Accurate approximated solution to the differential inclusion based on the ordinary differential equation
UDC 517.9 Many problems in applied mathematics can be transformed and described by the differential inclusion $\dot x\in f(t, x)-N_Qx$ involving $N_Qx,$ which is a normal cone to a closed convex set $Q \in \mathbb R^n$ at $x\in Q.$ The Cauchy problem of this inclusion is studied in the paper. Since...
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Many problems in applied mathematics can be transformed and described by the differential inclusion $\dot x\in f(t, x)-N_Qx$ involving $N_Qx,$ which is a normal cone to a closed convex set $Q \in \mathbb R^n$ at $x\in Q.$ The Cauchy problem of this inclusion is studied in the paper. Since the change of $x$ leads to the change of $N_Qx,$ solving the inclusion becomes extremely complicated. In this paper, we consider an ordinary differential equation containing a control parameter $K.$ When $K$ is large enough, the studied equation gives a solution approximating to a solution of the inclusion above. The theorem about the approximation of these solutions with arbitrary small error (this error can be controlled by increasing $K$) is proved in this paper.
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| doi_str_mv | 10.37863/umzh.v73i1.889 |
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DOI: 10.37863/umzh.v73i1.889
UDC 517.9
T. H. Nguyen (Hanoi Univ. Industry, Vietnam)
ACCURATE APPROXIMATED SOLUTION TO THE DIFFERENTIAL
INCLUSION BASED ON THE ORDINARY DIFFERENTIAL EQUATION
ТОЧНИЙ НАБЛИЖЕНИЙ РОЗВ’ЯЗОК ДИФЕРЕНЦIАЛЬНОГО ВКЛЮЧЕННЯ
НА ОСНОВI ЗВИЧАЙНОГО ДИФЕРЕНЦIАЛЬНОГО РIВНЯННЯ
Many problems in applied mathematics can be transformed and described by the differential inclusion \.x \in f(t, x) - NQx
involving NQx, which is a normal cone to a closed convex set Q \in \BbbR n at x \in Q. The Cauchy problem of this inclusion
is studied in the paper. Since the change of x leads to the change of NQx, solving the inclusion becomes extremely
complicated. In this paper, we consider an ordinary differential equation containing a control parameter K. When K is
large enough, the studied equation gives a solution approximating to a solution of the inclusion above. The theorem about
the approximation of these solutions with arbitrary small error (this error can be controlled by increasing K ) is proved in
this paper.
Багато задач у прикладнiй математицi можна трансформувати та описати за допомогою диференцiального включен-
ня \.x \in f(t, x) - NQx, в яке входить NQx, що є нормальним конусом для замкненої опуклої множини Q \in \BbbR n
у точцi x \in Q. У цiй роботi вивчається задача Кошi для такого включення. Оскiльки змiна x обумовлює змiну
NQx, розв’язання цього включення стає надто складним. Тут розглядається звичайне диференцiальне рiвняння, яке
мiстить керуючий параметр K. Коли K є достатньо великим, це рiвняння дає розв’язок, який наближає розв’язок
дослiджуваного включення. Доведено теорему про наближення цих розв’язкiв з будь-якою точнiстю (вiдповiдна
похибка контролюється за допомогою зростання K ).
1. Introduction. Theory of differential inclusions has gained great interest in recent decades. Under
some assumptions, differential inclusions are equivalent to equations in contigent [1, 2] and equations
in paratingent [3]. Particularly, A. F. Filippov [4] provided some research on the following differential
inclusion:
\.x \in F (t, x), (1)
where t \in [t0, T ], x \in \BbbR n and F (t, x) is a closed convex set of \BbbR n.
A solution of inclusion (1) is a locally absolutely continuous function x = x(t) satisfying (1) for
almost all t \in [t0, T ] (see [4, p. 54]). In addition, if x(t), t \in [t0, T ], is a solution of (1), then there
exists a vector function \varphi (t, x) \in F (t, x) such that \.x = \varphi (t, x) for almost all t \in [t0, T ].
Using differential inclusions that can be considered as a mathematical model for some problems
in the field of circuit theory [5] and general ecosystems [6], a specific problem modeled by differential
inclusions can be found in Section 2.
