The green supply chain should integrate environmental factors into product design, procurement, manufacturing, sales, and distribution (Yu and Khan, 2021). Khan et al. (2021e) believed that the application of advanced technology and the establishment of contact with the external environment can break through multiple supply chain management barriers. In order to achieve more effective green innovation, companies use their social capital to generate additional competitive advantages through collaboration (Chen and Hung, 2014). Collaborative innovation provides a new direction for the development of GSCs (Zhang and He, 2018), and is an important link to gain competitive advantage and achieve sustainable development goals (Zhou et al., 2020). GSC collaborative innovation is to use the specific advantages of individual organizations to jointly solve the problem of green management through a collaborative approach (Hong et al., 2019).

Hong et al. (2019) revealed the positive impact of organization-organization collaborative innovation, organization-government collaborative innovation, and organization-institution collaborative innovation on innovation performance. Collaborative innovation partner selection is an important factor affecting the green innovation of supply chain enterprises. The technical capabilities, resource integration capabilities, and learning capabilities of partners should be focused on (Melander, 2018; Xia et al., 2020). Hao and Li (2020) explored the collaborative innovation between enterprises in a green supply chain composed of a manufacturer and a supplier. The research found that the cooperative game is the optimal scenario for collaborative innovation. In terms of the impact of consumer preferences on green innovation, He et al. (2019) found that changes in consumer preferences are a key factor in promoting green innovation by companies in the supply chain. Li Q. et al. (2020) studied the decision-making mechanism and determinants of green collaborative innovation from the perspective of transaction cost economics and social exchange theory. Yang and Lin (2020) confirmed the importance of supply chain collaboration to green innovation, and found that environmental regulations, top management commitment and social recognition are the main driving forces for the implementation of green innovation. Zhai et al. (2021) studied the positive effects of governmental incentives and punishments, the trust between enterprises on the collaborative innovation of green technology in enterprises. Song et al. (2020) analyzed the mediating role of absorptive capacity between green KS and green innovation, and suggested that companies develop their absorptive capacity to achieve green innovation.

Collaborative innovation improves the environmental performance of the GSC and is conducive to promoting the sustainable development of the GSC (Abbas and Sagsan, 2019; Shi et al., 2021). It can be found that the existing research seldom pays attention to the influence of knowledge on GSC collaborative innovation, and seldom studies the decision-making mechanism of KS behavior of suppliers and manufacturers. What is the impact mechanism between KS and LC technology innovation? It is necessary to study LC technology innovation in the green supply chain from the perspective of knowledge sharing. This is essential to improve the efficiency of LC technology innovation.

The innovation and promotion of renewable energy can curb carbon dioxide emissions, help the government develop clean industries, and achieve carbon neutrality goals (Balsalobre-Lorente et al., 2021), while having a positive impact on economic development (Khan et al., 2021b). Bekun et al. (2021) believes that sustainable economic growth can be achieved by developing clean technologies and increasing the share of renewable energy in the energy structure. Krarti and Aldubyan (2021) advocated the use of photovoltaics and wind as renewable energy technologies to achieve carbon neutrality in residential communities. Cao et al. (2021) studied the impact of cleaner production standards on the total factor productivity and resource allocation efficiency of Chinese energy companies. They found that cleaner production regulations reduce the efficiency of resource allocation in the energy industry, but increase the level of enterprise total factor productivity. Gunarathne and Sankalpani (2021) believed that clean technology should be promoted as a management technology. Gupta et al. (2021) established a framework based on the concepts of circular economy, sustainable clean production and industry 4.0 standards to evaluate the sustainable performance of manufacturing companies. Wei et al. (2022) suggested the use of photovoltaic power generation and electrolysis of hydrogen to achieve carbon emission neutrality in industrial parks, and used examples to prove that this solution would reduce emissions by 61% compared with the solution relying on the grid and natural gas. Fortes et al. (2022) analyzed the impact of climate change on the demand for renewable resources and electricity. The results showed that climate change reduces hydropower generation by 20%, which will also reduce the cost-effectiveness of solar photovoltaics.

The application and innovation of cleaner production technologies can help achieve carbon neutrality. However, it can be found that the research on knowledge management in cleaner production has not received much attention.

