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# The Optimization of Mathematics Teaching Models in Colleges and Universities Based on Nonlinear Differential Equations

###### Accepté: 17 Apr 2022
Détails du magazine
Format
Magazine
eISSN
2444-8656
Première parution
01 Jan 2016
Périodicité
2 fois par an
Langues
Anglais
Introduction

Mathematical modeling plays a particularly important role in both scientific and technological research and national economic construction. Mathematical modeling mainly uses mathematical language to describe actual problems and phenomena in social and economic activities. It establishes a mathematical model described by functions or equations by looking for the relationship between variables. Economists believe that economic variables are dynamic. It will change over time [1]. Therefore, mathematical modeling and dynamic analysis of economic systems are very necessary. Dynamic economic modeling and dynamic analysis are among the most popular research directions in the study and practice of mathematical modeling courses for graduate or college students. Mathematical modeling and dynamic analysis of dynamic economics play a particularly important guiding role in cultivating college students' mathematical thinking abilities such as logical reasoning, spatial imagination, and scientific calculation, or promoting the development of science, technology, and economy.

Dynamic economics mathematical modeling analysis
Dynamic economics mathematical modeling process

The mathematical modeling of dynamic economics will not only apply mathematical theoretical knowledge and methods. The content includes ordinary differential equations, functional differential equations, linear algebra, probability theory, Lyapunov stability methods, economic principles, modern control theory, computer software programming, and other related professional knowledge [2]. Because the economic system is dynamic, economic variables change over time. When the change in product price is proportional to the difference between supply and demand, the following nonlinear price adjustment model for n products described by ordinary differential equations can be obtained: $dpidt=1ki[Di(p1,p2, …, pn)−Si(p1,p2, …, pn)]def__1kiEi,i=1, 2, …, n$ {{d{p_i}} \over {dt}} = {1 \over {{k_i}}}\left[{{D_i}\left({{p_1},{p_2},\, \ldots,\,{p_n}} \right) - {S_i}\left({{p_1},{p_2},\, \ldots,\,{p_n}} \right)} \right]\underline{\underline {def}} {1 \over {{k_i}}}{E_i},i = 1,\,2,\, \ldots,\,n pi represents the price of the i commodity. Ei represents the difference between the supply and demand of the i commodity. ki > 0 is a constant describing the reaction rate.

In the modeling process, consider that both Di (p1, p2, …, pn) and Si (p1, p2, …, pn) are nonlinear functions about price discontinuities. Its description is as follows: $Di(p1, p2, …, pn)=∑j=1naijfj(pj)+ci$ {D_i}\left({{p_1},\,{p_2},\, \ldots,\,{p_n}} \right) = \sum\limits_{j = 1}^n {{a_{ij}}{f_j}\left({{p_j}} \right) + {c_i}} $Si(p1, p2, …, pn)=∑j=1nbijgj(pj)+di$ {S_i}\left({{p_1},\,{p_2},\, \ldots,\,{p_n}} \right) = \sum\limits_{j = 1}^n {{b_{ij}}{g_j}\left({{p_j}} \right) + {d_i}} aij, bij, ci, di is constant. Functions fj (pj) and gj (pj) are discontinuous concerning the state pj, respectively [3]. Considering that economic control variables can have an important impact on the economic system, a control strategy variable ui (t) that can affect demand can be further introduced in the demand function. Hypothesis $Di(p1, p2, …, pn)=∑j=1naijfj(pj)+ci+ui(t)$ {D_i}\left({{p_1},\,{p_2},\, \ldots,\,{p_n}} \right) = \sum\limits_{j = 1}^n {{a_{ij}}{f_j}\left({{p_j}} \right) + {c_i} + {u_i}\left(t \right)}

