2022
273
11072021
College Kannauj, India
College Kannauj, India
Abstract: The objective of our paper is to investigate the optimal control of semilinear population dynamics system with diffusion using semigroup theory. The semilinear population dynamical model with the nonlocal birth process is transformed into a standard abstract semilinear control system by identifying the state, control, and the corresponding function spaces. The state and control spaces are assumed to be Hilbert spaces. The semigroup theory is developed from the properties of the population operators and Laplacian operators. Then the optimal control results of the system are obtained using the ..-semigroup approach, fixed point theorem, and some other simple conditions on the nonlinear term as well as on operators involved in the model.
Keywords: population dynamics, diffusion, optimal control, Gronwall’s inequality.
Articles
New discussion concerning to optimal control for semilinear population dynamics system in Hilbert spaces
Recepción: 11 Julio 2021
Revisado: 15 Diciembre 2021
Publicación: 25 Febrero 2022
Let us consider 
, a bounded domain in 
along the smooth boundary region 
We considered that a biological population is independent to move in the 
environment. Let 
be the population dissemination of human beings of age 
at location 
for time 
We considered that the population flow is 
where ∇ is the gradient vector with respect to the spatial variable 
and 
is a constant. The life expectancy of individual is denoted by 
is known as the natural mortality rate, and 
is known as the natural fertility rate corresponding to individual of age 
at location 
for the time 
In this paper, we have considered that the population distribution 
is differentiable in variables 
and 
. Moreover, the natural fertility and mortality rate depends only on age.
Let 
be a Hilbert space, and let 
be a function space. The symbol 
is norm in 
We have considered the following semilinear population dynamics model with nonlocal birth process:
where ∆ is Laplacian operator, 
is the control function in 
, which is a supply or removal of population, and 
is the characteristic function of 
is a bounded linear operator, the nonlinear function 
represents an infusion of population due to some natural or unnatural reasons, and 
is the initial population distribution. In some models, 
is assumed instead of zero population at the boundary region, where 
denotes the exterior normal derivatives on 
and 
means that the population inflow or outflow of the region 
is zero.
In recent years, the existence and uniqueness of mild solution, optimal control, timeoptimal control approximate control, and exact control for fractional-order, integer-order, integro –differential system, neutral system, etc. have been studied by many researcher’s articles [1–3,7–9,11–24,26–30,32–35]. In [6], the authors obtained the existence and optimal control results using Krasnoselskii’s fixed point theorem and minimizing sequence concept for the second-order stochastic differential equations having mixed fractional Brownian motion. In [22], the authors discussed the existing result for the mild solution and optimal control for fractional-order 
(1,2] semilinear control system by using .order sine and cosine family theory, Banach fixed point theorem, and certain assumptions on nonlinearity.
Motivations and contributions
In [1, 11, 19], the authors discussed the optimal control for the population of dynamics by different techniques. Taking the consideration of ideas from the above literature review and motivated by the fact, the optimal control of semilinear population dynamics system (1) is studied in the semigroup framework approach.
We have converted the population dynamic system into a abstract control problem.
The mild solution is defined in terms of integral equation.
We have discussed the existence and uniqueness of the assumed system by employing the contraction principle.
Optimal control results are obtained using Cauchy–Schwarz’s inequality and Gronwall’s lemma.
The structure of this article is as follows. Section 1 discusses some literature related to population dynamics and optimal control theory. Some new notations, important facts, lemmas, vital definitions, and theoretical results are recalled and problem formulation is done in Section 2. In Section 3, we assumed some conditions and proved the existence and uniqueness of mild solution for system (1). Section 4 deals with the optimal control for the population dynamics system with diffusion. Finally, in the last Section 5, we discussed the time optimal control.
Definition 1. (See [31].) A one-parameter family 
of bounded linear operators mapping the Hilbert space 
into itself is said to be a strongly continuous cosine family if and only if
(i) 
(ii) 
(iii) 
(iv) 
is a strongly continuous cosine family in 
, then 
, is the one-parameter family of operators in 
defined by
The infinitesimal generator of a strongly continuous cosine family 
is the operator 
defined by
where 
is defined as 
is a twice continuously differentiable function of 
Definition 2. The characteristic function of a subset
of a set 
is a function 
defined as
We are focusing here on the subsequent two linear population dynamics models with diffusion from the literature.