Let Q \subset \BbbR n be a fixed closed convex set and x \in Q. The normal cone to Q at x is defined by
NQx =
\bigl\{
z \in \BbbR n : (z, \xi - x) \leq 0 \forall \xi \in Q
\bigr\}
, (2)
where (\cdot , \cdot ) denotes the scalar product in \BbbR n. Some properties of NQx will be studied in Section 2.
Fix (t0, x0) \in \BbbR \times Q and T > 0, and let f : [t0, T ]\times Q\rightarrow \BbbR n be a continuous function satisfying
a Lipschitz condition with some Lipschitz constant L > 0 and bounded by a constant C, that is,\bigm\| \bigm\| f(t, x) - f(t, y)
\bigm\| \bigm\| \leq L\| x - y\| \forall x \in Q \forall t \in [t0, T ], (3)
c\bigcirc T. H. NGUYEN, 2021
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 1 117
118 T. H. NGUYEN\bigm\| \bigm\| f(t, x)\bigm\| \bigm\| \leq C \forall x \in Q \forall t \in [t0, T ], (4)
where \| \cdot \| denotes the norm defined by \| \cdot \| =
\sqrt{}
(\cdot , \cdot ).
Under the conditions (3), (4) we consider the differential inclusion
\.x \in f(t, x) - NQx, t \in [t0, T ], x(t) \in Q. (5)
Both existence and uniqueness of solutions of the Cauchy problem
\.x \in f(t, x) - NQx,
x(t0) = x0 \in Q
(6)
will be shown in Section 4. For this, we recall a solution of (6) as an absolutely continuous function
x = x(t) which satisfies (5) for almost all t \in [t0, T ] and the initial condition x(t0) = x0 \in Q.
Obviously, we see that differential inclusions are usually more difficult and complicated than
differential equations. Therefore, we propose a form of an ordinary differential equation whose right-
hand side is a continuous function in order to find a solution approximated to a solution of (5). This
is the main objective of the paper.
Now, let Q \subset \BbbR n be a convex closed set and y \in \BbbR n. An element y \in Q is called point of best
approximation to y in Q and denoted y = PQ(y) if
\| y - y\| = \mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(y,Q) := \mathrm{i}\mathrm{n}\mathrm{f}
\bigl\{
\| y - z\| : z \in Q
\bigr\}
. (7)
The element y is called projection of y onto Q.
With the mentioned goal above, we consider the following ordinary differential equation, where
the right-hand side is a continuous function involving a sufficiently large control parameter K :
\.y = f(t, y) - K(y - y),
y(t0) = x0 \in Q.
(8)
Obviously, for each K the Cauchy problem (8) has at most one continuously differentiable
solution yK(t) on [t0, T ]. The contribution of this paper is to evaluate the error between the solution
obtained by solving (8) and (6). We show that an arbitrary small error can be obtained by increasing
the control parameter K to a sufficiently large value.
2. Preliminaries. In \BbbR n let us consider cones in the sense of the following definition.
Definition 1. A subset S \subset \BbbR n is called a cone, if
x, y \in S and s, t \geq 0 \Rightarrow sx+ ty \in S.
Besides, the adjoint cone of a cone S \subset \BbbR n is a special normal cone, which is defined as follows.
Definition 2. Let S \subset \BbbR n be a cone. The adjoint cone to S is defined by
S\ast := NS0 =
\bigl\{
z \in \BbbR n : (z, y) \leq 0 \forall y \in S
\bigr\}
.
Let Q \subset \BbbR n be a closed convex set and x \in Q, the normal cone NQ(x) to Q at x is defined
by (2). Next, the tangent cone to Q at x is defined by
TQx = \{ z \in \BbbR n : (z, \xi ) \leq 0 \forall \xi \in NQx\} . (9)
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 1
ACCURATE APPROXIMATED SOLUTION TO THE DIFFERENTIAL INCLUSION . . . 119
A comparison of (2) and (9) shows that
TQx = (NQx)
\ast .
Now, the important properties of NQ are described in two following propositions.
Proposition 1. Let Q \subset \BbbR n be a closed convex set. Then the normal cone NQx, x \in Q, is a
cone in the sense of Definition 1, and it is a closed set.
Proof. The proof relies on the linearity and continuity of the scalar product. Given \xi \in Q,
z1, z2 \in NQx, and t1, t2 \geq 0, we have
(t1z1 + t2z2, \xi - x) = t1(z1, \xi - x) + t2(z2, \xi - x) \leq 0.
