Optimal control of stochastic system with Fractional Brownian Motion

1.   Introduction

In the biological population system, it is affected by a variety of external factors, which they are likely to change the population's amount. In order to control the development of biological population reasonably, it is necessary to select appropriate control variables and establish reasonable performance indicators to study the optimal control of stochastic population systems. In the paper, a nonlinear population system equation (the harvesting equation) is discussed. The typical harvesting system can be described in the form:

where $(r, t)\in Q$, $t\in(0, T)$, $r\in(0, A)$, $0 < A < \infty$. Since the system with the external factors, we are going to introduce the stochastic harvesting equations with Fractional Brownian Motion (FBM) as follows:

where $p(r, t)$ is the density of the population of age $r$ at time $t$, $A$ is the life expectancy, and $p(A, t) = 0$. $\lambda_{1}$ is the average mortality ratio of the population of age $r$ at time $t$, $\beta_{1}$ is the average fertility ratio of the population of age $r$ at time $t$, $u_{1}(r, t)$ is harvesting effort function, which is the control variable in the model and satisfies: $0\leq u(t)\leq u_{max}$.$f_{1}(r, t, P(t))+g_{1}(r, t, P(t))\frac{d B_{H}}{d t}$ is the stochastic perturbation, effecting of external environment on the population system, such as earthquakes, emigration, impacts of extra terrestrial objects, and so on.

The stochastic model has aroused concern in the recent years. Abel Cadenikls ^[1] used a stochastic maximum principle for systems with jumps, with applications to finance systems. Zhang ^[2,3,4,5,6] investigated the stability of numerical solutions for the stochastic age-dependent system. Zhang ^[7,8] provided the different methods for the numerical solutions in the stochastic system with age-dependent. Pei ^[10] focuses on asymptotic mean-square boundedness of several numerical methods applied to a class of stochastic age-dependent population equations with Poisson jumps. Emel Savku et al. ^[11,12] illustrated that they contributed to modern OR by hybrid (continuous-discrete) dynamics of stochastic differential equations with jumps and the optimal control. However, compared with stochastic system driven by the classical Brownian motion, FBM is a family of centered Gaussian random process indexed by the Hurst parameter $H\in(0, 1)$ with continuous sample paths. Some special kinds of dynamical systems require both Wiener process and FBM to model their dynamics. Meanwhile, few has been done because classical methods to solve stochastic problems can not be used directly, since Fractional Brownian Motion (FBM) is not a semi-martingale and not a Markov process. Ma ^[13] developed a numerical scheme and show the convergence of the numerical approximation solution to the analytic solution for stochastic age-dependent population equations with FBM. Kloeden ^[14] used the multilevel Monte Carlo method introduced by Giles ^[15] to stochastic differential equations with Fractional Brownian Motion of Hurst parameter $H > 1/2$ and achieved a prescribed root mean square error of order $\varepsilon$ with a computational effort of order $\varepsilon^{-2}$. Duncan ^[16] discussed the solutions semi-linear stochastic systems with FBM. Zhou ^[17] investigated the stability for the delayed neural networks with FBM.

On the other hand, optimal control problems have also attracted wide attention, due to their several applications in population system, economic system, finance system ^{[11,12,18,19,20]}. Luo ^[18] studied optimal harvesting control problem for an age-dependent competing system of n-dimension competing species. Zhao ^[19] and Chen ^[20] talked about optimal control of different stochastic system. He ^[21] investigated optimal harvesting problem for age-structured species. However, an optimal control problem requires the minimization of a criterion function of the states and control inputs of the system over a set of admissible control functions ^[22], which creates huge troublesome. Stochastic optimal control problem driven by FBM is the bottleneck problem. In this paper, all the previous fields are combined to consider the optimal control problem of stochastic harvesting population system with FBM, the necessary and sufficient conditions for the optimization are obtained, and the example for the obtained theoretical results is illustrated. We provide below a brief summary of our results.