Preferential policy activities for household appliances to the countryside is a control strategy variable that can affect demand [4]. Therefore, according to formula (1), formula (3), and formula (4), the following price adjustment model described by the discontinuous differential equation at the right end can be established: $dpidt=1ki[∑j=1naijfj(pj)−∑j=1nbijgj(pj)+ci−di]+1kiui(t)$ {{d{p_i}} \over {dt}} = {1 \over {{k_i}}}\left[{\sum\limits_{j = 1}^n {{a_{ij}}{f_j}\left({{p_j}} \right) - \sum\limits_{j = 1}^n {{b_{ij}}{g_j}\left({{p_j}} \right) + {c_i} - {d_i}}}} \right] + {1 \over {{k_i}}}{u_i}\left(t \right)

Assume that the discontinuous functions fj (pj) and gj (pj) satisfy the following conditions:

H1 : Except for at most several isolated points, fj and gj are continuous at other points on R, and there are left and right limits at these isolated discontinuous points. In addition, fj and gj have at most finite discontinuities in each compact subinterval of R.

H2 : For any j = 1, 2, …, n, there are non-negative constants lj and sj such that

$supξj∈co¯[fj(u)], ζj∈co ¯[fj(υ)]|ξj−ζj|≤lj|u−υ|+sj, ∀u, υ ∈R$ \mathop {\sup}\limits_{{\xi _j} \in \overline {co} \left[{{f_j}\left(u \right)} \right],\,{\zeta _j} \in \overline {co\,} \left[{{f_j}\left(\upsilon \right)} \right]} \left| {{\xi _j} - {\zeta _j}} \right| \le {l_j}\left| {u - \upsilon} \right| + {s_j},\,\forall u,\,\upsilon \, \in R

Where $co¯[fj(θ)]=[min{fj−(θ), fj+(θ)}, max{fj−(θ), fj+(θ)}]$ \overline {co} \left[{{f_j}\left(\theta \right)} \right] = \left[{\min \left\{{f_j^ - \left(\theta \right),\,f_j^ + \left(\theta \right)} \right\},\,\max \left\{{f_j^ - \left(\theta \right),\,f_j^ + \left(\theta \right)} \right\}} \right] , θR. $supγj∈co¯[gj(u)], ηj∈co ¯[gj(υ)]|γj−ηj|≤αj|u−υ|+βj, ∀u, υ ∈R$ \mathop {\sup}\limits_{{\gamma _j} \in \overline {co} \left[{{g_j}\left(u \right)} \right],\,{\eta _j} \in \overline {co\,} \left[{{g_j}\left(\upsilon \right)} \right]} \left| {{\gamma _j} - {\eta _j}} \right| \le {\alpha _j}\left| {u - \upsilon} \right| + {\beta _j},\,\forall u,\,\upsilon \, \in R

The dynamic economic model (5) obtained through mathematical modeling is essentially described by the differential equation on the right end of state discontinuity [5]. This discontinuous and state-dependent dynamic economic model has rich, dynamic behavior. However, there are not many research results on discontinuous dynamic economic models at this stage, which requires more effort from related researchers.

Stability control of the dynamic economic model

A booming market and stable prices are what people desire. At the price equilibrium point $P*=P1*, P2*,…, Pn*)T$ {\left. {{P^*} = P_1^*,\,P_2^*, \ldots,\,P_n^*} \right)^T} in the dynamic economic model (5), it is necessary to design a suitable control strategy U (t) = (u1 (t), u2 (t), …, un (t))T to keep the market price in the system (5) stable. First introduce the basic concepts and lemmas of differential inclusion, set value analysis and non-smooth analysis [6]. We assume that ‖ ‖ represents any vector norm and ‖ ‖2 represents Euclidean norm.

Where meas ( ) represents the Lebesgue measure of set N. ρ(X, δ) = {YRn : ‖ YX ‖ ≤ δ} is a ball with X as the center and δ as the radius. $co¯[ ]$ \overline {co} \left[{} \right] means to take the closed convex hull of the set. $dpidt∈1ki[∑j=1naijco¯[fj(pj)]−∑j=1nbijco¯[gj(pj)]+ci−di]+1kiuidef__F¯i(P)$ {{d{p_i}} \over {dt}} \in {1 \over {ki}}\left[{\sum\limits_{j = 1}^n {{a_{ij}}\overline {co} \left[{{f_j}\left({{p_j}} \right)} \right] - \sum\limits_{j = 1}^n {{b_{ij}}\overline {co} \left[{{g_j}\left({{p_j}} \right)} \right] + {c_i} - {d_i}}}} \right] + {1 \over {ki}}{u_i}\underline{\underline {def}} {\bar F_i}\left(P \right)