Hence, t1z1 + t2z2 \in NQx.
Now, if (zk)
+\infty
k=1 is a sequence in NQx converging to z \in \BbbR n, we get
(z, \xi - x) = \mathrm{l}\mathrm{i}\mathrm{m}
k\rightarrow +\infty
(zk, \xi - x) \leq 0
for all \xi \in Q. Hence, z \in NQx.
Proposition 1 is proved.
Proposition 2. Let Q \subset \BbbR n be a closed convex set. Then the multivalued map NQ : x \mapsto \rightarrow NQ(x)
is closed. That is, if (xk)
+\infty
k=1 is any sequence in Q and zk \in NQ(xk), then the relations
\mathrm{l}\mathrm{i}\mathrm{m}
k\rightarrow +\infty
xk = x, \mathrm{l}\mathrm{i}\mathrm{m}
k\rightarrow +\infty
zk = z
for some z \in \BbbR n imply that z \in NQ(x).
Proof. The relation zk \in NQ(xk) means that
(zk, \xi - xk) \leq 0 \forall \xi \in Q.
Passing in this inequality to the limit as k \rightarrow +\infty yields
(z, \xi - x) \leq 0 \forall \xi \in Q
which means that z \in NQ(x).
Proposition 2 is proved.
Based on these concepts we study a problem that is modeled by the differential inclusion of the
form (5). Let us consider the electrical circuit on Fig. 1. Recall that the inputs L (the inductance), R
(the resistance), and E (the electromotive force source) of the circuit are related by the equations
L
diL
dt
= uL, RiR = uR, uE = - e(t).
The parameters L (measured in Henry) and R (measured in Ohm) are given constants, while the
known time-dependent function e(t) is measured in Volt. The functions i (measured in Ampere) and
u (measured in Volt) with corresponding subscripts are the current and voltage, respectively; their
direction is indicated in Fig. 1 by the arrow.
An ideal diode D is an element which leads the current in the direction of the arrow (i.e., from
the anode to the cathode), but not in the reverse direction. For negative voltage (uD < 0), the current
is zero (iD = 0), while for vanishing voltage (uD = 0), the current is positive or zero (iD \geq 0); a
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120 T. H. NGUYEN
Fig. 1. An example of electrical circuits.
positive voltage is not possible. So the dependence of the current on the voltage in an ideal diode
may be described formally as a set of the following relations:
iD \geq 0, uD \leq 0, iD \cdot uD = 0. (10)
From (10) it follows iD \in Q :=
\bigl[
0,+\infty
\bigr)
and
uD \in U :=
\left\{ \{ 0\} , if iD > 0,
( - \infty , 0], if iD = 0.
By using (2), we get
uD \in NQiD. (11)
For such a circuit the second Kirchhoff rule states that
uL + uR + uD + uE = 0,
i.e., the sum of the voltages of all elements is zero. Using the fact that the current of all elements in
a non-ramified circuit (a circuit is called non-ramified (or non-bifurcating) if every knot joins exactly
two electrical elements) is the same (iL = iR = iD =: i), we arrive at the differential equation
L
di
dt
+Ri = e(t) - u
for the unknown functions i and u := uD. For a complete description of the circuit we also have to
take into account the relations (10) between u and i (or, equivalently, the affirmation in (11)). Then
the mathematical model leads to the differential inclusion
L
di
dt
\in e(t) - Ri - NQi.
It is convenient to rewrite this differential inclusion in the form
di
dt
\in g(t) - NQi, (12)
where g(t) =
e(t)
L
- R
L
i.
Thus, the mathematical model (12) for the reseach circuit is differential inclusion of form (5).
This is an illustration for a simple problem, and much more complex problems of electrical circuit
theory can be modelled by using differential inclusion (5).
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 1
ACCURATE APPROXIMATED SOLUTION TO THE DIFFERENTIAL INCLUSION . . . 121
3. Auxiliary lemma. Let us denote E = NQx and E = TQx.
Lemma 1. For each c \in \BbbR n, the following two relations are equivalent:
c = a+ b, a \in E, b \in E, (a, b) = 0, (13)
a = PEc, b = PEc. (14)
Proof. (13) \Rightarrow (14). Suppose that the relation (13) holds, we need to prove that (14) holds also.
