Tradable Credit Scheme for Control of Evolutionary Traffic Flows to System Optimum: Model and its Convergence

Abstract

We propose a dynamic tradable credit scheme for control of the day-to-day evolution process of traffic flows towards the system optimum (SO) state in a traffic network with elastic demand. In the scheme, the distribution and charge of travel credits are adjusted from period to period. The interacting dynamics among day-to-day traffic flows, period-to-period credit adjustment, and day-to-day credit price is formulated as an evolutionary game model. We mathematically prove two properties of the model, i.e., the consistence of the stationary state with the SO state and the convergence of the evolutionary trajectory. Finally, numerical results on a middle-size network are presented to validate the dynamic tradable credit scheme and to demonstrate the properties and application of the model.

Proofs of theorems

1.1 Proof of Theorem 1

Proof

First, we prove the necessity, i.e., if the trajectory of the dynamical system has arrived at the stationary state on day m₀M, then the corresponding flow and demand pattern $\left({\mathbf{x}}^{\left({m}_0M\right)},{\mathbf{f}}^{\left({m}_0M\right)},{\mathbf{d}}^{\left({m}_0M\right)}\right)$ is an SO flow and demand pattern, the credit charge ${\mathbf{k}}^{\left({m}_0M\right)}=\nabla \mathbf{c}{\left({\mathbf{x}}^{\left({m}_0M\right)}\right)}^{\mathrm{T}}{\mathbf{x}}^{\left({m}_0M\right)}$, and the unit credit price ${p}^{\left({m}_0M\right)}=1$. By formula (10), if ${\mathbf{k}}^{\left({m}_0M\right)}={\mathbf{k}}^{\left({m}_0M+1\right)}={\mathbf{k}}^{\left({m}_0M+2\right)}=\cdots$, then

$$ {\mathbf{k}}^{\left({m}_0M\right)}={\left[{\mathbf{k}}^{\left({m}_0M\right)}+{\rho}^{\left({m}_0M\right)}\left(\nabla \mathbf{c}{\left({\mathbf{x}}^{\left({m}_0M\right)}\right)}^{\mathrm{T}}{\mathbf{x}}^{\left({m}_0M\right)}-{\mathbf{k}}^{\left({m}_0M\right)}\right)\right]}_{+}. $$

(37)

According to Lemma 1, for any a ∈ L, the following inequalities hold

$$ \left({k}_a^{\left({m}_0M\right)}-\left({k}_a^{\left({m}_0M\right)}+{\rho}^{\left({m}_0M\right)}\left({c}_a^{\prime}\left({x}_a^{\left({m}_0M\right)}\right){x}_a^{\left({m}_0M\right)}-{k}_a^{\left({m}_0M\right)}\right)\right)\right)\left({k}_a^{\left({m}_0M\right)}-0\right)\le 0, $$

(38)

$$ \left({k}_a^{\left({m}_0M\right)}-\left({k}_a^{\left({m}_0M\right)}+{\rho}^{\left({m}_0M\right)}\left({c}_a^{\prime}\left({x}_a^{\left({m}_0M\right)}\right){x}_a^{\left({m}_0M\right)}-{k}_a^{\left({m}_0M\right)}\right)\right)\right)\left({k}_a^{\left({m}_0M\right)}-2{k}_a^{\left({m}_0M\right)}\right)\le 0, $$

(39)

$$ \left({k}_a^{\left({m}_0M\right)}-\left({k}_a^{\left({m}_0M\right)}+{\rho}^{\left({m}_0M\right)}\left({c}_a^{\prime}\left({x}_a^{\left({m}_0M\right)}\right){x}_a^{\left({m}_0M\right)}-{k}_a^{\left({m}_0M\right)}\right)\right)\right)\left({k}_a^{\left({m}_0M\right)}-\left({k}_a^{\left({m}_0M\right)}+1\right)\right)\le 0. $$

(40)

It follows from inequalities (38) and (39) that

$$ \left({c}_a^{\prime}\left({x}_a^{\left({m}_0M\right)}\right){x}_a^{\left({m}_0M\right)}-{k}_a^{\left({m}_0M\right)}\right){k}_a^{\left({m}_0M\right)}=0, $$

(41)

and it follows from inequality (40) that

$$ {c}_a^{\prime}\left({x}_a^{\left({m}_0M\right)}\right){x}_a^{\left({m}_0M\right)}-{k}_a^{\left({m}_0M\right)}\le 0. $$

(42)

For any a ∈ L, ${c}_a^{\prime}\left({x}_a^{\left({m}_0M\right)}\right){x}_a^{\left({m}_0M\right)}\ge 0$ under the assumption of non-decreasing link cost function c_a. When either ${c}_a^{\prime}\left({x}_a^{\left({m}_0M\right)}\right){x}_a^{\left({m}_0M\right)}=0$ or >0, combining relationships (41) and (42) yields ${k}_a^{\left({m}_0M\right)}={c}_a^{\prime}\left({x}_a^{\left({m}_0M\right)}\right){x}_a^{\left({m}_0M\right)}$.

If ${\mathbf{x}}^{\left({m}_0M\right)}={\mathbf{x}}^{\left({m}_0M+1\right)}={\mathbf{x}}^{\left({m}_0M+2\right)}=\cdots$ and ${\mathbf{k}}^{\left({m}_0M\right)}={\mathbf{k}}^{\left({m}_0M+1\right)}={\mathbf{k}}^{\left({m}_0M+2\right)}=\cdots$, then it is obtained that ${\varPhi}^{\left({m}_0M+1\right)}=0$ by formulae (11) and (13). Therefore, according to formula (12), if ${p}^{\left({m}_0M\right)}={p}^{\left({m}_0M+1\right)}={p}^{\left({m}_0M+2\right)}=\cdots$, then

$$ {p}^{\left({m}_0M\right)}={\left[{p}^{\left({m}_0M\right)}+{\zeta}^{\left({m}_0M+1\right)}\left(1-{p}^{\left({m}_0M\right)}\right)\right]}_{+}. $$

(43)

Similar to the derivation of formulae (38) to (42), by Lemma 1, we have ${p}^{\left({m}_0M\right)}=1$.

