maps (23).
Theorem 1. Suppose that F is concave on Rq+ , irreducible at zero and the con-
dition (14) is satisfied. If u ∈ U , then u ∈
/ D if and only if λ((Ψx̄u )0 ) < 1.
Proof. Necessity. Using (15) and Lemma 1 for u ∈ U \D, we obtain that Nv+ 6= ∅
for some v > u. Then, denoting x̄u − x̄v > 0 by x̄, we get: Ψx̄u (x̄) = F (x̄u ) −
F (x̄u −x̄) = F (x̄u )−F (x̄v ) = (x̄u +u)−(x̄v +v) = x̄+(u−v) < x̄, i. e. Ψx̄u (x̄) < x̄.
It then follows, by (11) due to convexity (and, hence, superhomogenity) of the
map Ψx̄u , that λ((Ψx̄u )0 ) < 1.
Sufficiency. If λ((Ψy )0 ) < 1, then, by (11), there exists x̄, such that 0 < x̄ < y
and x ∈ Q+ Ψy . Hence, due to the assertion (2) of Lemma 2, we obtain y − x̄ ∈ PFu ,
+
that is, Fu (y − x̄) > y − x̄. But this means, by Lemma 1, that u ∈ U \ D, as
required.
Now we can obtain a characterization of the set D in terms of the maps (23).
Corollary 1. Suppose that F is concave on Rq+ , irreducible at zero and the
condition (14) is satisfied. If u ∈ U , then u ∈ D if and only if λ((Ψx̄u )0 ) ≥ 1.
The following statement gives a sufficient condition for an admissible vector
not to belong to the set D.
Theorem 2. Suppose that F is concave on Rq+ , globally irreducible and the con-
dition (14) is satisfied. If u ∈ U and λ((Φx )0 ) > 1 for some x ∈ Nu , then
u ∈ U \ D.
Proof. By (10) and Lemma 3, there exists x̄ > 0 such that Φx (x̄) > x̄. It then
follows, since assertion (1) of Lemma 2, that Fu (x0 ) > x0 for x0 = y + x̄. But this
means, by Lemma 1, that u ∈ U \ D. The proof is complete.
Corollary 2. Suppose that F is concave on Rq+ , globally irreducible and the
condition (14) is satisfied. If u ∈ D and y ∈ Nu then λ((Φy )0 ) ≤ 1.
Combining Theorem 1, Corollary 1 and Corollary 2, we obtain the following
characteristic of U for the differentiable map F .
Corollary 3. Suppose that F is concave, differentiable on Rq+ , and the condition
(14) is satisfied. If the map F is also irreducible at zero, then
λ(F 0 (x̄u )) < 1 (∀u ∈ U \ D). (26)
If, in addition, the map F is globally irreducible, then
λ(F 0 (x̄u )) = 1 (∀u ∈ U ∩ D). (27)
Admissible Set of Optimal Non-destructive System Exploitation Problem 367
Proof. If u ∈ U \D, then the inequality (26) follows from Theorem 1 due to (25).
If u ∈ U ∩ D, then, by Corollary 1, we obtain λ(F 0 (x̄u )) ≥ 1. On the other hand,
according to Corollary 2, we have λ(F 0 (x̄u )) ≤ 1, so λ(F 0 (x̄u )) = 1. The proof
is complete.
Using the property 5 of Lemma 2 and Lemma 3, we obtain the following im-
portant property about the impossibility of a partial coincidence for coordinates
of fixed points of Fu in the case of the globally irreducible map F :
∀x, y ∈ Nu : x y ⇒ x < y. (28)
We will give now one more characteristic feature of the elements of Nu for
u ∈ U \ D. The following assertion supplements the conclusion of Theorem 2.
Theorem 3. Suppose that F is concave on Rq+ , global irreducible and the con-
dition (14) is satisfied. Then the following properties are valid:
λ((Φx̄u )0 ) < 1, λ((Φx )0 ) > 1 (∀ u ∈ U \ D, x ∈ Nu \ {x̄u }).