KS and collaborative innovation are effective models for the development of LC technologies. LC technology innovation and knowledge accumulation are mutually reinforcing (Hoette et al., 2021; Vinholis et al., 2021). Technology innovation can promote knowledge accumulation (Vinholis et al., 2021), and LC technology innovation depends on knowledge accumulation (Hoette et al., 2021). KS has a significant impact on the performance of LC technology innovation (Brandao Vinholis et al., 2021; Effendi et al., 2021). Gu et al. (2021) studied the impact of global LC technology transfer on carbon emission reduction. The study found that the sharing of LC technologies has achieved a significant increase in carbon emission reduction, while the emission reduction in developing countries is mainly limited by the stock of knowledge and research and development investment. KS in the process of LC technology transfer and innovation will be affected by organizational mechanisms (Hayashi, 2018). Abbas and Sagsan (2019) studied the effect of knowledge management on the green innovation and sustainable development of enterprises. Through structural equation models, they proved that KS has a significant impact on green innovation, and green innovation can promote the sustainable development of enterprises. The willingness of enterprises to share knowledge is related to the breadth and depth of their own knowledge. Enterprises with a wide range of knowledge are more willing to share knowledge, while enterprises with deep knowledge are unwilling to share knowledge (Lyu C. et al., 2020).

From the literature review, it can be found that the current research on LC technology innovation and KS mainly focuses on the impact of KS on LC technology innovation performance, the role of KS in promoting LC technology innovation and some obstacles to the implementation of KS. However, scholars seldom pay attention to the evolution of KS strategies in LC technology innovation and the dynamic changes of knowledge stock and emission reduction benefits under different sharing strategies. As time changes, knowledge will be updated and attenuated, and the KS strategy in LC technology innovation should also be constantly changing. Therefore, from the perspective of dynamic evolution, it is necessary to study the relationship between KS and LC technology innovation and emission reduction benefits.

The differential game model is a combination of optimal control and game theory. It studies the problems of cooperation and conflict between game subjects in dynamic systems (Chen et al., 2012). In a continuous time frame, multiple participants play a continuous game in an effort to optimize their respective goals. Through the model solution, the strategy of the game player’s evolution over time can be obtained and the Nash equilibrium can be reached. The differential game model has certain advantages for studying the optimal problem of dynamic control, and is widely used in the research of LC technology innovation and supply chain emission reduction problems. Hao and Li (2020) used the differential game model to study the collaborative innovation mechanism between enterprises in a GSC composed of a manufacturer and a supplier. Deng et al. (2021) and Wang et al. (2019b) established a differential game model from the perspective of carbon tax and studied the optimal solutions under different decision-making modes. The research results showed that carbon tax can promote LC technology innovation. Hou et al. (2020) used the differential game model to explore the dynamic emission reduction technology investment decision-making problem in the binary supply chain composed of manufacturers and retailers. Si et al. (2020) established a time-lag differential price game model, and analyzed the equilibrium strategy of price competition between technology supply and demand companies and the local asymptotic stability of the game system at the equilibrium point. Yin and Li (2018) used stochastic differential game to analyze the LC technology sharing problem in enterprise collaborative innovation. The results showed that random interference factors have the most significant impact on the level of LC technology in cooperative games. You and Zhu (2016) used the differential game model to study the LC supply chain R&D, promotion and pricing issues. Zhang et al. (2019) explored the joint dynamic green innovation policy and pricing strategy in the hybrid manufacturing and remanufacturing system. The results showed that too high or too low a carbon emission limit will reduce the manufacturer’s revenue.

In addition, differential games are also used in the study of KS problems. Lin and Wang (2019) used differential game theory to establish a dynamic incentive model to study the dynamic KS of construction project teams. Zhu et al. (2017) considered random interference factors and established a stochastic differential game model to study the problem of KS in industry-university-research collaborative innovation. It can be found that few scholars use the differential game model to study the problem of KS and emission reduction benefits in the collaborative innovation of LC technologies in the GSC. In the GSC, the knowledge level of suppliers and manufacturers is constantly changing over time. Therefore, the differential game model is used to study the optimal shared effort level, emission reduction benefits and knowledge changes over time between suppliers and manufacturers in the GSC.

In summary, the following research gaps can be found through literature review: 1) there is a lack of research on KS strategies in the dynamic process of collaborative innovation between suppliers and manufacturers. 2) In LC technology innovation, the dynamic relationship between KS strategy and knowledge stock and emission reduction benefits has not received attention. Therefore, this paper establishes a differential game model from the perspective of KS to dynamically study the issue of LC technology collaborative innovation in the GSC. This research aims to explore the evolutionary laws of knowledge stock under different KS strategies, as well as the optimal degree of effort and optimal emission reduction benefits of KS between suppliers and manufacturers. This research is of great significance for promoting the KS of GSC, improving the efficiency of LC technology innovation and the ability of supply chain to reduce carbon emissions.

Assumption 1

The amount of knowledge generated by the collaborative innovation of LC technologies in the GSC is determined by the KS effort of the supplier and the KS effort of the manufacturer. At the same time, knowledge is attenuated. As technology is iterated and updated, existing knowledge will be replaced by new knowledge. Therefore, the stock of knowledge is a dynamic variable that changes with time. Differential equations can be used to describe the law of the stock of knowledge with time (Plambeck, 2012; Ma et al., 2020): x ˙ ( t ) = α E M ( t ) + β E S ( t ) − δ x ( t )

Among them, x ( t ) represents the knowledge stock in the GSC at time t , and it satisfies x ( 0 ) = x 0 > 0 at the initial time. E M ( t ) and E S ( t ) , respectively represent the KS efforts of manufacturers and suppliers at time t . α and β represent the knowledge creation level of manufacturers and suppliers, respectively, and refer to the research and development capabilities of new technologies. δ represents the natural decay rate of knowledge.

Assumption 2

When the degree of effort increases, the sharing cost also shows an increasing trend. Therefore, the shared cost C S can be regarded as a convex function of the effort level E S ( t ) (Breton et al., 2005). The KS cost of suppliers at time t can be expressed as: C S ( E S ( t ) ) = 1 2 η S E S 2 ( t )

Among them, η S > 0 represents the cost coefficient of the supplier’s KS effort. Similarly, the manufacturer’s KS cost C M can be expressed as: C M ( E M ( t ) ) = 1 2 η M E M 2 ( t )

Among them, η M > 0 represents the effort cost coefficient of the manufacturer’s KS.

Assumption 3

In order to encourage suppliers and manufacturers to actively share knowledge and promote LC technology innovation, a government subsidy mechanism is introduced to subsidize the shared costs of suppliers and manufacturers. It is assumed that the government’s subsidy coefficients of KS costs for suppliers and manufacturers at time t are φ S ( t ) and φ M ( t ) , respectively.

Assumption 4

LC technology can promote carbon emission reduction and obtain certain economic benefits. The size of the benefits is related to the stock of knowledge in the GSC and the price of carbon allowances in the carbon trading market (Wang et al., 2019c). Therefore, the benefits of KS are ultimately manifested as emissions reduction benefits brought about by LC technology innovation. Let D ( x ( t ) ) represents the carbon emission reduction benefits created by LC technology innovation, D ( x ( t ) ) = P [ μ + ε x ( t ) ] . μ + ε x ( t ) represents the carbon emission reduction created by LC technology innovation. P represents the price of carbon allowances in the carbon trading market. ε represents the influence coefficient of knowledge stock on carbon emission reduction. μ represents a constant. Assume that the revenue distribution ratios of suppliers and manufacturers are γ ( 0 < γ ≤ 1 ) and 1 − γ , respectively.

Assumption 5

Suppliers and manufacturers make decisions based on complete information. And at any time t , both have the same discount rate, denoted as ρ ( ρ > 0 ) (Zhao et al., 2014). In the game process, the goal of the two is to seek the best effort coefficient and the best strategy to maximize their benefits.