It follows from formulae (7) and (8) that, if ${\mathbf{x}}^{\left({m}_0M+1\right)}={\mathbf{x}}^{\left({m}_0M\right)}$ and ${\mathbf{d}}^{\left({m}_0M+1\right)}={\mathbf{d}}^{\left({m}_0M\right)}$, then

$$ {\left(\mathbf{y}-{\mathbf{x}}^{\left({m}_0M\right)}\right)}^{\mathrm{T}}\left(\mathbf{c}\left({\mathbf{x}}^{\left({m}_0M\right)}\right)+{p}^{\left({m}_0M+1\right)}{\mathbf{k}}^{\left({m}_0M+1\right)}\right)-{\left(\mathbf{e}-{\mathbf{d}}^{\left({m}_0M\right)}\right)}^{\mathrm{T}}\mathbf{B}\left({\mathbf{d}}^{\left({m}_0M\right)}\right)\ge 0, $$

(44)

$\forall \left(\mathbf{y},\mathbf{e}\right)\in {\varPsi}_{\mathrm{LD}}^{\left({m}_0M\right)}$. Namely, $\left({\mathbf{x}}^{\left({m}_0M\right)},{\mathbf{d}}^{\left({m}_0M\right)}\right)$ solves the following mathematical programming problem

$$ \underset{\left(\mathbf{y},\mathbf{e}\right)\in {\varPsi}_{\mathrm{LD}}^{\left({m}_0M\right)}}{\min }{\left(\mathbf{c}\left({\mathbf{x}}^{\left({m}_0M\right)}\right)+{p}^{\left({m}_0M+1\right)}{\mathbf{k}}^{\left({m}_0M+1\right)}\right)}^{\mathrm{T}}\mathbf{y}-\mathbf{B}{\left({\mathbf{d}}^{\left({m}_0M\right)}\right)}^{\mathrm{T}}\mathbf{e}. $$

(45)

The period length M ≥ 2 and also the total number of credits consumed in each period at the stationary state is positive. Thus, we have

$$ {T}^{\left({m}_0+1\right)}=M{\left({\mathbf{k}}^{\left({m}_0M+1\right)}\right)}^{\mathrm{T}}{\mathbf{x}}^{\left({m}_0M\right)}>{\left({\mathbf{k}}^{\left({m}_0M+1\right)}\right)}^{\mathrm{T}}{\mathbf{x}}^{\left({m}_0M\right)}. $$

(46)

That is to say, the flow ${\mathbf{x}}^{\left({m}_0M\right)}$ is not on the surface ${\left({\mathbf{k}}^{\left({m}_0M+1\right)}\right)}^{\mathrm{T}}\mathbf{y}={T}^{\left({m}_0+1\right)}$. It immediately follows that $\left({\mathbf{x}}^{\left({m}_0M\right)},{\mathbf{d}}^{\left({m}_0M\right)}\right)$ also solves the following linear programming problem

$$ \underset{\left(\mathbf{y},\mathbf{e}\right)\in {\varOmega}_{\mathrm{LD}}}{\min }{\left(\mathbf{c}\left({\mathbf{x}}^{\left({m}_0M\right)}\right)+{p}^{\left({m}_0M+1\right)}{\mathbf{k}}^{\left({m}_0M+1\right)}\right)}^{\mathrm{T}}\mathbf{y}-\mathbf{B}{\left({\mathbf{d}}^{\left({m}_0M\right)}\right)}^{\mathrm{T}}\mathbf{e}. $$

(47)

Therefore, $\left({\mathbf{x}}^{\left({m}_0M\right)},{\mathbf{d}}^{\left({m}_0M\right)}\right)$ satisfies the following Karush–Kuhn–Tucker (KKT) conditions:

$$ {\left({\boldsymbol{\Delta}}^{\mathrm{T}}\left(\mathbf{c}\left({\mathbf{x}}^{\left({m}_0M\right)}\right)+{p}^{\left({m}_0M+1\right)}{\mathbf{k}}^{\left({m}_0M+1\right)}\right)-{\boldsymbol{\Lambda}}^{\mathrm{T}}\boldsymbol{\upmu} \right)}^{\mathrm{T}}{\mathbf{f}}^{\left({m}_0M\right)}=0, $$

(48)

$$ {\boldsymbol{\Delta}}^{\mathrm{T}}\left(\mathbf{c}\left({\mathbf{x}}^{\left({m}_0M\right)}\right)+{p}^{\left({m}_0M+1\right)}{\mathbf{k}}^{\left({m}_0M+1\right)}\right)-{\boldsymbol{\Lambda}}^{\mathrm{T}}\boldsymbol{\upmu} \ge \mathbf{0},{\mathbf{f}}^{\left({m}_0M\right)}\ge \mathbf{0}, $$

(49)

$$ {\left(-\mathbf{B}\left({\mathbf{d}}^{\left({m}_0M\right)}\right)+\boldsymbol{\upmu} +\tilde{\boldsymbol{\upmu}}\right)}^{\mathrm{T}}{\mathbf{d}}^{\left({m}_0M\right)}=0, $$

(50)

$$ -\mathbf{B}\left({\mathbf{d}}^{\left({m}_0M\right)}\right)+\boldsymbol{\upmu} +\tilde{\boldsymbol{\upmu}}\ge \mathbf{0},{\mathbf{d}}^{\left({m}_0M\right)}\ge \mathbf{0} $$

(51)

$$ {\left(\overline{\mathbf{d}}-{\mathbf{d}}^{\left({m}_0M\right)}\right)}^{\mathrm{T}}\tilde{\boldsymbol{\upmu}}=0, $$

(52)

$$ \overline{\mathbf{d}}-{\mathbf{d}}^{\left({m}_0M\right)}\ge \mathbf{0},\tilde{\boldsymbol{\upmu}}\ge \mathbf{0}, $$

(53)

where μ = (μ_w, w ∈ W)^T and $\tilde{\boldsymbol{\upmu}}={\left({\tilde{\mu}}_w,w\in W\right)}^{\mathrm{T}}$ are the Lagrange multipliers associated with the constraints d = Λf and $\mathbf{d}\le \overline{\mathbf{d}}$, respectively. Applying ${\mathbf{k}}^{\left({m}_0M+1\right)}=\nabla \mathbf{c}{\left({\mathbf{x}}^{\left({m}_0M\right)}\right)}^{\mathrm{T}}{\mathbf{x}}^{\left({m}_0M\right)}$ and ${p}^{\left({m}_0M+1\right)}=1$ to conditions (48) and (49) then leads to

$$ {\left({\boldsymbol{\Delta}}^{\mathrm{T}}\left(\mathbf{c}\left({\mathbf{x}}^{\left({m}_0M\right)}\right)+\nabla \mathbf{c}{\left({\mathbf{x}}^{\left({m}_0M\right)}\right)}^{\mathrm{T}}{\mathbf{x}}^{\left({m}_0M\right)}\right)-{\boldsymbol{\Lambda}}^{\mathrm{T}}\boldsymbol{\upmu} \right)}^{\mathrm{T}}{\mathbf{f}}^{\left({m}_0M\right)}=0, $$

(54)

$$ {\boldsymbol{\Delta}}^{\mathrm{T}}\left(\mathbf{c}\left({\mathbf{x}}^{\left({m}_0M\right)}\right)+\nabla \mathbf{c}{\left({\mathbf{x}}^{\left({m}_0M\right)}\right)}^{\mathrm{T}}{\mathbf{x}}^{\left({m}_0M\right)}\right)-{\boldsymbol{\Lambda}}^{\mathrm{T}}\boldsymbol{\upmu} \ge \mathbf{0},{\mathbf{f}}^{\left({m}_0M\right)}\ge \mathbf{0}, $$

(55)

Obviously, conditions (50) to (55) are just an alternative expression of the SO conditions, with $\left({\mathbf{x}}^{\left({m}_0M\right)},{\mathbf{f}}^{\left({m}_0M\right)},{\mathbf{d}}^{\left({m}_0M\right)}\right)$ representing an SO flow and demand pattern. Thus, the necessity holds.

Second, we prove the sufficiency, i.e., if the corresponding flow and demand pattern $\left({\mathbf{x}}^{\left({m}_0M\right)},{\mathbf{f}}^{\left({m}_0M\right)},{\mathbf{d}}^{\left({m}_0M\right)}\right)$ is an SO flow and demand pattern, the credit charge ${\mathbf{k}}^{\left({m}_0M\right)}=\nabla \mathbf{c}{\left({\mathbf{x}}^{\left({m}_0M\right)}\right)}^{\mathrm{T}}{\mathbf{x}}^{\left({m}_0M\right)}$, and the unit credit price ${p}^{\left({m}_0M\right)}=1$, then the trajectory of the dynamical system has arrived at the stationary state on day m₀M. It immediately follows from formula (10) that, if ${\mathbf{k}}^{\left({m}_0M\right)}=\nabla \mathbf{c}{\left({\mathbf{x}}^{\left({m}_0M\right)}\right)}^{\mathrm{T}}{\mathbf{x}}^{\left({m}_0M\right)}$, then

$$ {\mathbf{k}}^{\left({m}_0M+1\right)}={\mathbf{k}}^{\left({m}_0M\right)}=\nabla \mathbf{c}{\left({\mathbf{x}}^{\left({m}_0M\right)}\right)}^{\mathrm{T}}{\mathbf{x}}^{\left({m}_0M\right)}. $$

It can be seen from formula (12) that ${p}^{\left({m}_0M+1\right)}={p}^{\left({m}_0M\right)}=1$.

If the flow and demand pattern $\left({\mathbf{x}}^{\left({m}_0M\right)},{\mathbf{f}}^{\left({m}_0M\right)},{\mathbf{d}}^{\left({m}_0M\right)}\right)$ is an SO flow and demand pattern, then it satisfies conditions (50) to (55). Substituting $\nabla \mathbf{c}{\left({\mathbf{x}}^{\left({m}_0M\right)}\right)}^{\mathrm{T}}{\mathbf{x}}^{\left({m}_0M\right)}={\mathbf{k}}^{\left({m}_0M+1\right)}$ and $1={p}^{\left({m}_0M+1\right)}$ into conditions (54) and (55) then results in conditions (48) and (49). Therefore, $\left({\mathbf{x}}^{\left({m}_0M\right)},{\mathbf{d}}^{\left({m}_0M\right)}\right)$ solves the linear programming problem (47). Associated with the fact that $\left({\mathbf{x}}^{\left({m}_0M\right)},{\mathbf{d}}^{\left({m}_0M\right)}\right)\in {\varPsi}_{\mathrm{LD}}^{\left({m}_0M\right)}\subset {\varOmega}_{\mathrm{LD}}$, it is obtained that $\left({\mathbf{x}}^{\left({m}_0M\right)},{\mathbf{d}}^{\left({m}_0M\right)}\right)$ also solves the mathematical programming problem (45), and hence ${\mathbf{x}}^{\left({m}_0M+1\right)}={\mathbf{x}}^{\left({m}_0M\right)}$ and ${\mathbf{d}}^{\left({m}_0M+1\right)}={\mathbf{d}}^{\left({m}_0M\right)}$.

It can be seen from formula (10) that ${\mathbf{k}}^{\left({m}_0M+2\right)}={\mathbf{k}}^{\left({m}_0M+1\right)}$. By formulae (11) and (13), we have

$$ {\varPhi}^{\left({m}_0M+1\right)}={\left({\mathbf{k}}^{\left({m}_0M+1\right)}\right)}^{\mathrm{T}}{\mathbf{x}}^{\left({m}_0M+1\right)}-\frac{M{\left({\mathbf{k}}^{\left({m}_0M+1\right)}\right)}^{\mathrm{T}}{\mathbf{x}}^{\left({m}_0M\right)}-{\left({\mathbf{k}}^{\left({m}_0M+1\right)}\right)}^{\mathrm{T}}{\mathbf{x}}^{\left({m}_0M+1\right)}}{M-1}=0 $$

Therefore, it immediately follows from formula (12) that ${p}^{\left({m}_0M+2\right)}={p}^{\left({m}_0M+1\right)}$. Similar to the derivation of ${\mathbf{x}}^{\left({m}_0M+1\right)}={\mathbf{x}}^{\left({m}_0M\right)}$ and ${\mathbf{d}}^{\left({m}_0M+1\right)}={\mathbf{d}}^{\left({m}_0M\right)}$, we can obtain ${\mathbf{x}}^{\left({m}_0M+2\right)}={\mathbf{x}}^{\left({m}_0M+1\right)}$ and ${\mathbf{d}}^{\left({m}_0M+2\right)}={\mathbf{d}}^{\left({m}_0M+1\right)}$. In this way, it follows that

$$ {\displaystyle \begin{array}{l}{\mathbf{k}}^{\left({m}_0M\right)}={\mathbf{k}}^{\left({m}_0M+1\right)}={\mathbf{k}}^{\left({m}_0M+2\right)}=\cdots, {p}^{\left({m}_0M\right)}={p}^{\left({m}_0M+1\right)}={p}^{\left({m}_0M+2\right)}=\cdots, \\ {}{\mathbf{x}}^{\left({m}_0M\right)}={\mathbf{x}}^{\left({m}_0M+1\right)}={\mathbf{x}}^{\left({m}_0M+2\right)}=\cdots, {\mathbf{d}}^{\left({m}_0M\right)}={\mathbf{d}}^{\left({m}_0M+1\right)}={\mathbf{d}}^{\left({m}_0M+2\right)}=\cdots, \end{array}} $$

Therefore, the sufficiency holds.

1.2 Proof of Theorem 2

Proof

On one hand, if (x^{((m − 1)M)}, d^{((m − 1)M)}, k^{((m − 1)M)}, p^{((m − 1)M)}) is a stationary link flow, travel demand, credit charge, and credit price pattern, i.e., the trajectory of the dynamical system has entered the stationary state on day (m − 1)M, then both sides of inequality (17) equal zero and the inequality holds. On the other hand, according to Lemma 1, it is obtained from formulae (10) and (12) that

$$ {\displaystyle \begin{array}{l}{\left({\mathbf{k}}^{\left(\left(m-1\right)M+1\right)}-{\mathbf{k}}^{\left(\left(m-1\right)M\right)}\right)}^{\mathrm{T}}\left({\mathbf{k}}^{\left(\left(m-1\right)M\right)}-\nabla \mathbf{c}{\left({\mathbf{x}}^{\left(\left(m-1\right)M\right)}\right)}^{\mathrm{T}}{\mathbf{x}}^{\left(\left(m-1\right)M\right)}\right)\\ {}\le -\frac{1}{\rho^{\left(\left(m-1\right)M\right)}}{\left\Vert {\mathbf{k}}^{\left(\left(m-1\right)M+1\right)}-{\mathbf{k}}^{\left(\left(m-1\right)M\right)}\right\Vert}^2\end{array}} $$

and

$$ {\displaystyle \begin{array}{l}\left({p}^{\left(\left(m-1\right)M+i\right)}-{p}^{\left(\left(m-1\right)M+i-1\right)}\right)\left({p}^{\left(\left(m-1\right)M+i-1\right)}-\exp \left(\xi {\varPhi}^{\left(\left(m-1\right)M+i-1\right)}\right)\right)\\ {}\le -\frac{1}{\zeta^{\left(\left(m-1\right)M+i-1\right)}}{\left({p}^{\left(\left(m-1\right)M+i\right)}-{p}^{\left(\left(m-1\right)M+i-1\right)}\right)}^2,i=2,3,\cdots, M.\end{array}} $$