Proof. The first inequality follows from Theorem 1 and inequalities (24). Further,
if x ∈ Nu \ {x̄u }, then, by assertion 5 of Lemma 2, x0 = x̄u − x is the fixed point
of Φx that is positive by (28). Hence we obtain λ((Φx )0 ) ≥ 1 [10, Theorem 10.3].
Suppose, by way of contradiction, that λ((Φx )0 ) = 1. Then due to [13, Theo-
rem 2.2.11], we get (0, 1) ⊂ MΦx (x0 ) (see (12)–(13)). We show that the condition
α ∈ MΦx (x0 ) is satisfied if and only if the following equality is true:
F ((1 − α)x + α(x + x0 )) = (1 − α)F (x) + αF (x + x0 ) (∀α ∈ (0, 1)). (29)
Indeed, we have: α ∈ MΦx (x0 ) ⇔ Φx (αx0 ) = αΦx (x0 ) ⇔ F (αx0 + x) − F (x) =
α(F (x0 + x) − F (x)) ⇔ F (αx0 + x) = (1 − α)F (x) + αF (x + x0 ). Since αx0 + x =
(1 − α)x + α(x + x0 ), we obtain the equality (29). Taking into account that
x + x0 = x̄u , we obtain from here the equality
F ((1 − α)x + αx̄u ) = (1 − α)F (x) + αF (x̄u ) (∀α ∈ (0, 1)). (30)
By assertion 6 of Lemma 2, x0 is a fixed point of Ψx̄u too. We show now that,
moreover, if λ((Φx )0 ) = 1, then all points αx0 (α ∈ (0, 1]) are also fixed points
of the map Ψx̄u .
Indeed, since x̄u − αx0 = (1 − α)x̄u + αx, we have Ψx̄u (αx0 ) = F (x̄u ) − F (x̄u −
0
αx ) = F (x̄u ) − F ((1 − α)x̄u + αx). The equality (30) holds for all α ∈ (0, 1), so
we get: Ψx̄u (αx0 ) = F (x̄u ) − (1 − α)F (x̄u ) − αF (x) = α(x̄u + u) − α(x + u) =
αx0 , so that really Ψx̄u (αx0 ) = αx0 (∀α ∈ (0, 1]). These equalities mean that
(Ψx̄u )0 (x0 ) = x0 , thus λ((Ψx̄u )0 ) ≥ 1 [10, Theorem 10.3]. But then it follows, by
Corollary 1, that u ∈ D. This contradiction shows that λ((Φx )0 ) > 1. The proof
is complete.
In conclusion, we give several assertions about a cardinality of set Nu with
u ∈ D. If x̄u is not unique in Nu , then it follows from (28) that x̄u > xu
(∀xu ∈ Nu \ {x̄u }). In this case, we can define the set
Lu = co{xu , x̄u },
368 V.D. Mazurov, A.I. Smirnov
where co M is the convex hull of a set M . We note that, as follows from the
assertion 5 of Lemma 2 and from Theorem 3, for u ∈ U \ D the set Nu cannot
contain the entire segment Lu , along with points xu , x̄u . But for the D0 the
situation is different. The following assertion shows that Nu with u ∈ D is either
a singleton or an infinite set.
Theorem 4. Suppose that F is concave on Rq+ , global irreducible and the con-
dition (14) is satisfied. Then the following property holds:
u ∈ D, xu ∈ Nu \ {x̄u } ⇒ Nu ⊇ Lu .