The set-valued map $P→F¯(P)=(F¯1(P), F¯2(P),…, F¯n(P))T$ P \to \bar F\left(P \right) = {\left({{{\bar F}_1}\left(P \right),\,{{\bar F}_2}\left(P \right), \ldots,\,{{\bar F}_n}\left(P \right)} \right)^T} has a non-empty compact convex value, so it is upper semi-continuous and measurable [7]. If P(t) is the solution of system (5). $dpi(t)dt=1ki[∑j=1naijξj(t)−∑j=1nbijγj(t)+ci−di]+1kiui(t)$ {{d{p_i}\left(t \right)} \over {dt}} = {1 \over {ki}}\left[{\sum\limits_{j = 1}^n {{a_{ij}}{\xi _j}\left(t \right) - \sum\limits_{j = 1}^n {{b_{ij}}{\gamma _j}\left(t \right) + {c_i} - {d_i}}}} \right] + {1 \over {ki}}{u_i}\left(t \right)

If there is Γ: (0, +∞) → R, there is Γ (σ) > 0 for σ ∈ [0, +∞) and t ≥ 0 for almost everywhere $dV(t)dt≤−Γ(V(t)) with ∫0v(0)1Γ(σ)dσ=t*<+∞$ {{dV\left(t \right)} \over {dt}} \le - \Gamma \left({V\left(t \right)} \right)\,{\rm{with}}\,\int_0^{v\left(0 \right)} {{1 \over {\Gamma \left(\sigma \right)}}d\sigma = {t^*} < + \infty}

Then there is V(t) = 0 for tt*. In particular, the following conclusions are established

If there is for Γ(σ) = l1σ + l2σμ all σ ∈ [0, +∞), among which μ ∈ (0,1), l1, l2 > 0, then $t*=1l1(1−μ)ln(l1V1−μ(0)+l2l2)$ {t^*} = {1 \over {{l_1}\left({1 - \mu} \right)}}\ln \left({{{{l_1}{V^{1 - \mu}}\left(0 \right) + {l_2}} \over {{l_2}}}} \right)

If Γ (σ) = μ, 1 > 0, μ ∈ (0,1), then

$t*=V1−μ(0)1(1−μ)$ {t^*} = {{{V^{1 - \mu}}\left(0 \right)} \over {1\left({1 - \mu} \right)}}

To converge the market price of the product to the equilibrium price of supply and demand P* in a limited time, firstly, we do variable substitution X(t) = P(t) − P*. Then the following price adjustment model with discontinuous nature is obtained $dxi(t)dt=1ki[∑j=1naij(fj(pj)−fj(pj*))$ {{d{x_i}\left(t \right)} \over {dt}} = {1 \over {ki}}\left[{\sum\limits_{j = 1}^n {{a_{ij}}\left({{f_j}\left({{p_j}} \right) - {f_j}\left({p_j^*} \right)} \right)}} \right. $dX(t)dt=K−1AF(X(t))−K−1BG(X(t))+K−1U(t)$ {{dX\left(t \right)} \over {dt}} = {K^{- 1}}AF\left({X\left(t \right)} \right) - {K^{- 1}}BG\left({X\left(t \right)} \right) + {K^{- 1}}U\left(t \right)

The discontinuous switching control strategy U (t) = (u1(t), u2(t), …, un(t))T is designed as follows $U(t)=−h1KX(t)−h2Ksign(X(t))$ U\left(t \right) = - {h_1}KX\left(t \right) - {h_2}Ksign\left({X\left(t \right)} \right)