If (x^{((m − 1)M)}, d^{((m − 1)M)}, k^{((m − 1)M)}, p^{((m − 1)M)}) is not a stationary link flow, travel demand, credit charge, and credit price pattern, i.e., the sequence of the dynamical system is in a non-stationary state till day (m − 1)M, then

$$ {\left({\mathbf{k}}^{\left(\left(m-1\right)M+1\right)}-{\mathbf{k}}^{\left(\left(m-1\right)M\right)}\right)}^{\mathrm{T}}\left({\mathbf{k}}^{\left(\left(m-1\right)M\right)}-\nabla \mathbf{c}{\left({\mathbf{x}}^{\left(\left(m-1\right)M\right)}\right)}^{\mathrm{T}}{\mathbf{x}}^{\left(\left(m-1\right)M\right)}\right)<0, $$

or equivalently

$$ \sum \limits_{i=2}^M\left({p}^{\left(\left(m-1\right)M+i\right)}-{p}^{\left(\left(m-1\right)M+i-1\right)}\right)\left({p}^{\left(\left(m-1\right)M+i-1\right)}-\exp \left(\xi {\varPhi}^{\left(\left(m-1\right)M+i-1\right)}\right)\right)<0, $$

or equivalently

$$ {\displaystyle \begin{array}{l}\sum \limits_{i=1}^M\Big({\left({\mathbf{x}}^{\left(\left(m-1\right)M+i\right)}-{\mathbf{x}}^{\left(\left(m-1\right)M+i-1\right)}\right)}^{\mathrm{T}}\left(\mathbf{c}\left({\mathbf{x}}^{\left(\left(m-1\right)M+i-1\right)}\right)+{p}^{\left(\left(m-1\right)M+i\right)}{\mathbf{k}}^{\left(\left(m-1\right)M+i\right)}\right)\\ {}-{\left({\mathbf{d}}^{\left(\left(m-1\right)M+i\right)}-{\mathbf{d}}^{\left(\left(m-1\right)M+i-1\right)}\right)}^{\mathrm{T}}\mathbf{B}\left({\mathbf{d}}^{\left(\left(m-1\right)M+i-1\right)}\right)\Big)<0.\end{array}} $$

It immediately follows that Z(x^{((m − 1)M)}, d^{((m − 1)M)}, k^{((m − 1)M)}, p^{((m − 1)M)}) < 0, i.e.,

$$ -\beta Z\left({\mathbf{x}}^{\left(\left(m-1\right)M\right)},{\mathbf{d}}^{\left(\left(m-1\right)M\right)},{\mathbf{k}}^{\left(\left(m-1\right)M\right)},{p}^{\left(\left(m-1\right)M\right)}\right)>0 $$

Thus, a group of credit charge k^(mM + 1) and credit prices p^(mM + 2), p^(mM + 3), ⋯, p^{((m + 1)M)} can be found so that inequality (17) holds.

To guarantee that inequality (17) holds, we define that a group of credit charge k^(mM + 1) and credit prices p^(mM + 2), p^(mM + 3), ⋯, p^{((m + 1)M)} (m = 1, 2, ⋯) satisfy

$$ {\displaystyle \begin{array}{l}{\left({\mathbf{k}}^{\left( mM+1\right)}-{\mathbf{k}}^{(mM)}\right)}^{\mathrm{T}}\left({\mathbf{k}}^{\left( mM+1\right)}+{p}^{(mM)}{\mathbf{x}}^{(mM)}+\nabla \mathbf{c}{\left({\mathbf{x}}^{(mM)}\right)}^{\mathrm{T}}{\mathbf{x}}^{(mM)}\right)\\ {}\le -\frac{\beta }{M}Z\left({\mathbf{x}}^{\left(\left(m-1\right)M\right)},{\mathbf{d}}^{\left(\left(m-1\right)M\right)},{\mathbf{k}}^{\left(\left(m-1\right)M\right)},{p}^{\left(\left(m-1\right)M\right)}\right)\end{array}} $$

(56)

and

$$ {\displaystyle \begin{array}{l}\left({p}^{\left( mM+i\right)}-{p}^{\left( mM+i-1\right)}\right)\left({p}^{\left( mM+i\right)}+{\left({\mathbf{k}}^{\left( mM+i-1\right)}\right)}^{\mathrm{T}}{\mathbf{x}}^{\left( mM+i-1\right)}+\exp \left(\xi {\varPhi}^{\left( mM+i-1\right)}\right)\right)\\ {}\le -\frac{\beta }{M}Z\left({\mathbf{x}}^{\left(\left(m-1\right)M\right)},{\mathbf{d}}^{\left(\left(m-1\right)M\right)},{\mathbf{k}}^{\left(\left(m-1\right)M\right)},{p}^{\left(\left(m-1\right)M\right)}\right),\mathrm{for}\;i=2,3,\cdots, M.\end{array}} $$

(57)

The link travel cost functions are non-decreasing, and hence ∇c(x)^Tx ≥ 0 for any x ∈ Ω_L. Let the parameter ρ^(mM) ∈ (0, 1] (m = 0, 1, 2, ⋯) in formula (10). Thus, if k⁽⁰⁾ ≥ 0, then k^(mM + 1) (m = 0, 1, 2, ⋯) can be written as

$$ {\mathbf{k}}^{\left( mM+1\right)}={\mathbf{k}}^{(mM)}+{\rho}^{(mM)}\left(\nabla \mathbf{c}{\left({\mathbf{x}}^{(mM)}\right)}^{\mathrm{T}}{\mathbf{x}}^{(mM)}-{\mathbf{k}}^{(mM)}\right). $$

(58)

Substituting formula (58) into inequality (56), inequality (56) can be expressed in another form

$$ {\overline{a}}^{(mM)}{\left({\rho}^{(mM)}\right)}^2+{\overline{b}}^{(mM)}{\rho}^{(mM)}+{\overline{c}}^{(mM)}\le 0,m=1,2,\cdots . $$

(59)

