1 Introduction

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.

2 Preliminaries

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

3 Basic assumptions, existence and uniqueness results

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

4 Optimal control

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

5 Time optimal control

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

6 Conclusion

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.

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