Where sign(X(t)) = (sign(x1(t)), (sign(x2(t)), L, (sign(xn(t)))T, K = diag(k1, k2, L, kn). h1, h2 is the undetermined control coefficient. If the existence of time t* for any given initial value X (0) = P0P* makes $limt→t*‖X(t)‖=0$ \mathop {\lim}\limits_{t \to {t^*}} \left\| {X\left(t \right)} \right\| = 0 , and there is ‖X(t)‖ = ‖P(t) − P*‖ = 0 for t > t*, then the price adjustment model (5) can be stabilized by a suitable controller U(t) to achieve finite-time stabilization control. The price control is converged to the equilibrium price of supply and demand in a limited time [8]. $dX(t)dt∈K−1Aco¯[F(X(t))]−K−1Bco¯[G(X(t))]+K−1U(t)⊆K−1A(co¯[F(P(t))]−co¯[F(P*)])−K−1Bco¯[G(P(t))]−co¯[G(P*)])+K−1U(t)$ \matrix{{{{dX\left(t \right)} \over {dt}} \in {K^{- 1}}A\overline {co} \left[{F\left({X\left(t \right)} \right)} \right] - {K^{- 1}}B\overline {co} \left[{G\left({X\left(t \right)} \right)} \right] +} \hfill \cr {{K^{- 1}}U\left(t \right) \subseteq {K^{- 1}}A\left({\overline {co} \left[{F\left({P\left(t \right)} \right)} \right] - \overline {co} \left[{F\left({{P^*}} \right)} \right]} \right) -} \hfill \cr \left. {{K^{- 1}}B\overline {co} \left[{G\left({P\left(t \right)} \right)} \right] - \overline {co} \left[{G\left({{P^*}} \right)} \right]} \right) + {K^{- 1}}U\left(t \right)\hfill \cr} or equivalently. Then there is a measurable function $ξj(t)∈co¯[fj(pj(t))]$ {\xi _j}\left(t \right) \in \overline {co} \left[{{f_j}\left({{p_j}\left(t \right)} \right)} \right] , $γj(t)∈co¯[gj(pj(t))]$ {\gamma _j}\left(t \right) \in \overline {co} \left[{{g_j}\left({{p_j}\left(t \right)} \right)} \right] , $ξj*∈co¯[fj(pj*)]$ \xi _j^* \in \overline {co} \left[{{f_j}\left({p_j^*} \right)} \right] , $γj*∈co¯[gj(pj*)]$ \gamma _j^* \in \overline {co} \left[{{g_j}\left({p_j^*} \right)} \right] such that there is almost everywhere t ≥ 0 $dX(t)dt=K−1Aξ(t)−K−1Bγ(t)+K−1U(t)$ {{dX\left(t \right)} \over {dt}} = {K^{- 1}}A\xi \left(t \right) - {K^{- 1}}B\gamma \left(t \right) + {K^{- 1}}U\left(t \right)

In ξ(t) = ξ(t) − ξ*, γ(t) = γ(t) − γ*, ξ(t) = (ξ1(t), ξ2(t), L, ξn(t))T, $ξ*=(ξ1*, ξ2*, L,ξn*)T$ {\xi ^*} = {\left({\xi _1^*,\,\xi _2^*,\,{\rm{L}},\xi _n^*} \right)^T} , γ(t) = (γ1(t), γ2(t), L γn(t))T, $γ*=(γ1*, γ2*, L,γn*)T$ {\gamma ^*} = {\left({\gamma _1^*,\,\gamma _2^*,\,{\rm{L}},\gamma _n^*} \right)^T} ,. The following notation is introduced for convenience. A = ΘK−1 A = (aij)n×n, $amax=max1≤i,j≤n{|aij|}$ {a^{\max}} = \mathop {\max}\limits_{1 \le i,j \le n} \{|{a_{ij}}|\} , B = ΘK−1 B = (bij)n×n, $bmax=max1≤i,j≤n{|aij|}$ {b^{\max}} = \mathop {\max}\limits_{1 \le i,j \le n} \{|{a_{ij}}|\} . For the constant lj, sj, αj, βj, given in the hypotheses H2 and H3, let $lmax=max1≤j≤n{lj}$ {l^{\max}} = \mathop {\max}\limits_{1 \le j \le n} \left\{{{l_j}} \right\} , $smax=max1≤j≤n{sj}$ {s^{\max}} = \mathop {\max}\limits_{1 \le j \le n} \left\{{{s_j}} \right\} , $amax=max1≤j≤n{aj}$ {a^{\max}} = \mathop {\max}\limits_{1 \le j \le n} \left\{{{a_j}} \right\} , $βmax=max1≤j≤n{βj}$ {\beta ^{\max}} = \mathop {\max}\limits_{1 \le j \le n} \left\{{{\beta _j}} \right\} . λmax(Θ) and λmin(Θ) denote the maximum and Θ minimum eigenvalues, respectively.

