Almost periodic dynamics for a delayed di ﬀ erential neoclassical growth model with discontinuous control strategy

: In this study, we are concerned with the existence and exponential stability issue of a delayed di ﬀ erential neoclassical growth model with discontinuous control strategy. By employing the Filippov ’ s theory and dichotomy theory, together with the Lyapunov functional method, novel criteria on existence and exponential stability are established for the addressed model. The established theoretical results extend and supplement the related results in the existing literature. Moreover, a simulation example is presented to verify the practicability of the proposed results.


Introduction
Day [1,2] used the nonlinear delay differential equation of the following form: x t αx t βx t τ e , n δ x tτ (1.1) where α β δ , , , and τ are positive constants, to describe the appearance of complex dynamics even though the economic structure is very simple.Economically, ( ) x t is the capital per capita at time t, α stands for the depreciation ratio of capital, δ suggests an intensity of a "negative effect" given rise by increasing condensation of capital, and τ means the delay in the production process.The dynamical behaviors of equation (1.1) have been widely investigated in [3][4][5], where this model was called as the differential neoclassical growth model.
The theory and application of the differential neoclassical growth model have made prominent progress in the last two decades and drawn considerable attentions among researchers (e.g., [6][7][8] and references therein).Many meaningful and important results on the dynamical properties including the well-posedness, positive (almost) periodic solutions, persistence, attractivity, and stability of equation (1.1) and its generalizations (in particular, to variable parameters and stochastic equations) have been obtained.For example, Wang and Chen [9] studied the existence and ultimate boundedness in mean for a delayed differential neoclassical growth model in random environments.Long and Wang [10] established a criterion ensuring the existence and global exponential stability of an almost periodic non-autonomous delayed differential neoclassical growth model with an oscillating death rate.For more interesting dynamical results for differential neoclassical growth models, one can refer [11][12][13][14][15] and references therein.
However, in real economic activities, under the influence of inventory changes, capital flows, financial fluctuations, and even financial crises, issues such as commodity pricing decisions and monetary policy regulation in modern economic and financial markets are highly susceptible to instantaneous impacts from external reality factors, it is necessary to implement some economic regulation policies under such circumstances.Therefore, the research on economic dynamics with control strategy is an interesting and important research topic in mathematical economics, which is in accordance with the optimal management of economic activities, according to the economic situation of different stages, the relevant departments will adopt different control strategies, even discontinuous.On the other hand, for most successful applications of various models in the real world, stability is usually a prerequisite and the convergence dynamics of models depends mainly on their parametric configuration (e.g., [16,17]).Until now, fewer results have been reported on the dynamics of positive almost periodic solutions of delayed differential neoclassical growth model with discontinuous control strategy.Inspired by the aforementioned discussion, in this work, we shall investigate the existence and exponential stability problems of positive almost periodic solutions for a discontinuous differential neoclassical growth model of the following type: where ( ( )) x t is the discontinuous control function and ( ) b t stands for the technological coefficient like control strength at time t.In order to ensure the sustainability of the economy, the economy is prohibited from controlling, which can also be made to be zero to reflect this fact.The other variables and parameters admit the same economic significance as described in (1.1) but it needs to consider the influence of time factor.
In consideration of the above analysis, the main purpose of this work is to study the almost periodic dynamics of delayed differential neoclassical growth model (1.2) with discontinuous control strategy.Compared with the results of previous literature, the distinct characteristics of this work are summarized as follows: (1) This work first considers the exponential stability problem of solutions for a delayed almost periodic differential neoclassical growth model with discontinuous control strategy, which expands the scope of the study.(2) Since the control strategy employed herein is discontinuous, by employing the concept of Filippov solutions and the set-valued mapping theory, a unified criterion ensuring the existence and exponential stability of almost periodic solutions for the considered discontinuous model is established.(3) The exponential stability criteria are independent of time delay, which enhanced the anti-interference ability and robustness against time delays.
The structure of this work is outlined as follows.Section 2 presents some preliminaries, Section 3 establishes the existence and exponential stability of solutions for the considered almost periodic model.Section 4 provides a numerical example to verify the feasibility of the established theoretical results.