Let the parameter ζ^{(mM + i − 1)} ∈ (0, 1] (m = 0, 1, 2, ⋯ and i = 2, 3, ⋯, M) in formula (12). Therefore, if p⁽⁰⁾ ≥ 0, then p^(mM + i) (m = 0, 1, 2, ⋯ and i = 2, 3, ⋯, M) can be written as

$$ {p}^{\left( mM+i\right)}={p}^{\left( mM+i-1\right)}+{\zeta}^{\left( mM+i-1\right)}\left(\exp \left(\xi {\varPhi}^{\left( mM+i-1\right)}\right)-{p}^{\left( mM+i-1\right)}\right). $$

(60)

Substituting formula (60) into inequality (57), inequality (57) can be expressed in another form

$$ {\tilde{a}}^{\left( mM+i-1\right)}{\left({\zeta}^{\left( mM+i-1\right)}\right)}^2+{\tilde{b}}^{\left( mM+i-1\right)}{\zeta}^{\left( mM+i-1\right)}+{\overline{c}}^{(mM)}\le 0,m=1,2,\cdots \mathrm{and}\;i=2,3,\cdots, M. $$

(61)

Thus, to guarantee that Assumption 2 holds, when k^(mM) ≠ ∇c(x^(mM))^Tx^(mM), the parameter ρ^(mM) (m = 1, 2, ⋯) should take a value simultaneously satisfying inequalities (59) and 0 < ρ^(mM) ≤ 1, i.e.,

$$ {\rho}^{(mM)}\in \left(0,\min \left\{\frac{-{\overline{b}}^{(mM)}+\sqrt{{\left({\overline{b}}^{(mM)}\right)}^2-4{\overline{a}}^{(mM)}{\overline{c}}^{(mM)}}}{2{\overline{a}}^{(mM)}},1\right\}\right]; $$

(62)

when k^(mM) = ∇c(x^(mM))^Tx^(mM), ρ^(mM) ∈ (0, 1]. When p^{(mM + i − 1)} ≠ exp(ξΦ^{(mM + i − 1)}), the parameter ζ^{(mM + i − 1)} (m = 1, 2, ⋯ and i = 2, 3, ⋯, M) should take a value simultaneously satisfying inequalities (61) and 0 < ζ^{(mM + i − 1)} ≤ 1, i.e.,

$$ {\zeta}^{\left( mM+i-1\right)}\in \left(0,\min \left\{\frac{-{\tilde{b}}^{\left( mM+i-1\right)}+\sqrt{{\left({\tilde{b}}^{\left( mM+i-1\right)}\right)}^2-4{\tilde{a}}^{\left( mM+i-1\right)}{\overline{c}}^{(mM)}}}{2{\tilde{a}}^{\left( mM+i-1\right)}},1\right\}\right]; $$

(63)

when p^{(mM + i − 1)} = exp(ξΦ^{(mM + i − 1)}), ζ^{(mM + i − 1)} ∈ (0, 1].

1.3 Proof of Theorem 3

Proof

First, consider the following function

$$ V\left(\mathbf{x},\mathbf{d},\mathbf{k},p\right)=\sum \limits_{a\in L}{\int}_0^{x_a}{c}_a(x)\mathrm{d}x+p\sum \limits_{a\in L}{k}_a{x}_a-\sum \limits_{w\in W}{\int}_0^{d_w}{B}_w(u)\mathrm{d}u, $$

(64)

where (x, d, k, p) ∈ Ω_LD × Γ_C × Γ_P. The function V is continuous with respect to (x, d, k, p) and Ω_LD × Γ_C × Γ_P is compact. Therefore, according to Theorem 5.2.11 in Trench (2003), the function V has a lower bound.

For each link a ∈ L, c_a is continuously differentiable, and for each OD pair w ∈ W, B_w is continuously differentiable. Thus, according to the Taylor formula, we have

$$ {\displaystyle \begin{array}{l}\sum \limits_{a\in L}{\int}_0^{x_a^{\left(i+1\right)}}{c}_a(x)\mathrm{d}x-\sum \limits_{w\in W}{\int}_0^{d_w^{\left(i+1\right)}}{B}_w(u)\mathrm{d}u=\sum \limits_{a\in L}{\int}_0^{x_a^{(i)}}{c}_a(x)\mathrm{d}x-\sum \limits_{w\in W}{\int}_0^{d_w^{(i)}}{B}_w(u)\mathrm{d}u\\ {}+\sum \limits_{a\in L}\left({x}_a^{\left(i+1\right)}-{x}_a^{(i)}\right){c}_a\left({x}_a^{(i)}\right)-\sum \limits_{w\in W}\left({d}_w^{\left(i+1\right)}-{d}_w^{(i)}\right){B}_w\left({d}_w^{(i)}\right)+o\left(\left\Vert \left({\mathbf{x}}^{\left(i+1\right)},{\mathbf{d}}^{\left(i+1\right)}\right)-\left({\mathbf{x}}^{(i)},{\mathbf{d}}^{(i)}\right)\right\Vert \right).\end{array}} $$

Furthermore,

$$ {\displaystyle \begin{array}{l}{p}^{\left(i+1\right)}\sum \limits_{a\in L}{k}_a^{\left(i+1\right)}{x}_a^{\left(i+1\right)}={p}^{(i)}\sum \limits_{a\in L}{k}_a^{(i)}{x}_a^{(i)}+{p}^{\left(i+1\right)}\sum \limits_{a\in L}\left({x}_a^{\left(i+1\right)}-{x}_a^{(i)}\right){k}_a^{\left(i+1\right)}\\ {}+\sum \limits_{a\in L}\left({k}_a^{\left(i+1\right)}-{k}_a^{(i)}\right){p}^{\left(i+1\right)}{x}_a^{(i)}+\sum \limits_{a\in L}\left({p}^{\left(i+1\right)}-{p}^{(i)}\right){k}_a^{(i)}{x}_a^{(i)}.\end{array}} $$