Corollary 1

If a condition H1 − H3 is established, further assume that the following conditions satisfied h1 > namaxlmax + nbmaxαmax and. h2 > namaxsmax + nbmaxβmax Among them, $amax=max1≤i,j≤n{aijki}$ {a^{\max}} = \mathop {\max}\limits_{1 \le i,j \le n} \left\{{{{{a_{ij}}} \over {ki}}} \right\} and $bmax=max1≤i,j≤n{bijki}$ {b^{\max}} = \mathop {\max}\limits_{1 \le i,j \le n} \left\{{{{{b_{ij}}} \over {ki}}} \right\} pass the control strategy (15). The price adjustment model (5) can realize stabilization control for a limited time [9]. The price control is converged to the equilibrium price of supply and demand in a limited time: $t1*≤1h1−namaxlmax−nbmaxamaxln(1+(h1−namaxlmax+nbmaxamax)‖X(0)‖2h2−namaxsmax−nbmaxβmax).$ t_1^* \le {1 \over {{h_1} - n{a^{\max}}{l^{\max}} - n{b^{\max}}{a^{\max}}}}\ln \left({1 + {{\left({{h_1} - n{a^{\max}}{l^{\max}} + n{b^{\max}}{a^{\max}}} \right){{\left\| {X\left(0 \right)} \right\|}_2}} \over {{h_2} - n{a^{\max}}{s^{\max}} - n{b^{\max}}{\beta ^{\max}}}}} \right).

Theorem 2

If the condition H1 – H3 holds, further assume that the following conditions are met.

H5: There is a positive definite matrix Θ = (qij)n×n such that $−h1Θ+na^maxlmaxE+nb^maxamaxE≤0$ - {h_1}\Theta + n{\hat a^{\max}}{l^{\max}}E + n{\hat b^{\max}}{a^{\max}}E \le 0 and $h2>namaxsmax+nbmax βmaxλmin(Θ)$ {h_2} > {{n{a^{\max}}{s^{\max}} + n{b^{\max}}\,{\beta ^{\max}}} \over {{\lambda _{\min}}\left(\Theta \right)}} .

Then the price adjustment model (5) can realize stabilization control in a limited time. The price control is converged to the equilibrium price of supply and demand in a limited time and $t2≤λmax(Θ)‖X(0)‖2h2λmin(Θ)−na^max smax−nb^maxβmax$ {t_2} \le {{{\lambda _{\max}}\left(\Theta \right){{\left\| {X\left(0 \right)} \right\|}_2}} \over {{h_2}{\lambda _{\min}}\left(\Theta \right) - n{{\hat a}^{\max}}\,{s^{\max}} - n{{\hat b}^{\max}}{\beta ^{\max}}}} .