By using (13) and Definition 2, we obtain
(c - a, \xi - a) = (b, \xi ) \leq 0 \forall \xi \in E,
and, consequently, \| c - a\| \leq \| c - \xi \| for all \xi \in E, because \| c - \xi \| is the length of a side of the
triangle. This is opposite to right angle or obtuse angle of the triangle, while the length of the other
two sides are \| c - a\| and \| \xi - a\| . From here and (7) directly implies a = PEc. Similarly, we can
easily prove b = PEc. Hence, (14) is proved.
(14) \Rightarrow (13). From (14) and (7) we will have a \in E, b \in E and
(c - a, \xi - a) \leq 0 \forall \xi \in E, (15)
(c - b, \mu - b) \leq 0 \forall \mu \in E. (16)
Indeed, we consider a function z(t) = t\xi + (1 - t)a, where t \in [0, 1], and according to Proposi-
tion 1 we have z(t) \in E. Let
\varphi (t) = \| c - z(t)\| 2.
It is easy to see that the function \varphi (t) has the smallest value at t = 0, so \.\varphi (0) \geq 0. On the
other-hand, we have
\.\varphi (t) = 2
\bigl(
c - z(t), - \.z(t)
\bigr)
= 2
\bigl(
c - t\xi - (1 - t)a, a - \xi
\bigr)
=
= - 2(c - a, \xi - a) + 2t\| \xi - a\|
which gives
\.\varphi (0) = - 2(c - a, \xi - a) \geq 0.
Therefore, the inequality (15) is proved. Similarly, we can prove the condition (16) is also true.
Now, suppose b2 = c - a and since the set E \subset \BbbR n is a cone in the sense of Definition 1, so
0 \in E. By virtue of (15) we get
(b2, a) \geq 0. (17)
Moreover, since a = PEc \in E, 2a \in E. Then
(b2, a) \leq 0. (18)
By virtue of (17) and (18) we can obtain (b2, a) = 0. From here and (15) it immediately follows
that
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 1
122 T. H. NGUYEN
(b2, \xi ) = (c - a, \xi - a) \leq 0 \forall \xi \in E.
Next, by using (9), we get b2 \in E. In addition, from Definition 2 it follows that
(c - b2, \mu - b2) = (a, \mu ) \leq 0 \forall \mu \in E.
Thus, the inequality \| c - b2\| \leq \| c - \mu \| holds for all \mu \in E, that is, \| c - b2\| is the smallest
length from c to the set E. And, by virtue of (7) we easily get b2 = PEc.
To finish the proof of Lemma 1, we have to show b = b2. Indeed, since b, b2 \in E and using (16)
we get that the vectors b, b2 are satisfied such that
(c - b, b2 - b) \leq 0 and (c - b2, b - b2) \leq 0. (19)
Besides, we can see that
(c - b, b2 - b) = - (c - b2, b - b2) + (b2 - b, b2 - b) \geq 0. (20)
Now, from (19) and (20) it follows that b = b2.
Lemma 1 is proved.
4. Main result.
Theorem 1. Let Q \subset \BbbR n be a closed bounded convex set, (t0, x0) \in \BbbR \times Q, T > t0 and f :
[t0, T ]\times Q\rightarrow \BbbR n a continuous function satisfying conditions (3), (4). Then the Cauchy problem (6)
has a unique solution x(t). Moreover,
\bigm\| \bigm\| x(t) - yK(t)
\bigm\| \bigm\| \leq CeL(T - t0)
\surd
L
1\surd
K
, t \in [t0, T ], (21)
where yK(t) is a solution of problem (8) for a fixed parameter K.
Proof. First, we prove that when K is large enough, solution yK(t) of (8) can not go far beyond
the set Q, namely yK(t) satisfies \bigm\| \bigm\| yK - yK
\bigm\| \bigm\| \leq C
K
, (22)
where C is defined by (4) and yK = PQ(yK).
To prove (22), we put
\psi (t) =
\bigm\| \bigm\| yK(t) - yK(t)
\bigm\| \bigm\| 2, t \in [t0, T ].