Thus, from the above two equations it is generated that

$$ {\displaystyle \begin{array}{l}V\left({\mathbf{x}}^{(nM)},{\mathbf{d}}^{(nM)},{\mathbf{k}}^{(nM)},{p}^{(nM)}\right)-V\left({\mathbf{x}}^{(0)},{\mathbf{d}}^{(0)},{\mathbf{k}}^{(0)},{p}^{(0)}\right)\\ {}=\sum \limits_{m=1}^n\sum \limits_{i=1}^M\Big(V\left({\mathbf{x}}^{\left(\left(m-1\right)M+i\right)},{\mathbf{d}}^{\left(\left(m-1\right)M+i\right)},{\mathbf{k}}^{\left(\left(m-1\right)M+i\right)},{p}^{\left(\left(m-1\right)M+i\right)}\right)\\ {}-V\left({\mathbf{x}}^{\left(\left(m-1\right)M+i-1\right)},{\mathbf{d}}^{\left(\left(m-1\right)M+i-1\right)},{\mathbf{k}}^{\left(\left(m-1\right)M+i-1\right)},{p}^{\left(\left(m-1\right)M+i-1\right)}\right)\Big)\\ {}=\sum \limits_{m=1}^n\sum \limits_{i=1}^M\Big({\left({\mathbf{x}}^{\left(\left(m-1\right)M+i\right)}-{\mathbf{x}}^{\left(\left(m-1\right)M+i-1\right)}\right)}^{\mathrm{T}}\left(\mathbf{c}\left({\mathbf{x}}^{\left(\left(m-1\right)M+i-1\right)}\right)+{p}^{\left(\left(m-1\right)M+i\right)}{\mathbf{k}}^{\left(\left(m-1\right)M+i\right)}\right)\\ {}-{\left({\mathbf{d}}^{\left(\left(m-1\right)M+i\right)}-{\mathbf{d}}^{\left(\left(m-1\right)M+i-1\right)}\right)}^{\mathrm{T}}\mathbf{B}\left({\mathbf{d}}^{\left(\left(m-1\right)M+i-1\right)}\right)\Big)\\ {}+\sum \limits_{m=1}^n{\left({\mathbf{k}}^{\left(\left(m-1\right)M+1\right)}-{\mathbf{k}}^{\left(\left(m-1\right)M\right)}\right)}^{\mathrm{T}}{p}^{\left(\left(m-1\right)M\right)}{\mathbf{x}}^{\left(\left(m-1\right)M\right)}\\ {}+\sum \limits_{m=1}^n\sum \limits_{i=2}^M\left({p}^{\left(\left(m-1\right)M+i\right)}-{p}^{\left(\left(m-1\right)M+i-1\right)}\right){\left({\mathbf{k}}^{\left(\left(m-1\right)M+i-1\right)}\right)}^{\mathrm{T}}{\mathbf{x}}^{\left(\left(m-1\right)M+i-1\right)}\\ {}+\sum \limits_{m=1}^n\sum \limits_{i=1}^Mo\left(\left\Vert \left({\mathbf{x}}^{\left(\left(m-1\right)M+i\right)},{\mathbf{d}}^{\left(\left(m-1\right)M+i\right)}\right)-\left({\mathbf{x}}^{\left(\left(m-1\right)M+i-1\right)},{\mathbf{d}}^{\left(\left(m-1\right)M+i-1\right)}\right)\right\Vert \right).\end{array}} $$

The term on the left hand side of the above equation has a lower bound. On the right hand side, (k⁽¹⁾ − k⁽⁰⁾)^Tp⁽⁰⁾x⁽⁰⁾ in the second term and ${\sum}_{i=2}^M\left({p}^{(i)}-{p}^{\left(i-1\right)}\right){\left({\mathbf{k}}^{\left(i-1\right)}\right)}^{\mathrm{T}}{\mathbf{x}}^{\left(i-1\right)}$ in the third term are fixed. Therefore, it is obtained that, as n →  + ∞, the following term

$$ {\displaystyle \begin{array}{l}\sum \limits_{m=1}^{+\infty}\sum \limits_{i=1}^M\Big({\left({\mathbf{x}}^{\left(\left(m-1\right)M+i\right)}-{\mathbf{x}}^{\left(\left(m-1\right)M+i-1\right)}\right)}^{\mathrm{T}}\left(\mathbf{c}\left({\mathbf{x}}^{\left(\left(m-1\right)M+i-1\right)}\right)+{p}^{\left(\left(m-1\right)M+i\right)}{\mathbf{k}}^{\left(\left(m-1\right)M+i\right)}\right)\\ {}-{\left({\mathbf{d}}^{\left(\left(m-1\right)M+i\right)}-{\mathbf{d}}^{\left(\left(m-1\right)M+i-1\right)}\right)}^{\mathrm{T}}\mathbf{B}\left({\mathbf{d}}^{\left(\left(m-1\right)M+i-1\right)}\right)\Big)\\ {}+\sum \limits_{m=2}^{+\infty }{\left({\mathbf{k}}^{\left(\left(m-1\right)M+1\right)}-{\mathbf{k}}^{\left(\left(m-1\right)M\right)}\right)}^{\mathrm{T}}{p}^{\left(\left(m-1\right)M\right)}{\mathbf{x}}^{\left(\left(m-1\right)M\right)}\\ {}+\sum \limits_{m=2}^{+\infty}\sum \limits_{i=2}^M\left({p}^{\left(\left(m-1\right)M+i\right)}-{p}^{\left(\left(m-1\right)M+i-1\right)}\right){\left({\mathbf{k}}^{\left(\left(m-1\right)M+i-1\right)}\right)}^{\mathrm{T}}{\mathbf{x}}^{\left(\left(m-1\right)M+i-1\right)}\\ {}+\sum \limits_{m=1}^{+\infty}\sum \limits_{i=1}^Mo\left(\left\Vert \left({\mathbf{x}}^{\left(\left(m-1\right)M+i\right)},{\mathbf{d}}^{\left(\left(m-1\right)M+i\right)}\right)-\left({\mathbf{x}}^{\left(\left(m-1\right)M+i-1\right)},{\mathbf{d}}^{\left(\left(m-1\right)M+i-1\right)}\right)\right\Vert \right)\end{array}} $$

(65)

has a lower bound.

Second, consider the following function

$$ \overline{V}\left(\mathbf{x},\mathbf{k}\right)={\mathbf{k}}^{\mathrm{T}}\left(\frac{\mathbf{k}}{2}-\nabla \mathbf{c}{\left(\mathbf{x}\right)}^{\mathrm{T}}\mathbf{x}\right), $$

(66)

where (x, k) ∈ Ω_L × Γ_C. The function $\overline{V}$ is continuous with respect to (x, k) and Ω_L × Γ_C is compact. Thus, the function $\overline{V}$ has a lower bound. We consider the following equation

$$ {\displaystyle \begin{array}{l}\overline{V}\left({\mathbf{x}}^{(nM)},{\mathbf{k}}^{(nM)}\right)-\overline{V}\left({\mathbf{x}}^{(0)},{\mathbf{k}}^{(0)}\right)=\sum \limits_{m=1}^n\left(\overline{V}\left({\mathbf{x}}^{(mM)},{\mathbf{k}}^{(mM)}\right)-\overline{V}\left({\mathbf{x}}^{\left(\left(m-1\right)M\right)},{\mathbf{k}}^{\left(\left(m-1\right)M\right)}\right)\right)\\ {}=\frac{1}{2}\sum \limits_{m=1}^n{\left({\mathbf{k}}^{\left(\left(m-1\right)M+1\right)}-{\mathbf{k}}^{\left(\left(m-1\right)M\right)}\right)}^{\mathrm{T}}\left({\mathbf{k}}^{\left(\left(m-1\right)M\right)}-\nabla \mathbf{c}{\left({\mathbf{x}}^{\left(\left(m-1\right)M\right)}\right)}^{\mathrm{T}}{\mathbf{x}}^{\left(\left(m-1\right)M\right)}\right)\\ {}+\frac{1}{2}\sum \limits_{m=1}^n{\left({\mathbf{k}}^{\left(\left(m-1\right)M+1\right)}-{\mathbf{k}}^{\left(\left(m-1\right)M\right)}\right)}^{\mathrm{T}}\left({\mathbf{k}}^{\left(\left(m-1\right)M+1\right)}+\nabla \mathbf{c}{\left({\mathbf{x}}^{\left(\left(m-1\right)M\right)}\right)}^{\mathrm{T}}{\mathbf{x}}^{\left(\left(m-1\right)M\right)}\right)\\ {}-{\left({\mathbf{k}}^{(nM)}\right)}^{\mathrm{T}}\nabla \mathbf{c}{\left({\mathbf{x}}^{(nM)}\right)}^{\mathrm{T}}{\mathbf{x}}^{(nM)}+{\left({\mathbf{k}}^{(0)}\right)}^{\mathrm{T}}\nabla \mathbf{c}{\left({\mathbf{x}}^{(0)}\right)}^{\mathrm{T}}{\mathbf{x}}^{(0)}.\end{array}} $$