Proof

According to formula (19), in the proof process of Theorem 1, we can get $XT(t)ΘK−1Aξ(t)−XT(t)ΘK−1Bγ(t)≤(namaxlmax+nbmaxamax)∑i=1n∑j=1nxi2(t)+(namaxsmax+nbmax βmax)∑i=1n|xi(t)|=(namax lmax+nbmaxamax)XT(t)+(namax smax+nbmax βmax)∑i=1n|xi(t)|$ \matrix{{{X^T}\left(t \right)\Theta {K^{- 1}}A\xi \left(t \right) - {X^T}\left(t \right)\Theta {K^{- 1}}B\gamma \left(t \right) \le} \hfill \cr {\left({n{a^{\max}}{l^{\max}} + n{b^{\max}}{a^{\max}}} \right)\sum\limits_{i = 1}^n {\sum\limits_{j = 1}^n {x_i^2\left(t \right) + \left({n{a^{\max}}{s^{\max}} +} \right.}}} \hfill \cr {\left. {n{b^{\max}}\,{\beta ^{\max}}} \right)\sum\limits_{i = 1}^n {\left| {{x_i}\left(t \right)} \right| =}} \hfill \cr {\left({n{a^{\max}}\,{l^{\max}} + n{b^{\max}}{a^{\max}}} \right){X^T}\left(t \right) + \left({n{a^{\max}}\,{s^{\max}} + n{b^{\max}}\,{\beta ^{\max}}} \right)\sum\limits_{i = 1}^n {\left| {{x_i}\left(t \right)} \right|}} \hfill \cr}

From equation (15), equation (17), and hypothesis H5, it can be deduced $dV(t)dt≤XT(t)(−2h1Θ+2namaxlmaxE+2nbmax amaxE)X(t)−(2h2λmin(Θ)−2namaxsmax−2abmax βmax)∑i=1n|xi(t)|≤−(2h2λmin(Θ)−2namaxsmax−2abmax βmax)∑i=1n|xi(t)|≤−(2h2λmin(Θ)−2namaxsmax−2nbmaxβmax)λmax12(Θ)gV12(t).$ \matrix{{{{dV\left(t \right)} \over {dt}} \le {X^T}\left(t \right)\left({- 2{h_1}\Theta + 2n{a^{\max}}{l^{\max}}E +} \right.} \hfill \cr {\left. {2n{b^{\max}}\,{a^{\max}}E} \right)X\left(t \right) - \left({2{h_2}{\lambda _{\min}}\left(\Theta \right) - 2n{a^{\max}}{s^{\max}} -} \right.} \hfill \cr {\left. {2a{b^{\max}}\,{\beta ^{\max}}} \right)\sum\limits_{i = 1}^n {\left| {{x_i}\left(t \right)} \right| \le - \left({2{h_2}{\lambda _{\min}}\left(\Theta \right) - 2n{a^{\max}}{s^{\max}} -} \right.}} \hfill \cr {\left. {2a{b^{\max}}\,{\beta ^{\max}}} \right)\sum\limits_{i = 1}^n {\left| {{x_i}\left(t \right)} \right| \le - \left({2{h_2}{\lambda _{\min}}\left(\Theta \right) - 2n{a^{\max}}{s^{\max}} -} \right.}} \hfill \cr {\left. {2n{b^{\max}}{\beta ^{\max}}} \right)\lambda _{\max}^{{1 \over 2}}\left(\Theta \right)g{V^{{1 \over 2}}}\left(t \right).} \hfill \cr}

According to the hypothesis and the situation of Lemma 2 (2), the price adjustment model (5) can realize the finite-time stabilization control. Noticed $Γ(σ)=1σ12$ \Gamma \left(\sigma \right) = 1{\sigma ^{{1 \over 2}}} , where $1=(2h2λmin(Θ)−2na^maxsmax−2nb^maxβmax)λmax12(Θ)>0$ 1 = \left({2{h_2}{\lambda _{\min}}\left(\Theta \right) - 2n{{\hat a}^{\max}}{s^{\max}} - 2n{{\hat b}^{\max}}{\beta ^{\max}}} \right)\lambda _{\max}^{{1 \over 2}}\left(\Theta \right) > 0 , so $t2=∫0V(0)1Γ(σ)dσ=V12(0)12((2h2 λmin(Θ)−2na^maxsmax−2nb^maxβmax)λmax12(Θ)≤λmin(Θ)‖X(0)‖2h2λmin(Θ)−na^maxsmax−2nb^maxβmax).$ \matrix{{{t_2} = \int_0^{V\left(0 \right)} {{1 \over {\Gamma \left(\sigma \right)}}} d\sigma = {{{V^{{1 \over 2}}}\left(0 \right)} \over {{1 \over 2}\left({\left({2{h_2}\,{\lambda _{\min}}\left(\Theta \right) - 2n{{\hat a}^{\max}}{s^{\max}} - 2n{{\hat b}^{\max}}{\beta ^{\max}}} \right)\lambda _{\max}^{{1 \over 2}}\left(\Theta \right)} \right.}}} \hfill \cr {\le {{{\lambda _{\min}}\left(\Theta \right){{\left\| {X\left(0 \right)} \right\|}_2}} \over {\left. {{h_2}{\lambda _{\min}}\left(\Theta \right) - n{{\hat a}^{\max}}{s^{\max}} - 2n{{\hat b}^{\max}}{\beta ^{\max}}} \right)}}.} \hfill \cr}