Proof. Due to the assertion 5 of Lemma 2, the map Φxu has a non-zero fixed
point x̄ = x̄u − xu . The map Φxu is global irreducible (see Lemma 3), therefore,
this fixed point is positive. Due to (8), we get: (Φxu )0 (x̄) ≥ Φxu (x̄) = x̄, hence
λ((Φxu )0 ) ≥ 1 [10, Theorem 10.3]. But, the Corollary 2 gives the opposite in-
equality, so that λ((Φxu )0 ) = 1. In this case, the equation αΦxu = 0 must be
fulfilled, where αΦxu is defined by (13) (see [13, Theorem 2.2.11]). This means
that Φxu (αx̄) = αΦxu (x̄) (∀α ∈ [0, 1]). Therefore, we obtain for the set of positive
fixed points of Φxu that NΦ+xu ⊇ {αx̄ | α ∈ [0, 1]}. By assertion 3 of Lemma 2 this
means that Nu ⊇ {xu + αx̄ | α ∈ [0, 1]} = {xu + α(x̄u − xu ) | α ∈ [0, 1]} = Lu .
The proof is complete.
Taking into account properties (20), (28), we obtain from the Theorem 4 the
following assertion.
Corollary 4. Suppose that F is concave on Rq+ , global irreducible and the con-
dition (14) is satisfied. Then the following properties hold:
|Nu | ∈ {1, +∞} (∀u ∈ D0 ), |Nu | = 1 (∀u ∈ D00 ). (31)
If F is also strictly concave on [0, x̄F ], then |Nu | = 1 (∀u ∈ D).
In conclusion, let us give an example showing the essentiality of the require-
ment of global irreducibility in the above statements, which used this assump-
tion. This example uses a generalization of so-called Leslie’s model [14] of the
following form:
(t+1) (t+1) (t)
xi,1 = fi (at ), xi,j+1 = αi,j xi,j (i ∈ 1, m, j ∈ 1, n − 1), (32)
Pm Pn (t) (t)
where at = i=1 j=1 βi,j xi,j , αi,j > 0, βi,j ≥ 0, xi,j ≥ 0 (∀i ∈ 1, m, j ∈ 1, n).
The functions fi (a) (i ∈ 1, m) will be assumed to be nonnegative and concave
on R+ . The step operator F (x) = (fi,j (x)) of this model has the following
components:
fi,1 (x) = fi (a(x)), fi,j+1 (x) = αi,j xi,j (i ∈ 1, m, j ∈ 1, n − 1), (33)
Pm Pn
where a(x) = i=1 j=1 βi,j xi,j . Thus, the map F inherits a concavity and
monotonicity of functions fi (a).
Admissible Set of Optimal Non-destructive System Exploitation Problem 369
The admissible set U in this case, in addition to the requirement of non-
negativity of variables, is given by the following constraints:
xi,1 = fi (a(x)) − ui,1 , xi,j+1 = αi,j xi,j − ui,j+1 (i ∈ 1, m, j ∈ 1, n − 1), (34)
Let us introduce the following notation:
m
X
σ(a) = σ (i) fi (a), q(u) = (q, u), µ(a) = σ(a) − a, (35)
i=1
n k−1
(i) P Q
where (· , ·) denotes the scalar product, qj = βi,k αi,` (by convention
k=j `=j
s
(i) (i) (i) (i)
a` = 1 for i > k), σ (i) = q1 , q = (q (1) , q (2) , . . . , q (m) ) , q (i) = (q1 , q2 , . . . , qn )
Q
`=r
(i ∈ 1, m, j ∈ 1, n) (k, `, r, s ∈ Z).
Multiplying equalities (34) by βi,j and summing, we obtain for a = a(x) the
equation
q(u) = µ(a). (36)
Thus, if xu ∈ Nu , then (36) holds, which for a given u can be regarded as an
equation for a. This equation, due to the concavity of µ(a), can have no more
than two solutions for µ(a) 6= µ∗ = maxa µ(a).
Example 1. Consider the system of constraints
1
xi,1 = fi (a) − ui,1 , xi,2 = xi,1 − ui,2 ,
2
a = x1,1 + x1,2 + x2,1 + x2,2 , xi,j ≥ 0, ui,j ≥ 0 (i, j = 1, 2),
−1
where f1 (a) = 6a(1 + a) , f2 (a) = min{a, 1} are non-negative monotone in-
creasing concave functions of a nonnegative argument. We find from (35):
3 3 9a 3
σ(a) = f1 (a) + f2 (a) = + min{a, 1},
2 2 1+a 2
3 3 9a 3
q(u) = u1,1 + u1,2 + u2,1 + u2,2 , µ(a) = + min{a, 1} − a.