In [5], the behavior of the semigroup of the following linear population dynamics model with Dirichlet boundary condition type was studied:
Suppose that 
is defined as
with domain of 
denoted by
and 
is the diffusion constant.
In this paper, it has been proven that 
is the infinitesimal generator of a strongly continuous semigroup (
semigroup) 
This enables us to reformulate system (2) in the following abstract form:
Note that by Theorem 2 of [5] the 
semigroup 
generated by 
is compact when 
otherwise it is not compact because the 
semigroup generated by the population operator
is not compact when 
Similarly, in [10] the formulation of 
semigroup for the following linear population dynamics system with diffusion in Neuman boundary condition type is studied:
We now determine 
as
In that paper, it is shown that the operator 
is the infinitesimal generator of a 
semigroup 
using 
dissipativity of the diffusion operator ∆ and the population operator without diffusion. That means the operator 
is characterized in the following way.
Let the operator 
be defined by
where
Then it is shown that 
is linear and .-dissipative.
Similarly, let the operator 
be defined by
Where
Then it is shown that 
is linear and 
dissipative operator. From the above two operators 
is presented as
where 
Therefore, 
is linear and 
dissipative since a sum of two 
dissipative operators is 
dissipative. Hence, 
is an infinitesimal generator of the 
semigroup 
[25].
Now, we reformulate the abstract form of system (2) as
This article deals mainly the optimal control of (1) using 
semigroup theory.
Now, we consider system (1) as an extension of model (2) to semilinear population dynamics control system with finite time interval and fixed life expectancy 
Therefore, the abstract form of the population dynamics system (1) can be rewritten as follows:
Wher
Assume that the integral cost function is presented as
subject to
is measurable, 
is defined as admissibleset set.
Also, it is clear that 
is nonvoid. 
is closed, convex, and bounded. Also, for every 
Let 
represent the mild solution, and let 
represent the control, then 
is a set of state-control pairs 
which are admissible. Hence, the optimal control problem is given by
Determine a pair 
such that
The linear system corresponding to (3) is given by
Then we will make use of the 
semigroup 
operator for the mild solution of (3) defined below:
Suppose 
is mild solution of (3), then we define the population distribution 
in the following manner:
Here we give the verification of the natural fertility rate of the population in relation to the mild solution of (1):
On the other hand, when we put 
in the integral equation (5), we have
From (5) and (6) we establish the following conditions:
which can be simplified as
In this article the following assumptions are important to obtain existence and uniqueness results.
For the biological significance of assumption (A1), one can refer to [1,11,19].
The interpretation of the integral condition in (A1)(ii) is that each individual dies before age 
. Hence, the approximate controllability could not hold at age 
.
Lemma 1. (See [28].) According to the control u, the mild solution of (3) is assumed to be (4) on [0,b.. Then there exists 
such that
Proof. Let 
described by (4) be the mild solution (3). Provided that 
by applying assumption (A5) and Cauchy–Schwartz inequality we have
Then
By referring Gronwall’s inequality we have 
Theorem 1. Assume that (A1).(A5) are fulfilled, then corresponding to each control function 
, system (3) has a unique mild solution in 
.
Proof. 
and consider 
the Banach space of all continuous functions from [0,b] to 
along with supremum norm.
We now define 
such that
Clearly, 
is well defined. We need to verify that (4) represents the mild solution on [0,b1]. It is enough to verify that 
has a unique fixed point in 
By applying the fixed point technique we are able to verify this discussion.
Assume that 
. Assume that 
denotes the closed ball in 
with radius 
:
Clearly, 
is bounded and closed subset of 
. For any 
, we have
For positive right-hand side,
Therefore, if (7) holds, the 
maps 
into itself.
Now, we will prove that 
is a contraction on 
. With the help of definition of 
, there exists 
such that 
. Hence, condition (A3) yields
Following the same method 
times, we have
Conclusion (8) is achieved easily by continuing . times the above procedure. There exist 
such that 
Hence, 
is contraction for sufficiently large 
. Using contraction principle in 
has a unique fixed point 
, which represents mild solution of (3). In the same manner, we will show that (4) is the mild solution on 
, 
Continuing this process, we will achieve that (4) represents mild solution on the maximal existence interval 
Our aim is here to verify the uniqueness. Let us consider any two solutions 
and 
, we have
By applying Gronwall’s inequality we conclude that 
The existence of optimal control for the considered system with diffusion is discussed in this section.
Define
with the following conditions:
(C1) 
is Borel measurable.
(C2) For all 
, the function 
is sequentially lower semicontinuous on 
.
(C3) For 
, convexity is fulfilled by 
.
(C4) There exists 
and 
with the condition 
.
Theorem 2. If (C1).(C4) and (A1).(A5) hold, then for system (3), there exi sts at least an optimal pair 
and
Proof. If greatest lower bound of 
is 
, then, obviously, we obtain the result. So, we will assume that greatest lower bound of 
as 
From above conditions (C1)–(C4) it is clear that 
With the help of greatest lower bound, there exist a sequence of state-control pair 
such that 
as minimizing sequence. We know that 
is a reflexive separable Banach space, 
is a bounded subset of 
and also 
so there is relabeled sequence {u
such that 
(weakly converges as 
in 
As we know that the admissible set 
is bounded, closed, and convex, so Mazur’s lemma forces us to conclude that 
Now, let us assume that corresponding to sequence of controls 
the sequence of solutions of system (3) be given by 
that is,
From definition of 
we can easily prove that there exists 
for which
Let corresponding to control 
denotes the mild solution, which is given by
Using (A3) and Cauchy–Schwarz inequality, we obtain
By referring Gronwall’s lemma
where 
is a constant.
Because 
is strongly continuous,
From (9) and (10) we conclude that
This implies that 
(all converge strongly).
By referring [4, Thm. 2.1], under (C1)–(C4), assumptions of Balder are fulfilled. So, with the help of theorem of Balder’s, we get
in the weak topology and strong topology of 
(11) is sequentially lower semicontinuous. Hence, on 
with condition (C4) and weakly lower semicontinuity of 
Greatest lower bound of 
is achieved at 
Let us assume two distinct members 
with the condition 
such that 
Define the transition time 
satisfying 
The time value 
of 
for which there exists an admissible control satisfying 
is said to be the optimal time, and a control 
such that 
is said to be the time optimal control. Now, we need to verify that there exist admissible control such that 
with respect to 
Theorem 3. Assume that (A1).(A5) are fulfilled, then corresponding to 
there exist a time optimal control.
Proof. By referring the methodologies presented in [14, 21, 35], with certain modifications, we present the existence of time optimal control.
Consider
Then there is a nonincreasing sequence 
which converges to 
and we consider 
a sequence of controls having state trajectories as
satisfying 
Note that 
since 
is bounded, weakly sequentially compact. Therefore, we fix 
Let 
has the dual space as 
So, the dual pair 
is given by
In view of (A5) and Lemma 1, we have
In the right-hand side of equation (12), the first and third term converge strongly to 
and 0, respectively. The convergence of second term is given with the help of weak convergence of 
and assumption (A4), we achieve
strongly in 
it becomes
Since 
was arbitrary, we have
Thus, the time optimal control is 
, and the corresponding trajectory is 
The primary focus of our article is to prove the optimal control of the semilinear population dynamics system using ..-semigroup theory. The main outcomes are proved by applying Lipschitz continuity, Banach contraction principle, and well-known Gronwall’s inequality. One may extend the optimal control outcomes of the assumed system, nonlocal and impulsive with suitable modifications. We can also study the optimal control of assumed population dynamics systems in stochastic and fractional-order systems by utilizing stochastic and fractional calculus.