Preliminaries
Given that the economic explanation of model (1.2), just positive solutions are meaningful and thus admissible, and so, we give the following initial condition: be the maximal right interval of existence of ( ) is decreasing with the range ( ) + n n 0, and increasing with the range ( ) + ∞ n n, , and there is a unique In addition, one can readily see that Next we always assume that the controlling function in model (1.2) admits the following discontinuous characteristics.
Definition 2.1.(Function class C) Let us call C ∈ , if meets the assumptions in the following: (1) ≥ 0 is a monotone nondecreasing function and admits at most finite discontinuous jump points on every compact interval of .(2) ( ) ( ) = = + 0 0 0, and there is a nonnegative constant N such that for any ≠ x y, for all in which Remark 2.1.Since the controlling function considered in model (1.2) is discontinuous, the usual Lipschitz condition is no longer satisfied, so the unilateral Lipschitz condition (2.3) is imposed to investigate the exponential stability of model (1.2), which has been successfully used to study the neural network and biomathematical models.Moreover, it is readily seen from Assumptions (1)-( 2) that In view of the proposed control strategy being discontinuous, there is a need to explain what does it mean of a solution to model (1.2).In this work, we shall employ the Filippov solution which was initially proposed by Filippov in [18].[19], for all C ∈ , the following set-valued map is said to admit an exponential dichotomy on , if there are positive constants k α , , and projection P such that the fundamental solution matrix ( ) X t of (2.7) meets in which I is the identity matrix.
Lemma 2.1.If system (2.7) has an exponential dichotomy, then the following almost periodic system admits a unique almost periodic solution ( ) x t , and x t X t PX s q s s X t I P X s q s s d d .
Lemma 2.2.Suppose c i is almost periodic, which is defined on and Then, the linear system satisfies exponential dichotomy on .
In order to obtain exponential stability criteria on almost periodic equation (1.2), the following Halanay inequality will be employed.
t τ s t 0 and > > β ζ 0, then as ≥ t t 0 , the following inequality holds: Otherwise, there is ≥ , we have

Main results
In this section, we shall establish new conditions on existence and exponential stability of solutions to almost periodic model (1.2). and Then, the set which is impossible.This proves (3.3).Second, we show that Then, accordingly, . Then, model (1.2) has a unique almost periodic solution in Proof.For each ( ) ∈ ϕ AP , , consider the following equation: x t e β s ϕ s τ s e bs γ s s d .which, and (3.12), implies that the mapping Γ is a self-mapping from Ω to Ω.
Dynamics for a delayed differential neoclassical growth model  7 In (3.13), we have used the following fact: Denote ( ) , it is easy to see that x e y e L x θ y x x y L n n x y x y κ θ , where , , , 0 1.

n x n y 1
Since < ρ 1 in (3.14), it means that Γ is contractive.We then conclude from the Banach fixed point theorem that Γ admits a unique fixed point which is exactly the solution to model (1.2) x t x t * .In view of the monotonicity of controlling function, one obtains This says that the almost periodic solution ( ) x t * is globally exponentially stable.This completes the proof.□ Corollary 3.4.Suppose that the delay ( ) τ t is a constant and all conditions in Theorem 3.2 hold, then the unique almost periodic solution of model (1.2) is globally exponential stable.Remark 3.1.It is interesting to note that the periodic or equilibrium dynamics would follow, if the economic parameters are periodic or constant in Theorems 3.1 and 3.2.
Remark 3.2.The exponential stability issue of model (1.2) has been well addressed in [5,12,14], however, one finds that the exponential stability results obtained by the Halanay inequality are different from these references, and the dynamics of discontinuous model (1.2) has seldom been considered up to now, which means that the established theoretical results are novel and extend some existing results.

Numerical example
In order to illustrate the validity of the obtained results, a numerical example is now given below.where Then, model (4.1) has a unique positive almost periodic solution which is globally exponentially stable.This fact is verified by numerical simulation with different initial values in Figure 1.
Dynamics for a delayed differential neoclassical growth model  9

μτ
the unique positive root to the following algebra equation:We are in a position to establish the criteria ensuring the global existence of positive solutions to model (1.2) with admissible initial conditions (2.1).Lemma 2.4.Let C ∈ , then the maximal right existence interval of each solution ( )

Lemma 3 . 1 . 1 .
(Positive invariance) Let < < n 0 If there is a positive constant > M κ show that model (1.2) has a unique almost periodic solution, one only needs to prove that Γ admits a fixed point in Ω.Let us first testify that the operator Γ is a self-mapping from Ω to itself.Actually, for each ∈ ϕ Ω, by(3

Example
Theorem 3.2.If the assumptions in Lemma 3.1 hold, assume further that 1which is a contradiction.Thus, (3.5) holds.(3.3) together with (3.5) implies that the main results of Lemma 3.1 hold.□ 4.1.Consider the following delayed neoclassical growth model with discontinuous controlling: