Analysis framework and assumptions
The implementation of the YHRWDP has changed the allocation of water resources and the distribution of pollution in the original water network. To further improve the water environment and ecological problems in the middle and lower reaches of the Han River basin resulting from the water resources replacement as well as to ensure the water demand of the public in the Han River basin. The central government, which is in charge of the construction of the national water network, should be involved as one of the participants and assume the responsibility of coordinating the water pollution control. The WSA, which diverts class III water from the main stream of the Yangtze River into the Han River to replace class II water, should also assume the role as one of the participants. In addition to protecting the water environment and ecology of the middle and lower reaches of the Han River, meanwhile to guarantee the water demand of the public in the Han River Basin, the PE and WDE should also be the parties involved. In order to encourage WSA to improve water quality, the central government borne part of the cost of water environmental protection and water pollution control. At the same time, the central government will share the cost of water pollution treatment in order to encourage PE to improve the standard of wastewater discharge and to decrease the amount of pollutants discharged. In order to alleviate the cost pressure of WDE, the central government will take part of the cost of water processing of WDE. The problem analysis framework is shown in Fig. 2.
Assumption 1
Assumed that the cost of water quality protection of WSA is in the form of quadratic function \(\frac{{r_{1} }}{2}S_{1}^{2}\), where \(r_{1}\) represents the cost coefficient of water quality protection. The effort degree of water quality protection of WSA is \(S_{1}\), \(0 < S_{1} < 1\), which represents the water environment protection, water pollution control, water quality monitoring and other inputs. Assume that \(q_{S}\) is the total amount of pollutants flowing into the Han River without any water quality protection measures.
Assumption 2
The WSA bears the dual responsibility of economic development and water quality improvement. While aiming to meet its own ecological benefits, the WSA tends to prioritize resources for local economic development, such as industrial production and urban expansion. This often leads to insufficient investment in pollution control, making it difficult to fully eliminate the negative impacts of water quality differences on the middle and lower reaches of the Han River. When pollution shifts downstream to the Han River, the pollutants entering the river create reduction pressures for PE and water treatment pressures for WDE. In this case, the WSA transfers part of its pollution control responsibilities to downstream enterprises. Since one party’s increased efforts in governance can reduce the other party’s responsibility, the WSA believes that the efforts of PE and WDE can partially replace its own responsibility. Assume that the pollutants flowing into the Han River create a reduction pressure on PE, represented by \(\alpha\), \(0 < \alpha < 1\). This parameter is used to measure the degree of responsibility transfer between the water supplier and water diversion enterprises. Similarly, the pollutants flowing into the Han River create water treatment pressure on WDE, represented by \(x\), \(0 < x < 1\), which measures the degree of responsibility transfer between the WSA and WDE. Additionally, the emissions from PE create water treatment pressure on WDE, represented by \(y\), \(0 < y < 1\), which measures the degree of responsibility transfer between PE and WDE.
Assumption 3
Assumed that the emission reduction cost of PE is \(\frac{{r_{2} }}{2}S_{2}^{2}\), where \(r_{2}\) is the emission reduction cost coefficient and \(S_{2}\) is the emission reduction effort of PE, \(0 < S_{2} < 1\), which represents the personnel and capital input of PE, and is affected by the amount of pollutants flowing into the Han River from WSA. Assumed that \(S_{0}\) is the initial emission reduction effort of PE when weighing its own benefits and costs without considering pressure factor.
Assumption 4
The amount of pollutant elimination in the watershed is the result of the joint efforts of WSA and PE, and it is a process that changes dynamically with time. Assume that the amount of pollutant elimination in the watershed can be expressed as the following dynamic differential equation:
$$\dot{q}(t) = \eta_{1} S_{1} + \eta_{2} S_{2} – \theta q(t)$$
(1)
Among them, \(\eta_{1}\) is the pollution elimination of WSA paying unit pollution control efforts. \(\eta_{2}\) is the pollution elimination of PE paying unit pollution control efforts, and \(\theta\) is the attenuation coefficient of emission reductions due to the aging of sewage treatment equipment and other factors.
Assumption 5
Assumed that the water treatment cost of WDE is \(\frac{{r_{3} }}{2}Z^{2}\), where \(r_{3}\) is the comprehensive water pollution treatment cost coefficient. \(Z\) is the degree of effort in water treatment of WDE. Assumed that \(Z_{0}\) is the initial emission reduction effort of PE when weighing its own benefits and costs without considering pressure factors.
Assumption 6
The effort of the WSA is not only affected by the pressure coefficients of pollutants on the PE and WDE, but also by the effort of the PE and WDE and the amount of pollutants eliminated per unit of pollution control effort, because the amount of pollutants eliminated per unit of pollution control effort to a certain extent represents the management capacity within the enterprise. It is assumed that the water supplier’s effort is a linear function of the PE’s and WDE’s effort and the amount of pollutant eliminated per unit of treatment effort, as well as the pressure coefficients generated by the water supplier on the PE and the WDE, expressed as:
$$S_{1} = S_{01} – \alpha \eta_{2} S_{2} – x\eta_{3} Z$$
(2)
Among them, \(S_{01}\) is the initial level of effort to protect water quality when the WSA does not transfer responsibility. Similarly, the effort of the PE is influenced by the pollutants flowing into the Han River as well as by the WSA and WDE. It is expressed as:
$$S_{2} = S_{0} + \alpha (q_{E} – \eta_{1} S_{1} ) – y\eta_{3} Z$$
(3)
The effort level of WDE is expressed as:
$$Z = Z_{0} + x(q_{E} – \eta_{1} S_{1} ) – y\eta_{2} S_{2}$$
(4)
Assumption 7
Assumed that the emission tax that PE needs to pay per unit of pollutant discharge is \(\omega\).
Assumption 8
The water quality protection behavior of WSA and the emission reduction behavior of PE can improve local social welfare. Social welfare can be expressed as:
$$W(t) = w_{0} + \beta_{1} S_{1} + \beta_{2} S_{2} + \varphi q(t)$$
(5)
Among them, \(w_{0} > 0\) represents the initial welfare status of the river basin, \(\beta_{1} > 0\) represents the impact coefficient of the pollution control work of WSA on social welfare, \(\beta_{2} > 0\) represents the impact coefficient of the emission reduction work of PE on social welfare, \(\varphi > 0\) represents the impact coefficient of pollutant removal in the watershed on social welfare.
Assumption 9
The coefficients of the social welfare effect of the river basin on the income of WSA, PE, WDE, and the central government are \(\pi_{1}\), \(\pi_{2}\), \(\pi_{3}\) and \(\pi_{4}\), respectively, \(\pi_{1} > 0\), \(\pi_{2} > 0\), \(\pi_{3} > 0\), \(\pi_{4} > 0\).
Assumption 10
Ignoring the impact of water prices on water resource demand, assume that the demand for water resources is \(z = u + \sigma q(t)_{3}\). Among them, \(u\) is the fixed demand for water resources, \(\sigma\) is the user’s experience of water quality, \(q(t)_{3}\) is the pollutant treatment capacity of WDE, which can be expressed as:
$$\dot{q}(t)_{3} = \eta_{3} Z – \theta q(t)_{3}$$
(6)
Among them, \(\eta_{3}\) is the pollution elimination of WDE paying unit pollution control efforts.
Assumption 11
The participating entities have the same discount rate \(\rho\), with \(\rho > 0\).
Model building and solving
Establish a differential game model based on a cost-sharing mechanism, in which the central government, as the beneficiary of the water resource exchange, provides WSA with cost compensation based on the volume of water transferred, as well as a certain proportion of water quality protection costs, covers a certain proportion of increased pollution control cost to PE and WDE. Game participants are all finite rational subjects. The common goal of WSA, PE and WDE is to seek the optimal pollution control strategy to maximize their own interests, while the goal of the central government is to maximize its own interests by coordinating pollution control efforts through financial support. The decision sequence is: the central government decides the cost-sharing ratio for WSA, PE and WDE, and then WSA, PE and WDE decide their own optimal strategies.
Using the inverse induction method, we first solve the optimal decisions of WSA, PE and WDE. The revenue functions are:
The revenue function of WSA is:
$$\int_{0}^{t} {[H_{0} } + m_{1} \eta_{1} S_{1} + \pi_{1} (w_{0} + \beta_{1} S_{1} + \beta_{2} S_{2} + \varphi q(t)) – (1 – \xi_{1} )\frac{1}{2}r_{1} S_{1}^{2} ]e^{ – \rho t} dt$$
(7)
where \(H_{0}\) is the cost compensation from the central government to WSA based on the amount of water transferred, \(m_{1}\) is the environmental benefit that WSA brings to the central government and itself by combating pollution, \(m_{1} > 0\), and \(\xi_{1}\) is the cost-sharing ratio from the central government to WSA.
The revenue function of PE is:
$$\begin{gathered} \int_{0}^{t} {[J + kS_{2} + \pi_{2} } (w_{0} + \beta_{1} S_{1} + \beta_{2} S_{2} + \varphi q(t)) – (1 – \xi_{2} )\frac{1}{2}r_{2} S_{2}^{2} – \frac{1}{2}\xi_{2} r_{2} S_{0}^{2} – (q_{E} – \eta_{2} S_{2} )\omega \hfill \\ \;\;\;\;\; + n_{1} \eta_{1} S_{1} ]e^{ – \rho t} dt \hfill \\ \end{gathered}$$
(8)
where \(J\) is the benefit gained by the production activities of PE. \(k\) represents the reputational benefit obtained through the emission reduction efforts of PE. \(n_{1}\) represents the benefit due to WSA’s pollution control to relieve pollution control pressure of PE \(n_{1} < r_{2}\), \(q_{E}\) is the pollutant emission when PE does not increase emission reduction, and \(\xi_{2}\) is the proportion of the central government’s cost-sharing to PE.
The revenue function of WDE is:
$$\begin{gathered} \int_{0}^{t} {[P(u + \sigma q(t)_{3} } ) + \pi_{3} (w_{0} + \beta_{1} S_{1} + \beta_{2} S_{2} + \varphi q(t)) – (1 – \xi_{3} )\frac{1}{2}r_{3} Z^{2} – \frac{1}{2}\xi_{3} r_{3} Z_{0}^{2} \hfill \\ \;\;\;\;\; + n_{2} \eta_{1} S_{1} + n_{3} \eta_{2} S_{2} ]e^{ – \rho t} dt \hfill \\ \end{gathered}$$
(9)
where \(P\) is the price of water resources, \(u\) is the fixed market demand for water resources, \(\sigma\) is the user’s experience of water quality, \(n_{2}\) represents the benefit due to WSA’s pollution control to relieve water treatment pressure of WDE, \(n_{3}\) represents the benefit due to PE efforts of reducing emissions to relieve water treatment pressure of WDE, and \(\xi_{3}\) is the proportion of the cost-sharing of the central government to WDE.
In order to make Eqs. (2) and (4) have unique continuous solutions \(q(t)\) and \(q(t{)}_{3}\), construct continuous bounded differential functions \({V}_{1}\), \({V}_{2}\), \({V}_{3}\), the Hamilton–Jacobi-Bellman-Fleming (HJB) equations for WSA, PE and WDE are respectively:
$$\rho V_{1} = m\mathop a\limits_{{S_{1} }} x\left\{ \begin{gathered} H_{0} + m_{1} \eta_{1} S_{1} + \pi_{1} (w_{0} + \beta_{1} S_{1} + \beta_{2} S_{2} + \varphi q_{(t)} ) – (1 – \xi_{1} )\frac{1}{2}r_{1} S_{1}^{2} \hfill \\ + V_{1}^{\prime} [\eta_{1} S_{1} + \eta_{2} S_{2} – \theta q(t)] \hfill \\ \end{gathered} \right\}$$
(10)
$$\rho V_{2} = m\mathop a\limits_{{S_{2} }} x\left\{ \begin{gathered} J + kS_{2} + \pi_{2} (w_{0} + \beta_{1} S_{1} + \beta_{2} S_{2} + \varphi q(t)) – (1 – \xi_{2} )\frac{1}{2}r_{2} S_{2}^{2} – \hfill \\ \frac{1}{2}\xi_{2} r_{2} S_{0}^{2} – (q_{E} – \eta_{2} S_{2} )\omega + n_{1} \eta_{1} S_{1} + V_{2}^{\prime} (\eta_{1} S_{1} + \eta_{2} S_{2} – \theta q(t)) \hfill \\ \end{gathered} \right\}$$
(11)
$$\rho V_{3} = m\mathop a\limits_{Z} x\left\{ \begin{gathered} P(u + \sigma q(t){}_{3}) + \pi_{3} (w_{0} + \beta_{1} S_{1} + \beta_{2} S_{2} + \varphi q(t)) – (1 – \xi_{3} )\frac{1}{2}r_{3} Z^{2} – \frac{1}{2}\xi_{3} r_{3} Z_{0}^{2} \hfill \\ + n_{2} \eta_{1} S_{1} + n_{3} \eta_{2} S_{2} + V_{3}^{\prime} (\eta_{1} S_{1} + \eta_{2} S_{2} – \theta q(t)) + V_{3}^{P} (\eta_{3} Z – \theta q(t)_{3} ) \hfill \\ \end{gathered} \right\}$$
(12)
In order to reflect the mutual influence of the efforts of each subject, we substitute formula (3) and (4) into formula (10), substitute formula (2) and (4) into formula (11), and substitute formula (2) and (3) into formula (12). By the maximization of first-order conditions of combining (10) and (11) and (12) for \(S_{1}\), \(S_{2}\), and \(Z\), we get
$$S_{1} = \frac{{m_{1} \eta_{1} + \pi_{1} \beta_{1} – \pi_{1} \beta_{2} \alpha \eta_{1} + V_{1}^{\prime} (\eta_{1} – \alpha \eta_{1} \eta_{2} )}}{{(1 – \xi_{1} )r_{1} }}$$
(13)
$$S_{2} = \frac{{k – \pi_{2} \beta_{1} \alpha \eta_{2} + \pi_{2} \beta_{2} + \eta_{2} \omega – \alpha n_{1} \eta_{1} \eta_{2} + V_{2}^{\prime} (\eta_{2} – \alpha \eta_{1} \eta_{2} )}}{{(1 – \xi_{2} )r_{2} }}$$
(14)
$$Z = \frac{{V_{3}^{P} \eta_{3} – \pi_{3} (x\beta_{1} \eta_{3} + y\beta_{2} \eta_{3} ) – xn_{2} \eta_{1} \eta_{3} – yn_{3} \eta_{2} \eta_{3} – V_{3}^{\prime} (x\eta_{1} \eta_{3} + y\eta_{2} \eta_{3} )}}{{(1 – \xi_{3} )r_{3} }}$$
(15)
The central government considers the efforts of WSA, PE and WDE to determine the cost-sharing rate. The revenue function of the central government is:
$$\begin{gathered} \int_{0}^{t} {[G – H_{0} + m_{1} } \eta_{1} S_{1} – m_{2} (q_{S} – \eta_{1} S_{1} ) + m_{3} \eta_{2}( S_{2} – S_{0}) + \pi_{4} (w_{0} + \beta_{1} S_{1} + \beta_{2} S_{2} \hfill \\ \;\;\;\; + \varphi q(t)) + m_{4} Z – \frac{1}{2}\xi_{1} r_{1} S_{1}^{2} – \frac{1}{2}\xi_{2} r_{2} (S_{2}^{2} – S_{0}^{2} ) – \frac{1}{2}\xi_{3} r_{3} (Z_{{}}^{2} – Z_{0}^{2} )]e^{ – \rho t} dt \hfill \\ \end{gathered}$$
(16)
where \(G\) is the benefit brought to the central government by the increased northward transfer of water from the SNWDP, \(m_{2}\) is the environmental loss brought to the central government by the pollutants of the Yangtze River flowing into the Han River, \(m_{3}\) is the environmental benefit brought to the central government by the increase in emissions reductions of PE, and \(m_{4}\) is the social benefit brought to central government of integrated water resources treatment of WDE.
Its objective function satisfies the HJB equation:
$$\rho V_{4} = \mathop {\max }\limits_{{\xi_{1} \xi_{{2}} \xi_{3} }} \mathop {\left\{ \begin{gathered} G – H_{0} + m_{1} \eta_{1} S_{1} – m_{2} (q_{S} – \eta_{1} S_{1} ) + m_{3} \eta_{2} (S_{2} – S_{0} ) + m_{4} Z \hfill \\ + \pi_{4} (w_{0} + \beta_{1} S_{1} + \beta_{2} S_{2} + \varphi q(t)) – \frac{1}{2}\xi_{1} r_{1} S_{1}^{2} – \frac{1}{2}\xi_{2} r_{2} (S_{2}^{2} – S_{0}^{2} ) \hfill \\ – \frac{1}{2}\xi_{3} r_{3} (Z^{2} – Z_{0}^{2} ) + V_{4}^{\prime} (\eta_{1} S_{1} + \eta_{2} S_{2} – \theta q(t)) \hfill \\ \end{gathered} \right\}}\limits_{{}}$$
(17)
Substituting Eqs. (13), (14), and (15) into the central government’s HJB equation, and solving the first-order maximization conditions for \(\xi_{1}\), \(\xi_{2}\), and \(\xi_{3}\), we get:
$$\xi_{1} = \frac{{2(m_{1} \eta_{1} + m_{2} \eta_{1} + \pi_{4} \beta_{1} + V_{4}^{\prime} \eta_{1} ) – m_{1} \eta_{1} – \pi_{1} \beta_{1} + \pi_{1} \beta_{2} \alpha \eta_{1} – V_{1}^{\prime} (\eta_{1} – \alpha \eta_{1} \eta_{2} )}}{{2(m_{1} \eta_{1} + m_{2} \eta_{1} + \pi_{4} \beta_{1} + V_{4}^{\prime} \eta_{1} ) + m_{1} \eta_{1} + \pi_{1} \beta_{1} – \pi_{1} \beta_{2} \alpha \eta_{1} + V_{1}^{\prime} (\eta_{1} – \alpha \eta_{1} \eta_{2} )}}$$
(18)
$$\xi_{2} = \frac{{2(m_{3} \eta_{2} + \pi_{4} \beta_{2} + V_{4}^{\prime} \eta_{2} ) – k + \pi_{2} \beta_{1} \alpha \eta_{2} – \pi_{2} \beta_{2} – \eta_{2} \omega + \alpha n_{1} \eta_{1} \eta_{2} – V_{2}^{\prime} (\eta_{2} – \alpha \eta_{1} \eta_{2} )}}{{2(m_{3} \eta_{2} + \pi_{4} \beta_{2} + V_{4}^{\prime} \eta_{2} ) + k – \pi_{2} \beta_{1} \alpha \eta_{2} + \pi_{2} \beta_{2} + \eta_{2} \omega – \alpha n_{1} \eta_{1} \eta_{2} + V_{2}^{\prime} (\eta_{2} – \alpha \eta_{1} \eta_{2} )}}$$
(19)
$$\xi_{3} = \frac{{2m_{4} – V_{3}^{P} \eta_{3} + \pi_{3} (x\beta_{1} \eta_{3} + y\beta_{2} \eta_{3} ) + xn_{2} \eta_{1} \eta_{3} + yn_{3} \eta_{2} \eta_{3} + V_{3}^{\prime} (x\eta_{1} \eta_{3} + y\eta_{2} \eta_{3} )}}{{2m_{4} + V_{3}^{P} \eta_{3} – \pi_{3} (x\beta_{1} \eta_{3} + y\beta_{2} \eta_{3} ) – xn_{2} \eta_{1} \eta_{3} – yn_{3} \eta_{2} \eta_{3} – V_{3}^{\prime} (x\eta_{1} \eta_{3} + y\eta_{2} \eta_{3} )}}$$
(20)
Assumed that the expressions of functions \({V}_{1}\), \({V}_{2}\), \({V}_{3}\), and \({V}_{4}\) are in linear form, the equilibrium solution of the game can be obtained:
$$S_{1}^{*} = \frac{{m_{1} \eta_{1} + \pi_{1} \beta_{1} – \pi_{1} \beta_{2} \alpha \eta_{1} + \frac{{\pi_{1} \varphi }}{\theta + \rho }(\eta_{1} – \alpha \eta_{1} \eta_{2} )}}{{(1 – \xi_{1} )r_{1} }}$$
(21)
$$S_{2}^{*} = \frac{{k – \pi_{2} \beta_{1} \alpha \eta_{2} + \pi_{2} \beta_{2} + \eta_{2} \omega – \alpha n_{1} \eta_{1} \eta_{2} + \frac{{\pi_{2} \varphi }}{\theta + \rho }(\eta_{2} – \alpha \eta_{1} \eta_{2} )}}{{(1 – \xi_{2} )r_{2} }}$$
(22)
$$Z^{*} = \frac{{\frac{P\sigma }{{\theta + \rho }}\eta_{3} – \pi_{3} (x\beta_{1} \eta_{3} + y\beta_{2} \eta_{3} ) – xn_{2} \eta_{1} \eta_{3} – yn_{3} \eta_{2} \eta_{3} – \frac{{\pi_{3} \varphi }}{\theta + \rho }(x\eta_{1} \eta_{3} + y\eta_{2} \eta_{3} )}}{{(1 – \xi_{3} )r_{3} }}$$
(23)
$$\xi_{1}^{*} = \frac{{2(m_{1} \eta_{1} + m_{2} \eta_{1} + \pi_{4} \beta_{1} + \frac{{\pi_{4} \varphi }}{\theta + \rho }\eta_{1} ) – m_{1} \eta_{1} – \pi_{1} \beta_{1} + \pi_{1} \beta_{2} \alpha \eta_{1} – \frac{{\pi_{1} \varphi }}{\theta + \rho }(\eta_{1} – \alpha \eta_{1} \eta_{2} )}}{{2(m_{1} \eta_{1} + m_{2} \eta_{1} + \pi_{4} \beta_{1} + \frac{{\pi_{4} \varphi }}{\theta + \rho }\eta_{1} ) + m_{1} \eta_{1} + \pi_{1} \beta_{1} – \pi_{1} \beta_{2} \alpha \eta_{1} + \frac{{\pi_{1} \varphi }}{\theta + \rho }(\eta_{1} – \alpha \eta_{1} \eta_{2} )}}$$
(24)
$$\xi_{2}^{*} = \frac{{2(m_{3} \eta_{2} + \pi_{4} \beta_{2} + \frac{{\pi_{4} \varphi }}{\theta + \rho }\eta_{2} ) – k + \pi_{2} \beta_{1} \alpha \eta_{2} – \pi_{2} \beta_{2} – \eta_{2} \omega + \alpha n_{1} \eta_{1} \eta_{2} – \frac{{\pi_{2} \varphi }}{\theta + \rho }(\eta_{2} – \alpha \eta_{1} \eta_{2} )}}{{2(m_{3} \eta_{2} + \pi_{4} \beta_{2} + \frac{{\pi_{4} \varphi }}{\theta + \rho }\eta_{2} ) + k – \pi_{2} \beta_{1} \alpha \eta_{2} + \pi_{2} \beta_{2} + \eta_{2} \omega – \alpha n_{1} \eta_{1} \eta_{2} + \frac{{\pi_{2} \varphi }}{\theta + \rho }(\eta_{2} – \alpha \eta_{1} \eta_{2} )}}$$
(25)
$$\xi_{3}^{*} = \frac{{2m_{4} – \frac{P\sigma }{{\theta + \rho }}\eta_{3} + \pi_{3} (x\beta_{1} \eta_{3} + y\beta_{2} \eta_{3} ) + xn_{2} \eta_{1} \eta_{3} + yn_{3} \eta_{2} \eta_{3} + \frac{{\pi_{3} \varphi }}{\theta + \rho }(x\eta_{1} \eta_{3} + y\eta_{2} \eta_{3} )}}{{2m_{4} + \frac{P\sigma }{{\theta + \rho }}\eta_{3} – \pi_{3} (x\beta_{1} \eta_{3} + y\beta_{2} \eta_{3} ) – xn_{2} \eta_{1} \eta_{3} – yn_{3} \eta_{2} \eta_{3} – \frac{{\pi_{3} \varphi }}{\theta + \rho }(x\eta_{1} \eta_{3} + y\eta_{2} \eta_{3} )}}$$
(26)
The optimal returns of WSA, PE, WDE and the central government under this equilibrium condition are respectively:
$$V_{1}^{*} = \frac{{\pi_{1} \varphi }}{\theta + \rho }q(t)^{*} + \frac{1}{\rho }\left[ \begin{gathered} H_{0} + \left( {m_{1} \eta_{1} + \pi_{1} \beta_{1} – (1 – \xi_{1} )\frac{1}{2}r_{1} S_{1}^{*} + \frac{{\pi_{1} \varphi }}{\theta + \rho }\eta_{1} } \right)S_{1}^{*} + \left( {\pi_{1} \beta_{2} + \frac{{\pi_{1} \varphi }}{\theta + \rho }\eta_{2} } \right)S_{2}^{*} \hfill \\ + \pi_{1} w_{0} \hfill \\ \end{gathered} \right]$$
(27)
$$V_{2}^{*} = \frac{{\pi_{2} \varphi }}{\theta + \rho }q(t)^{*} + \frac{1}{\rho }\left[ \begin{gathered} J + \left( {k + \pi_{2} \beta_{2} – (1 – \xi_{2} )\frac{1}{2}r_{2} S_{2}^{*} + \eta_{2} \omega + \frac{{\pi_{2} \varphi }}{\theta + \rho }\eta_{2} } \right)S_{2}^{*} \hfill \\ + \left( {\pi_{2} \beta_{1} + n_{1} \eta_{1} + \frac{{\pi_{2} \varphi \eta_{1} }}{\theta + \rho }} \right)S_{1}^{*} + \pi_{2} w_{0} – \frac{1}{2}\xi_{2} r_{2} S_{0}^{2} – q_{E} \omega \hfill \\ \end{gathered} \right]$$
(28)
$$V_{3}^{*} = \frac{{\pi_{3} \varphi }}{\theta + \rho }q(t)^{*} + \frac{\sigma P}{{\theta + \rho }}q(t)_{3}^{*} + \frac{1}{\rho }\left[ \begin{gathered} Pu + \left( {\frac{p\sigma }{{\theta + \rho }}\eta_{3} – (1 – \xi_{3} )\frac{1}{2}r_{3} Z^{*} } \right)Z^{*} + \left( {\pi_{3} \beta_{1} + n_{2} \eta_{1} } \right. \hfill \\ \left. { + \frac{{\pi_{3} \varphi \eta_{1} }}{\theta + \rho }} \right)S_{1}^{*} + \left( {\pi_{3} \beta_{2} + n_{3} \eta_{2} + \frac{{\pi_{3} \varphi \eta_{2} }}{\theta + \rho }} \right)S_{2}^{*} + \pi_{3} w_{0} – \frac{1}{2}\xi_{3} r_{3} Z_{0}^{2} \hfill \\ \end{gathered} \right]$$
(29)
$$V_{4}^{*} = \frac{{\pi_{4} \varphi }}{\theta + \rho }q(t)^{*} + \frac{1}{\rho }\left[ \begin{gathered} G – H_{0} + \left( {m_{1} \eta_{1} + m_{2} \eta_{1} + \pi_{4} \beta_{1} – \frac{1}{2}\xi_{1} r_{1} S_{1}^{*} + \frac{{\pi_{4} \varphi }}{\theta + \rho }\eta_{1} } \right)S_{1}^{*} \hfill \\ + \left( {m_{3} \eta_{2} + \pi_{4} \beta_{2} – \frac{1}{2}\xi_{2} r_{2} S_{2}^{*} + \frac{{\pi_{4} \varphi \eta_{2} }}{\theta + \rho }} \right)S_{2}^{*} + \left( {m_{4} – \frac{1}{2}\xi_{3} r_{3} Z^{*} } \right)Z^{*} \hfill \\ – m_{2} q_{S} – m_{3} \eta_{2} S_{0} + \pi_{4} w_{0} + \frac{1}{2}\xi_{2} r_{2} S_{0}^{2} + \frac{1}{2}\xi_{3} r_{3} Z_{0}^{2} \hfill \\ \end{gathered} \right]$$
(30)