The term on the left hand side of the above equation has a lower bound. On the right hand side, (k⁽¹⁾ − k⁽⁰⁾)^T(k⁽¹⁾ + ∇c(x⁽⁰⁾)^Tx⁽⁰⁾) in the second term is fixed, the third term is bounded, and the fourth term is fixed. Therefore, it is obtained that, as n →  + ∞, the following term

$$ {\displaystyle \begin{array}{l}\sum \limits_{m=1}^{+\infty }{\left({\mathbf{k}}^{\left(\left(m-1\right)M+1\right)}-{\mathbf{k}}^{\left(\left(m-1\right)M\right)}\right)}^{\mathrm{T}}\left({\mathbf{k}}^{\left(\left(m-1\right)M\right)}-\nabla \mathbf{c}{\left({\mathbf{x}}^{\left(\left(m-1\right)M\right)}\right)}^{\mathrm{T}}{\mathbf{x}}^{\left(\left(m-1\right)M\right)}\right)\\ {}+\sum \limits_{m=2}^{+\infty }{\left({\mathbf{k}}^{\left(\left(m-1\right)M+1\right)}-{\mathbf{k}}^{\left(\left(m-1\right)M\right)}\right)}^{\mathrm{T}}\left({\mathbf{k}}^{\left(\left(m-1\right)M+1\right)}+\nabla \mathbf{c}{\left({\mathbf{x}}^{\left(\left(m-1\right)M\right)}\right)}^{\mathrm{T}}{\mathbf{x}}^{\left(\left(m-1\right)M\right)}\right)\end{array}} $$

(67)

has a lower bound.

Third, consider the following function

$$ \tilde{V}\left(p,\varPhi \right)=p\left(\frac{p}{2}-\exp \left(\xi \varPhi \right)\right), $$

(68)

where (p, Φ) ∈ Γ_P × Γ_E. The function $\tilde{V}$ is continuous with respect to (p, Φ) and Γ_P × Γ_E is compact. Thus, the function $\tilde{V}$ has a lower bound. We consider the following equation

$$ {\displaystyle \begin{array}{l}\tilde{V}\left({p}^{(nM)},{\varPhi}^{(nM)}\right)-\tilde{V}\left({p}^{(0)},{\varPhi}^{(0)}\right)\\ {}=\sum \limits_{m=1}^n\sum \limits_{i=1}^M\left(\tilde{V}\left({p}^{\left(\left(m-1\right)M+i\right)},{\varPhi}^{\left(\left(m-1\right)M+i\right)}\right)-\tilde{V}\left({p}^{\left(\left(m-1\right)M+i-1\right)},{\varPhi}^{\left(\left(m-1\right)M+i-1\right)}\right)\right)\\ {}=\frac{1}{2}\sum \limits_{m=1}^n\sum \limits_{i=2}^M\left({p}^{\left(\left(m-1\right)M+i\right)}-{p}^{\left(\left(m-1\right)M+i-1\right)}\right)\left({p}^{\left(\left(m-1\right)M+i-1\right)}-\exp \left(\xi {\varPhi}^{\left(\left(m-1\right)M+i-1\right)}\right)\right)\\ {}+\frac{1}{2}\sum \limits_{m=1}^n\sum \limits_{i=2}^M\left({p}^{\left(\left(m-1\right)M+i\right)}-{p}^{\left(\left(m-1\right)M+i-1\right)}\right)\left({p}^{\left(\left(m-1\right)M+i\right)}+\exp \left(\xi {\varPhi}^{\left(\left(m-1\right)M+i-1\right)}\right)\right)\\ {}-{p}^{(nM)}\exp \left(\xi {\varPhi}^{(nM)}\right)+{p}^{(0)}\exp \left(\xi {\varPhi}^{(0)}\right).\end{array}} $$

The term on the left hand side of the above equation has a lower bound. On the right hand side, ${\sum}_{i=2}^M\left({p}^{(i)}-{p}^{\left(i-1\right)}\right)\left({p}^{(i)}+\exp \left(\xi {\varPhi}^{\left(i-1\right)}\right)\right)$ in the second term is fixed, the third term is bounded, and the fourth term is fixed. Therefore, it is obtained that, as n →  + ∞, the following term

$$ {\displaystyle \begin{array}{l}\sum \limits_{m=1}^{+\infty}\sum \limits_{i=2}^M\left({p}^{\left(\left(m-1\right)M+i\right)}-{p}^{\left(\left(m-1\right)M+i-1\right)}\right)\left({p}^{\left(\left(m-1\right)M+i-1\right)}-\exp \left(\xi {\varPhi}^{\left(\left(m-1\right)M+i-1\right)}\right)\right)\\ {}+\sum \limits_{m=2}^{+\infty}\sum \limits_{i=2}^M\left({p}^{\left(\left(m-1\right)M+i\right)}-{p}^{\left(\left(m-1\right)M+i-1\right)}\right)\left({p}^{\left(\left(m-1\right)M+i\right)}+\exp \left(\xi {\varPhi}^{\left(\left(m-1\right)M+i-1\right)}\right)\right)\end{array}} $$

(69)

has a lower bound.

Fourth, by Assumption 2, combining terms (65), (67), and (69) leads to

$$ {\displaystyle \begin{array}{l}\sum \limits_{m=1}^{+\infty }{\left({\mathbf{k}}^{\left( mM+1\right)}-{\mathbf{k}}^{(mM)}\right)}^{\mathrm{T}}\left({\mathbf{k}}^{\left( mM+1\right)}+{p}^{(mM)}{\mathbf{x}}^{(mM)}+\nabla \mathbf{c}{\left({\mathbf{x}}^{(mM)}\right)}^{\mathrm{T}}{\mathbf{x}}^{(mM)}\right)\\ {}+\sum \limits_{m=1}^{+\infty}\sum \limits_{i=2}^M\left({p}^{\left( mM+i\right)}-{p}^{\left( mM+i-1\right)}\right)\left({p}^{\left( mM+i\right)}+{\left({\mathbf{k}}^{\left( mM+i-1\right)}\right)}^{\mathrm{T}}{\mathbf{x}}^{\left( mM+i-1\right)}+\exp \left(\xi {\varPhi}^{\left( mM+i-1\right)}\right)\right)\\ {}+\sum \limits_{m=1}^{+\infty }Z\left({\mathbf{x}}^{\left(\left(m-1\right)M\right)},{\mathbf{d}}^{\left(\left(m-1\right)M\right)},{\mathbf{k}}^{\left(\left(m-1\right)M\right)},{p}^{\left(\left(m-1\right)M\right)}\right)\\ {}+\sum \limits_{m=1}^{+\infty}\sum \limits_{i=1}^Mo\left(\left\Vert \left({\mathbf{x}}^{\left(\left(m-1\right)M+i\right)},{\mathbf{d}}^{\left(\left(m-1\right)M+i\right)}\right)-\left({\mathbf{x}}^{\left(\left(m-1\right)M+i-1\right)},{\mathbf{d}}^{\left(\left(m-1\right)M+i-1\right)}\right)\right\Vert \right)\\ {}\le \left(1-\beta \right)\sum \limits_{m=1}^{+\infty }Z\left({\mathbf{x}}^{\left(\left(m-1\right)M\right)},{\mathbf{d}}^{\left(\left(m-1\right)M\right)},{\mathbf{k}}^{\left(\left(m-1\right)M\right)},{p}^{\left(\left(m-1\right)M\right)}\right)\\ {}+\sum \limits_{m=1}^{+\infty}\sum \limits_{i=1}^Mo\left(\left\Vert \left({\mathbf{x}}^{\left(\left(m-1\right)M+i\right)},{\mathbf{d}}^{\left(\left(m-1\right)M+i\right)}\right)-\left({\mathbf{x}}^{\left(\left(m-1\right)M+i-1\right)},{\mathbf{d}}^{\left(\left(m-1\right)M+i-1\right)}\right)\right\Vert \right)\end{array}} $$

It can be seen from previous explanations of Assumption 2 that Z is continuous with respect to $\left(\mathbf{x},\mathbf{d},\mathbf{k},p\right)\in {\overline{E}}_{\alpha }$ and Z(x, d, k, p) < 0 holds for all $\left(\mathbf{x},\mathbf{d},\mathbf{k},p\right)\in {\overline{E}}_{\alpha }$. Moreover, the set ${\overline{E}}_{\alpha }$ is compact. Therefore, according to Theorem 5.2.12 in Trench (2003), there exists a constant h > 0 such that Z(x, d, k, p) <  − h for all $\left(\mathbf{x},\mathbf{d},\mathbf{k},p\right)\in {\overline{E}}_{\alpha }$.

In addition, it holds that

$$ o\left(\left\Vert \left({\mathbf{x}}^{\left(i+1\right)},{\mathbf{d}}^{\left(i+1\right)}\right)-\left({\mathbf{x}}^{(i)},{\mathbf{d}}^{(i)}\right)\right\Vert \right)/\left\Vert \left({\mathbf{x}}^{\left(i+1\right)},{\mathbf{d}}^{\left(i+1\right)}\right)-\left({\mathbf{x}}^{(i)},{\mathbf{d}}^{(i)}\right)\right\Vert \to 0, $$

as ‖(x^(i + 1), d^(i + 1)) − (x⁽ⁱ⁾, d⁽ⁱ⁾)‖ → 0 for any i = 0, 1, 2, ⋯. Thus, under the precondition that the set Ω_LD is compact, there exists a constant $\overline{h}>0$ such that

$$ o\left(\left\Vert \left({\mathbf{x}}^{\left(i+1\right)},{\mathbf{d}}^{\left(i+1\right)}\right)-\left({\mathbf{x}}^{(i)},{\mathbf{d}}^{(i)}\right)\right\Vert \right)<\left(1-\beta \right)h/(2M) $$

if $\left\Vert \left({\mathbf{x}}^{\left(i+1\right)},{\mathbf{d}}^{\left(i+1\right)}\right)-\left({\mathbf{x}}^{(i)},{\mathbf{d}}^{(i)}\right)\right\Vert \le \overline{h}$ for any i = 0, 1, 2, ⋯. Further, from Assumption 1, it is obtained that

$$ o\left(\left\Vert \left({\mathbf{x}}^{\left(i+1\right)},{\mathbf{d}}^{\left(i+1\right)}\right)-\left({\mathbf{x}}^{(i)},{\mathbf{d}}^{(i)}\right)\right\Vert \right)<\left(1-\beta \right)h/(2M), $$

for any i = 0, 1, 2, ⋯.

Thus, if the trajectory of the dynamical system model always belongs to the set ${\overline{E}}_{\alpha }$, then

$$ {\displaystyle \begin{array}{l}\left(1-\beta \right)Z\left({\mathbf{x}}^{\left(\left(m-1\right)M\right)},{\mathbf{d}}^{\left(\left(m-1\right)M\right)},{\mathbf{k}}^{\left(\left(m-1\right)M\right)},{p}^{\left(\left(m-1\right)M\right)}\right)\\ {}+\sum \limits_{i=1}^Mo\left(\left\Vert \left({\mathbf{x}}^{\left(\left(m-1\right)M+i\right)},{\mathbf{d}}^{\left(\left(m-1\right)M+i\right)}\right)-\left({\mathbf{x}}^{\left(\left(m-1\right)M+i-1\right)},{\mathbf{d}}^{\left(\left(m-1\right)M+i-1\right)}\right)\right\Vert \right)<-\left(1-\beta \right)h/2,\end{array}} $$

(70)

for m = 0, 1, 2, ⋯. Furthermore,

$$ {\displaystyle \begin{array}{l}\left(1-\beta \right)\sum \limits_{m=1}^{+\infty }Z\left({\mathbf{x}}^{\left(\left(m-1\right)M\right)},{\mathbf{d}}^{\left(\left(m-1\right)M\right)},{\mathbf{k}}^{\left(\left(m-1\right)M\right)},{p}^{\left(\left(m-1\right)M\right)}\right)\\ {}+\sum \limits_{m=1}^{+\infty}\sum \limits_{i=1}^Mo\left(\left\Vert \left({\mathbf{x}}^{\left(\left(m-1\right)M+i\right)},{\mathbf{d}}^{\left(\left(m-1\right)M+i\right)}\right)-\left({\mathbf{x}}^{\left(\left(m-1\right)M+i-1\right)},{\mathbf{d}}^{\left(\left(m-1\right)M+i-1\right)}\right)\right\Vert \right)<-\infty .\end{array}} $$

(71)

This contradicts the fact that all terms (65), (67), and (69) have a lower bound. Therefore, the sequence (x⁽ⁿ⁾, d⁽ⁿ⁾, k⁽ⁿ⁾, p⁽ⁿ⁾) (n = 0, 1, 2, ⋯) must leave ${\overline{E}}_{\alpha }$ and enter E_α for some n₀. Once the trajectory enters the set E_α, it is impossible that the trajectory leaves and enters E_α endlessly or leaves E_α forever. Otherwise, eq. (71) still holds because the term on the left hand side of eq. (70) is non-positive under Assumption 1 (no matter whether the trajectory is in the set ${\overline{E}}_{\alpha }$ or in E_α). Thus, the trajectory will enter and stay in the set E_α after some n₁ (≥n₀).