2 2 1+a 2
As shown in [14], condition (14) for the model (32) is equivalent to condition
σ 0 (+∞) < 1 < σ 0 (0).
It is easy to see that in our particular case this condition is satisfied. Next, here
F ∈ {R4+ 7→ R4+ } has the following form: F (x) = (f1 (a(x), 21 x1,1 , f2 (a(x)), 21 x2,1 ).
This map clearly satisfies all assumptions of previous propositions, with the ex-
ception of the global irreducibility. The function µ(a) reaches its maximum value
µ∗ = 11/2 for a∗ = 2.
We take for illustration u = (16/5, 0; 1/10, 0) and show that ∈ D00 . Calcu-
late q(u) = 27/5 for this vector. Solving the equation µ(a) = 27/5, we obtain
370 V.D. Mazurov, A.I. Smirnov
solutions a1 = 3/2, a2 = 13/5. Further, we find x(a1 , u) = (2/5, 1/5; 9/10, 0),
x(a2 , u) = (17/15, 17/30; 9/10, 0), so that Nu = {x(a1 , u), x(a2 , u)}. Therefore,
x̄u = x(a2 , u), and from (19) it follows that u ∈ D00 . Thus, we get |Nu | = 2
for u ∈ D00 , so that the second part of the conclusion (31) of Corollary 4 is not
satisfied. In addition, we see that the property (28) is also violated for these
vectors.
The matrix F 0 (x) has the form of a generalized Leslie’s matrix:
0
f1 (a) f10 (a) f10 (a) f10 (a)
1/2 0 0 0
L(a) = f20 (a) f20 (a) f20 (a) f20 (a) ,
0 0 1/2 0
where a = a(x) (see (33)). To compare the dominant eigenvalue of this matrix
with unity, it is not necessary to find this value, because the followig equation
is valid [14]:
sgn(λ(L(a)) − 1) = sgn(σ 0 (a) − 1),
where sgn(x) = |x|/x for x 6= 0, and sgn(0) = 0.
For x = x(a1 , u) we obtain λ(F 0 (x)) = λ(L(a1 )) > 1 as σ(a1 ) > 1. We see
that the conclusions of Theorem 2 and of Corollary 2 for the chosen vector u ∈ D
also do not hold.
Thus, we have demonstrated that the requirement of the global irreducibility
in the above statements is essential.
4 Conclusion
In this paper we investigated the properties of the admissible controls of the
problem (3) that can be optimal with appropriate presetting of the objective
function. In the context of the interpretation of this problem, from which we
proceeded, the question of presence of positive equilibrium x for the iterative
process (2) with given admissible control u is essential. We showed that such
controls must be contained in the set D0 defined by (18). In the case of the
differentiability of the map F , these controls are distinguished from the others,
by Corollary 3, with simple characteristic property (27). It turns out that the
dominant eigenvalue at the equilibrium point x̄u should equal unity. In addition,
as Corollary 4 shows, only for u ∈ D0 the set Nu can contain an infinite number
of elements.
The question whether in our assumptions this set is non-empty remains gen-
erally open. But in some cases, for some known (and sufficiently general) math-
ematical models used in the modeling of biological communities, it is sometimes
possible to establish that D0 6= ∅. Preliminary studies show that this takes place
for the above generalization of Leslie’s model, as was the case in Example 1. It
can be shown that in the assumption of concavity of all functions fi (x), the set
D0 in this model is always nonempty. Particularly important and, to a degree,
Admissible Set of Optimal Non-destructive System Exploitation Problem 371
unexpected is the fact that the set D0 here is part of some hyperplane. Other
key properties of the admissible set of problem (3) for this and other models call
for further investigations.
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