We have
\.\psi (t) = 2
\bigl(
yK - yK , \.yK - \.yK
\bigr)
= 2
\Bigl(
yK - yK , f(t, yK) - K(yK - yK) - \.yK
\Bigr)
,
consequently,
\.\psi (t) = 2
\bigl(
yK - yK , f(t, yK)
\bigr)
- 2K\| yK - yK\| 2 - 2
\bigl(
yK - yK , \.yK
\bigr)
. (23)
Moreover, we can prove that
\bigl(
yK - yK , \.yK
\bigr)
= 0. Indeed, assuming the contrary, we conclude
that there exists t1 \in [t0, T ] such that\bigl(
yK(t1) - yK(t1), \.yK(t1)
\bigr)
= \alpha \not = 0. (24)
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 1
ACCURATE APPROXIMATED SOLUTION TO THE DIFFERENTIAL INCLUSION . . . 123
If \alpha > 0, then with \Delta t > 0 small enough, we obtain
yK(t1 +\Delta t) - yK(t1) = \.yK(t1) \cdot \Delta t+ o(\Delta t).
From (24) it follows that\biggl(
yK(t1) - yK(t1),
yK(t1 +\Delta t) - yK(t1)
\Delta t
\biggr)
= \alpha +
\biggl(
yK(t1) - yK(t1),
o(\Delta t)
\Delta t
\biggr)
. (25)
When \Delta t \rightarrow 0+, the right-hand side of (25) will go to \alpha > 0, while using inequality (15) the
left-hand side is smaller than 0. We thus arrive at a contradiction.
If \alpha < 0, then with \Delta t < 0 small enough, we have
yK(t1 +\Delta t) - yK(t1) = - \.yK(t1)( - \Delta t) + o(\Delta t),
consquently,\biggl(
yK(t1) - yK(t1),
yK(t1 +\Delta t) - yK(t1)
- \Delta t
\biggr)
= - \alpha +
\biggl(
yK(t1) - yK(t1),
o(\Delta t)
- \Delta t
\biggr)
.
Arguing as in case \alpha > 0, the contradiction is derived.
Therefore, from the definition of function \psi (t) and (23) we have
\.\psi (t) \leq 2C
\sqrt{}
\psi (t) - 2K\psi (t). (26)
Setting u(t) =
\sqrt{}
\psi (t), we get
\.u \leq C - Ku. (27)
From (27) we assert that u(t) =
\bigm\| \bigm\| yK(t) - yK(t)
\bigm\| \bigm\| satisfies
\.u = - Ku+ C - a(t), (28)
where a(t) a continuous, non-negative function. Also, since u(t0) =
\sqrt{}
\psi (t0) = 0, we have
u(t) =
t\int
t0
eK(s - t)
\bigl[
C - a(s)
\bigr]
ds.
For all t \in [t0, T ], we have
\bigm\| \bigm\| yK - yK
\bigm\| \bigm\| = \| u\| \leq C
t\int
t0
eK(s - t)ds \leq C
K
,
which proves (22). Consequently, when K is sufficiently large, the solution yK(t) of (8) is inside of
set Q1 defined by
Q1 =
\bigl\{
z \in \BbbR n : \mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(z,Q) \leq 1
\bigr\}
.
Then the solution yK(t) is bounded on [t0, T ].
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 1
124 T. H. NGUYEN
Moreover, from (4) and (22) it follows that\bigm\| \bigm\| \.yK\bigm\| \bigm\| \leq 2C. (29)
The important point is now that the conditions (22) and (29) show that the set of all solutions
(yK) of (8) is equicontinuous and bounded such that the Arzela compactness criterion implies that
there is a subsequence of (yK) (which for simplicity is denoted by (yK)n) converging uniformly on
[t0, T ] to some continuous x.
We are now going to prove that the limit function x takes its values in Q, and therefore it is a
solution of the system (6) in [t0, T ]. Using (22) and the fact that
\| yK - x\| \leq \| yK - yK\| + \| yK - x\| ,
the subsequence (yK) converges uniformly on [t0, T ] to the function x. From yK = PQ(yK) \in Q it
directly follows x \in Q.
Moreover, from (29) it follows that the solution x(t) satisfies the Lipschitz condition, conse-
quently, x(t) is an absolutely continuous function in [t0, T ]. Therefore, \.x(t) exists for almost all
t \in [t0, T ] and we have
t\int
t0
\.x(s)ds = x(t) - x0, t \in [t0, T ].
Next, it can be easily seen that \.yK converges weakly to \.x for K \rightarrow +\infty . Indeed,
t\int
t0
\.yK(s)ds = yK(t) - x0 - \rightarrow x(t) - x0 =
t\int
t0
\.x(s)ds for K \rightarrow +\infty .
Besides, by using (15), we have
yK - yK \in NQ(yK),
and, by using the properties NQ(\cdot ) in Proposition 1, we get
(yK - yK) \in NQ(yK).
Hence,
\.yK \in f(t, yK) - NQ(yK). (30)
By virtue of Proposition 2 and passing in (30) to the limit as K \rightarrow +\infty , we can obtain that the
function x satisfies (5).
Now, we are going to prove the uniqueness of solution of problem (6). Suppose that x1 = x1(t)
and x2 = x2(t) are two solutions of the differential inclusion
\.x \in f(t, x) - NQ(x), t \in [t0, T ].
The function t \mapsto \rightarrow \mu (t) :=
\bigm\| \bigm\| x1(t) - x2(t)
\bigm\| \bigm\| 2 is absolutely continuous. At any point t \in [t0, t1] of
differentiability of \mu we get
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ACCURATE APPROXIMATED SOLUTION TO THE DIFFERENTIAL INCLUSION . . . 125
\.\mu (t) = 2
\bigl(
x1(t) - x2(t), \.x1(t) - \.x2(t)
\bigr)
,
consequently,
\.\mu (t) = 2
\bigl(
x1(t) - x2(t), f(t, x1(t)) - f(t, x2(t))
\bigr)
- 2
\bigl(
x1(t) - x2(t), p1 - p2
\bigr)
(31)
for suitable points p1 \in NQ(x1(t)) and p2 \in NQ(x2(t)). Moreover, we have\bigl(
x1(t) - x2(t), p1 - p2
\bigr)
= -
\bigl(
p1, x2(t) - x1(t)
\bigr)
-
\bigl(
p2, x1(t) - x2(t)
\bigr)
\geq 0.
Dropping the last term
\bigl(
x1(t) - x2(t), p1 - p2
\bigr)
of (31) we get
\.\mu (t) \leq 2
\bigm\| \bigm\| x1(t) - x2(t)
\bigm\| \bigm\| \bigm\| \bigm\| f(t, x1(t)) - f(t, x2(t))
\bigm\| \bigm\| ,
and using the Lipschitz condition (3) we obtain
\.\mu (t) \leq 2L\mu (t).
Then there exists a continuous function b(t) \leq 0 such that \mu (t) satisfies the differential equation
\.\mu (t) = 2L\mu (t) + b(t).
So, we get
\mu (t) = \mu (t0)e
2L(t - t0) + e2Lt
t\int
t0
e - 2Lsb(s) ds \leq z(t0)e
2L(t - t0),
since eat > 0, t \geq t0, and e - asb(s) \leq 0.
Hence \mu (t0) :=
\bigm\| \bigm\| x1(t0) - x2(t0)
\bigm\| \bigm\| 2 = 0, we obtain x1(t) \equiv x2(t) for t0 \leq t \leq T, which show
the uniqueness of solution.
Now, we are going to prove the estimate (21). For this let us set
\rho (t) =
1
2
\bigm\| \bigm\| x(t) - yK(t)
\bigm\| \bigm\| 2, t \in [t0, T ].
Since x(t) is the solution of (6), there exists a vector v(t) \in NQx(t) such that
\.x(t) = f(t, x) - v(t), t \in
\bigl[
t0, T
\bigr]
. (32)
Then we have
\.\rho (t) =
\bigl(
\.x - \.yK , x - yK
\bigr)
=
\Bigl(
f(t, x) - f(t, yK) - v +K(yK - yK), x - yK
\Bigr)
,
consequently,
\rho (t) =
\bigl(
f(t, x) - f(t, yK), x - yK
\bigr)
+
\bigl(
v, yK - yK
\bigr)
+
+
\bigl(
v, yK - x
\bigr)
+K
\bigl(
yK - yK , x - yK
\bigr)
+K
\bigl(
yK - yK , yK - yK
\bigr)
. (33)
By virtue of (15) we get
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 1
126 T. H. NGUYEN\bigl(
yK - yK , x - yK
\bigr)
\leq 0.
From (2) it follows that the function \rho (t) can be estimated by
\rho (t) \leq
\bigl(
f(t, x) - f(t, yK), x - yK
\bigr)
+
\bigl(
v, yK - yK
\bigr)
.
Combining this with condition (3) gives
\rho (t) \leq L\| x - yK\| 2 + \| v\| \| yK - yK\| (34)
(here we can use the inequality \| x - yK\| \leq \| x - yK\| , because\| x - yK\| is the length of a side of
the triangle, which is opposite to right angle or obtuse angle of the triangle).
In addition, since yK is the solution of equation (8), if yK \in Q, then \rho (t) is estimated as follows:
\rho (t) \leq L\| x - yK\| 2 (35)
(because yK = yK ).
We now consider the value of \rho (t) in the case yK \not \in Q.
As the next step, to evaluate the function \rho (t) we evaluate \| v\| in the right-hand side of (34).
We first show that
\.x \in TQx(t) and ( \.x, v) = 0 for almost all t. (36)
Let t be a point in [t0, T ] where \.x(t) exists. Given any \varepsilon > 0 small enough and 0 \leq s < \varepsilon , we
have
x(t+ s) \in Q\Rightarrow \.x+(t) \in TQx(t), (37)
x(t - s) \in Q\Rightarrow - \.x - (t) \in TQx(t). (38)
Indeed, since
\.x+(t) = \mathrm{l}\mathrm{i}\mathrm{m}
\delta \rightarrow 0+
x(t+ \delta ) - x(t)
\delta
, (39)
by (2) and for every \xi \in NQx(t) we obtain\bigl(
x(t+ \delta ) - x(t), \xi
\bigr)
\leq 0. (40)
Then, from (39), (40) and (9) it follows condition (37) and also \.x(t) \in TQx(t). Similarly, (38)
is also proved. Combining (37), (38) and Definition 2 of adjoint cone, we obtain: on the one
hand ( \.x, v) =
\bigl(
\.x+(t), v
\bigr)
\leq 0, on the other hand ( - \.x, v) = -
\bigl(
\.x - (t), v
\bigr)
\leq 0, and, consquently,
( \.x, v) = 0, that is, (36) is proved.
By using (32), (36) and Lemma 1, we obtain that the vector \.x is the projections of f(t, x) on the
set TQx(t). Since v(t) = f(t, x) - \.x(t) (see (32)), 0 \in TQ(x(t) and by virtue of (7), we get
\| v\| = \| f - \.x\| \leq \| f(t, x)\| .
From this fact and the condition (4) it follows
\| v\| \leq C. (41)
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 1
ACCURATE APPROXIMATED SOLUTION TO THE DIFFERENTIAL INCLUSION . . . 127
Therefore, from (34), (22) and (41) it follows that
\rho (t) \leq L\| x - yK\| 2 + C2
K
. (42)
From (35) and (42) we have (42) for two case: y \in Q and y \not \in Q.
Since (42), the absolutely continuous function \gamma (t) :=
\bigm\| \bigm\| x(t) - yK(t)
\bigm\| \bigm\| 2 satisfies the equation
\.\gamma (t) = 2L\gamma (t) + 2
C2
K
- p(t),
where p(t) is a non-negative continuous function. From the initial condition in (6) and (8) we have
\gamma (t0) = 0. Then we obtain
\gamma (t) = 2
t\int
t0
e2L(t - s)
\biggl[
C2
K
- p(s)
2
\biggr]
ds.
Finally, with t \in [t0, T ] we get
\bigm\| \bigm\| x(t) - yK(t)
\bigm\| \bigm\| 2 \leq 2C2
K
t\int
t0
e2L(t - s)ds \leq C2
KL
e2L(T - t0),
which leads to the estimate (21).
Theorem 1 is proved.
5. Conclusions. Some of technical problems have been mathematically modeled by the dif-
ferential inclusion (6). Nevertheless, finding exact solutions to the differential inclusion (6) is very
difficult. Therefore, studying the ordinary differential equation in form (8) has high scientific interest.
Firstly, we can use the ordinary differential equation (8) to obtain the approximated solutions of the
differential inclusion. Secondly, we can find and present the solutions of (8) by using such software
as Mathematica, Matlab, and Maple.
References
1. A. Wazewski, On a condition equivalent to the quota equation in contingent, Bull. Pol. Acad. Sci., Ser. Math., Astron.
and Phys., 9, № 12, 865 – 867 (1961).