Suppose Θ = E then λmax (Θ) = λmin (Θ) = 1. From Theorem 1, the following conclusions can be obtained.

Corollary 2

If the condition H1 – H3 is established, further assume that the following conditions are met $−h1E+namaxlmaxE+nbmaxamaxE≤0 and h2>namaxsmax+2nbmaxβmax,$ - {h_1}E + n{a^{\max }}{l^{\max }}E + n{b^{\max }}{a^{\max }}E \le 0\,\,\,{\rm{and}}\,\,{h_2} > n{a^{\max }}{s^{\max }} + 2n{b^{\max }}{\beta ^{\max }},

Among them, amax and bmax have been given in Corollary 1. Then the price adjustment model (5) can realize stabilization control in a limited time. In a limited time, the price control is converged to the equilibrium price of supply and demand and $t2*≤‖X(0)‖2h2−na^maxsmax−nb^maxβmax$ t_2^* \le {{{{\left\| {X\left(0 \right)} \right\|}_2}} \over {{h_2} - n{{\hat a}^{\max}}{s^{\max}} - n{{\hat b}^{\max}}{\beta ^{\max}}}} .

Numerical simulation analysis

This section will use Matlab to carry out numerical simulation experiments. Considering the price adjustment model (5), we control the product price to the equilibrium price P* = 0 of supply and demand [10]. This situation often occurs in real life. For example, a large retailer engages in promotional activity, giving a product as a gift to customers without charging any fees. That is, the price is zero. In model (5), we take n = 2, k1 = k2 = 1, a11 = a21 = 2.5, a12 = b11 = 0.1, and a22 = b21 = 1, b22 = 2 the control coefficient h1 = h2 = 5. $fj(θ)=gj(θ)={0.2 sin(θ−1)+0.5,0≤θ<10.2sin(θ−1)+1,θ≥1$ {f_j}\left(\theta \right) = {g_j}\left(\theta \right) = \left\{{\matrix{{0.2\,\sin \left({\theta - 1} \right) + 0.5,} & {0 \le \theta < 1} \cr {0.2\sin \left({\theta - 1} \right) + 1,} & {\theta \ge 1} \cr}} \right.

The initial value condition is P(0) = (10, 2)T. It is not difficult to verify that all the conditions in Corollary 1 are satisfied [11]. Therefore, the price adjustment model (5) can realize finite time stabilization control under the control strategy (15). Converge price control to the equilibrium price of supply and demand in a limited time P* = 0. Numerical simulation through MATLAB software programming has also verified that the conclusion is correct (Figure 1 and Figure 2).

Conclusion

Considering that the demand function and supply function are affected by discontinuous factors, we introduce a switching control strategy to perform mathematical modeling and finite-time stabilization control analysis on the nonlinear price adjustment model. Mathematical modeling itself is closely related to economic development and the progress of science and technology. In mathematical modeling research and teaching, classic mathematical theories and methods should be used to conduct scientific research and solve problems in economic life. Through analysis, there are several suggestions for the research of dynamic economics, the training of talents in universities, and the teaching reform of mathematical modeling: (1) Mathematical modeling should be closely linked to the current social and economic needs. (2) Strengthen the professional construction in the teaching process of mathematical modeling to improve the comprehensive quality of full-time teachers and scientific researchers in colleges and universities. (3) Mathematical modeling and dynamic analysis of dynamic economics should be carried out in talent training for graduates and undergraduates in related majors.

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