$\bullet$ We introduce the fractional Brownian noise into a class of stochastic harvesting population system and establish necessary as well as sufficient conditions of optimal control, which has not been studied before;

$\bullet$ Using the stochastic maximum principle, Hamilton function and $It\hat{o}$ formula to stochastic harvesting equations with Fractional Brownian Motion and study the optimal control of the system;

$\bullet$ The example is presented, and it supports our theoretical results.

The paper is divided into five sections. The assumption, notations and some basic definition are given in section 2. In section 3, we establish necessary as well as sufficient conditions of optimal control. In Section 4, an example is provided to illustrate the theoretical results. The conclusions are given in section 5.

2.   Preliminaries of the problem

Definition 2.1. (Fractional Brownian Motion)

For $0 < H < 1$ Fractional Brownian Motion (FBM) $B^{H} = {B^{H}(t), t\in \bf{R}}$, for Hurst parameter $H\in(0, 1)$ is the Gaussian process with mean 0 for all $t: E[B^{H}] = 0$ and covariance

We take $B^{H}(0) = 0$. For $H = \frac{1}{2}$, $B^{\frac{1}{2}}$ is standard Brownian Motion.

Definition 2.2.

Let $f(x)$ is continuous functions, its $H$ order fractional derivative is defined as:

let $H > 0$,

such that

Where $D$ is differential operator, $H$ is Hurst parameter.

Lemma 2.1

Let $f(t)$ is continuous functions, such that

In this paper, we discuss stochastic optimal control problems driven by fractional Brownian motion (fBm), and consider the following stochastic control harvesting population system with FBM interval [0, A]:

Getting motivation from the above facts, we discuss the optimal control problems in the system $(2.1)$ is

Were $E(\cdot)$ is expectation operator.

As the standing hypotheses, we always assume that the following conditions are satisfied:

$(A_{1}) \; \lambda\in C(Q \times R^{+})$ is nonnegative measurable function, where $\int_0^r\lambda(\tau)d\tau < +\infty$, $r < A$, $\int_0^A\lambda(\xi)d\xi = +\infty$.

$(A_{2}) \; \beta\in C(Q\times R^{+})$ is nonnegative measurable function, where $\sup\limits_{\xi\in(0, A)}\int_0^A\beta(\xi)d\xi\leq 1$.

$(A_{3}) \; u\in U_{ad} = U$ which is non-empty convex subset, where $U = L^{2}(Q)$.

$(A_{4})$ All $x_{k}, y_{k}\in R^{n}$, where $\|x_{k}\|\vee\|y_{k}\|\leq d(k = 1, 2)$, and there exists a constant $c_{d} > 0$, such that

$(A_{5})$ All $x, y\in R^{n}$, and there exists a constant $L > 0$, such that all $t\in [0, T]$ satisfied

$(A_{6})$ Let $f(t, y, u), g(t, y, u), h(t, y, u)$ are linear functions, we introduce

The Hamiltonian function is given by

Now, we introduce the adjoint equation for our problem. The adjoint equation can be written as:

Note that the couple$(q, \gamma)$ is the adjoint process corresponding to the stochastic system $p(r, t)$. The adjoint equation admits one and only one $\mathscr{F}_{t}-$ adapted solution $(q, \gamma)$, where $L(t, y, u) = -uy^{u}$.

Moreover, to ensure that the above stochastic differential equation make sense, we shall consider only those $\mathscr{F}_{t}-$ predictable control processes $u: u\in U_{ad}$ that satisfy

3.   The main results

This is the main result of this paper, in this section, we derive necessary conditions for a control to be optimal.

Lemma 3.1. $J$ is G$\hat{a}$teaux-differentiable with differential given by

To obtain Equation (3.1), we use It$\hat{o}$ formula ^[24], Gronwall's inequality ^[25], and equivalently the formula of integration by parts.

The above equation may be rewritten as

Where we denote for every $u\in U$; $t\in[0, T]$:

We have to consider the following tow cases:

Case 1: For every $u\in U$: $E[\bar{R}_{t}^{u}]\leq[R_{t}^{u}]$.