2. A. Marchaud, Sur les champs de demi-cones convexes, Bull. Sci. Math., 62, № 8, 229 – 240 (1938).
3. S. C. Zaremba, On paratingent equations, Bull. Sci. Math., 6, № 5, 139 – 160 (1936).
4. A. F. Filippov, Differential equations with a discontinuous right-hand side, Science, Moscow (1985).
5. R. V. Nesterenko, B. N. Sadovskii, Forced vibrations of two-dimensional cone, Autom. and Remote Control., № 2,
181 – 188 (2002).
6. T. H. Nguyen, On auto-oscillations in the generalized system “predator-prey”, Systems and Inform. Technology, 2,
22 – 24 (2013).
Received 27.01.18,
after revision — 20.08.19
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 1
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| id | umjimathkievua-article-889 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:05:58Z |
| publishDate | 2021 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/3d/ead24e1a44391ac4bc53c5ed1a4cec3d.pdf |
| spelling | umjimathkievua-article-8892025-03-31T08:49:21Z Accurate approximated solution to the differential inclusion based on the ordinary differential equation Accurate approximated solution to the differential inclusion based on the ordinary differential equation Accurate approximated solution to the differential inclusion based on the ordinary differential equation Nguyen, T. H. Nguyen , T. H. Differential inclusion differential equation normal cone projection Differential inclusion differential equation normal cone projection UDC 517.9 Many problems in applied mathematics can be transformed and described by the differential inclusion $\dot x\in f(t, x)-N_Qx$ involving $N_Qx,$ which is a normal cone to a closed convex set $Q \in \mathbb R^n$ at $x\in Q.$ The Cauchy problem of this inclusion is studied in the paper. Since the change of $x$ leads to the change of $N_Qx,$ solving the inclusion becomes extremely complicated. In this paper, we consider an ordinary differential equation containing a control parameter $K.$ When $K$ is large enough, the studied equation gives a solution approximating to a solution of the inclusion above. The theorem about the approximation of these solutions with arbitrary small error (this error can be controlled by increasing $K$) is proved in this paper. &nbsp; UDC 517.9 Точний наближений розв’язок диференцiального включення на основi звичайного диференцiального рiвняння Багато задач у прикладній математиці можна трансформувати та описати за допомогою диференціального включення $\dot x\in f(t, x)-N_Qx,$ в яке входить $N_Qx,$ що є нормальним конусом для замкненої опуклої множини $Q \in \mathbb R^n$ у точці $x\in Q.$ У цій роботі вивчається задача Коші для такого включення. Оскільки зміна $x$ обумовлює зміну $N_Qx,$ розв'язання цього включення стає надто складним. Тут розглядається звичайне диференціальне рівняння, яке містить керуючий параметр $K.$ Коли $K$ є достатньо великим, це рівняння дає розв'язок, який наближає розв'язок досліджуваного включення. Доведено теорему про наближення цих розв'язків з будь-якою точністю (відповідна похибка контролюється за допомогою зростання $K$). Institute of Mathematics, NAS of Ukraine 2021-01-22 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/889 10.37863/umzh.v73i1.889 Ukrains’kyi Matematychnyi Zhurnal; Vol. 73 No. 1 (2021); 117 - 127 Український математичний журнал; Том 73 № 1 (2021); 117 - 127 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/889/8908 |
| spellingShingle | Nguyen, T. H. Nguyen , T. H. Accurate approximated solution to the differential inclusion based on the ordinary differential equation |
| title | Accurate approximated solution to the differential inclusion based on the ordinary differential equation |
| title_alt | Accurate approximated solution to the differential inclusion based on the ordinary differential equation Accurate approximated solution to the differential inclusion based on the ordinary differential equation |
| title_full | Accurate approximated solution to the differential inclusion based on the ordinary differential equation |
| title_fullStr | Accurate approximated solution to the differential inclusion based on the ordinary differential equation |
| title_full_unstemmed | Accurate approximated solution to the differential inclusion based on the ordinary differential equation |
| title_short | Accurate approximated solution to the differential inclusion based on the ordinary differential equation |
| title_sort | accurate approximated solution to the differential inclusion based on the ordinary differential equation |
| topic_facet | Differential inclusion differential equation normal cone projection Differential inclusion differential equation normal cone projection |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/889 |
| work_keys_str_mv | AT nguyenth accurateapproximatedsolutiontothedifferentialinclusionbasedontheordinarydifferentialequation AT nguyenth accurateapproximatedsolutiontothedifferentialinclusionbasedontheordinarydifferentialequation |