Case 2: For every $u\in U$: $E[\bar{R}_{t}^{u}]\geq[R_{t}^{u}]$.

Let us consider the function $\tilde{H}$: $[0, T]\times \Omega \times U\rightarrow R$ defined by

We note that $H(t, u)$ is convex.

Theorem 3.1. If case 1 hold, then a necessary condition for a control $u^{*}$ to be optimal for Problem $(2.2)$ is that for every $u^{*}\in U$:

On the other hand, if case 2 holds, then inequality $(3.2)$ is a sufficient condition of optimality for a control $u^{*}$.

Proof: Here, we apply previous knowledge and methods to obatain the results, such as Young inequality ^[2], Itô integral (Lemma 2.1) and the Hölder, Burkholder-Davis-Gundy (BDG) inequalities ^[22]. According to (2.3), $u^{*}$ is an optimal control if and only if $u^{*}\in U$:

In case 1, we see that for every $u\in U$:

Thus, in case 1 and in conjunction with (3.2), a necessary condition for a control $u^{*}$ to be optimal is that $\forall u\in U$

Which is equivalent to (3.1).

On the other hand, in case 2, for every $\forall u\in U$:

Thus, in case 2, a su1cient condition for a control $u^{*}$ to be optimal is that (3.6), or equivalently (3.2), holds for every $u^{*}\in U$.

4.   An example

For convenience, we still adopt the notation introduced in Section 2.

Example 4.1. Let the admissible control domain be [0, 1], consider the following optimal control problem

subject to

Here, $B_{H}$ stands for Fractional Brownian Motion (FBM). Take $T = 1$, $A = 1$ in Eq. (4.1). We can set this problem in our formulation by taking $H = L^{2}([0, 1]\times[0, 1])$, $V = W_{0}^{1}([0, 1])$(a Sobolev space with elements satisfying the boundary condition above), $f(r, t, P) = 2pt$, $\lambda(r, t) = \frac{1}{1-r^{2}}$, $\beta(r, t) = t^{2}$, $g(r, t, P) = p$, and $p(r, 0) = exp(-\frac{1}{1-r})$.

Clearly, the operators $f$, $g$ and $\lambda$ satisfy the assumption.

To solve this problem, we must write down Hamiltonian function ^[26]:

And adjoint equation ^[22] is

Solving equation (4.1) and (4.2), for any admissible control $u(t, a)\in\mathcal {U}_{ad}$ and $t\in[0, 1]$, if we have the results that the equation (3.2) is valid, which is the necessary and sufficient conditions for optimality in this control problem.

The corresponding Hamiltonian

and

If $u^{*}(r, t)$ is optimal, the necessary condition for $(4.1)$ is

Moreover, because of the solution for state equation $p(r, t)$ is the function of $u(r, t)$, $\lambda(r, t)$, $f(r, t, p)$ and $g(r, t, p)$ satisfying the assumption, we can conclude that

So the Eq. (4.3) is also the sufficient condition for Eq. (4.1).

5.   Conclusions

Existence and optimal control results of the stochastic model with Fractional Brownian Motion (FBM) is studied in this paper. Firstly, we introduce the fractional Brownian noise into a class of stochastic harvesting population system and establish necessary as well as sufficient conditions of optimal control, which has not been studied before. Secondly, Using the stochastic maximum principle, Hamilton function and $It\hat{o}$ formula to stochastic harvesting equations with Fractional Brownian Motion and study the optimal control of the system. finally, the obtained theoretical results are verified by an illustrative example. As further direction, researchers are invited to investigate the optimal control problem for stochastic model by including G$\hat{a}$teaux-differentiable with differential.

Acknowledgments

The authors would like to thank the editor and reviewers for their very helpful suggestions which greatly improved this paper. The research was supported by the Funding scheme of the young backbone teachers of Henan's higher education institutions (No.2015GGJS-206)(China), and the Program for Natural Scientific Research Foundation of Ningxia (2020AAC03062).

Conflict of interest

The authors declare that they have no conflict of interest.