When suppliers and manufacturers aim at maximizing individual benefits, this situation is a decentralized decision-making without cost sharing. This helps to study the minimum constraints for suppliers and manufacturers to participate in KS (Wang et al., 2019c). In order to promote the KS of suppliers, manufacturers introduce a cost sharing mechanism to share the cost of KS of suppliers. This situation is a decentralized decision-making of cost sharing. The collaborative sharing of suppliers and manufacturers is an ideal state of KS, and both parties aim to maximize the overall benefits of the GSC. This situation is centralized decision-making. The purpose of analyzing these three game situations is to explore the minimum constraints of KS, the optimal cost-sharing mechanism of manufacturers and the optimal KS strategy.

In the case of decentralized decision-making without cost sharing, suppliers and manufacturers form a Nash non-cooperative game. Both parties independently share knowledge and choose their own efforts to maximize profits. At this time, the supplier’s optimal profit function P S N and the manufacturer’s optimal profit function P M N are as follows, respectively P S N = max E S N ∫ 0 ∞ e − ρ t [ γ D ( x ( t ) ) − ( 1 − φ S ( t ) ) C S ( E S ( t ) ) ] d t (1) P M N = max E M N ∫ 0 ∞ e − ρ t [ ( 1 − γ ) D ( x ( t ) ) − ( 1 − φ M ( t ) ) C M ( E M ( t ) ) ] d t (2)

It can be seen from the above formula that the benefits of suppliers and manufacturers at time t are jointly determined by the benefits of carbon emission reduction, the cost of KS, and the government subsidy coefficient. In this paper, the Hamilton-Jacobi-Bellman equation (HJB equation) is used to solve the above equation to determine the equilibrium strategy of both parties. The equilibrium result is shown in Proposition 1. For the convenience of expression and writing, the time t is no longer displayed during the solution process.

Proposition 1

The equilibrium results in the case of decentralized decision-making without cost sharing are as follows:

(1) The optimal trajectory of the knowledge stock x N ∗ ( t ) in the GSC is:

Proof: According to the optimal control theory, from Eq. 1, the objective function of the supplier’s optimal decision-making at time t can be obtained as P S N ∗ ( x , t ) = max E S ∫ t ∞ e − ρ t [ γ D ( x ) − ( 1 − φ S ) C S ( E S ) ] d r = e − ρ t m a x E S N ∫ t ∞ [ γ D ( x ) − ( 1 − φ S ) C S ( E S ) ] d r (3)

Make U S N ( x ) = m a x E S ∫ t ∞ e − ρ ( r − t ) [ γ D ( x ) − ( 1 − φ S ) C S ( E S ) ] d r (4)

The Eq. 3 can be expressed as P S N ∗ ( x , t ) = e − ρ t U S N ( x ) (5)

At this time, the supplier’s optimal decision objective function can be transformed into the following HJB equation ρ U S N ( x ) = m a x E S [ γ D ( x ) − ( 1 − φ S ) C S ( E S ) + U S N ′ ( x ) x ˙ ] (6)

Expand Eq. 6 ρ U S N ( x ) = m a x E S [ γ P [ μ + ε x ] − 1 2 ( 1 − φ S ) η S E S 2 + U S N ′ ( x ) ( α E M + β E S − δ x ) ] (7)

Equation 7 is about the concave function of E S . In order to solve E S , this study takes the first-order partial derivative of E S and makes it equal to zero, and the supplier’s KS effort can be obtained as E S = β U S N ′ ( x ) ( 1 − φ S ) η S (8)

Similarly, according to the proof process of Eqs 3–5, the manufacturer’s optimal decision objective function P M N ∗ ( x , t ) at time t can be obtained as P M N ∗ ( x , t ) = e − ρ t U M N ( x ) (9)

In Eq. 9, U M N ( x ) = m a x E M ∫ t ∞ e − ρ ( r − t ) [ ( 1 − γ ) D ( x ) − ( 1 − φ M ) C M ( E M ) ] d r . At this time, the manufacturer’s optimal decision objective function can be transformed into the following HJB equation ρ U M N ( x ) = m a x M [ ( 1 − γ ) D ( x ) − ( 1 − φ M ) C M ( E M ) + U M N ′ ( x ) x ˙ ] (10)

Expand Eq. 10 ρ U M N ( x ) = m a x E S [ ( 1 − γ ) P ( μ + ε x ) − 1 2 ( 1 − φ M ) η M E M 2 + U M N ′ ( x ) ( α E M + β E S − δ x ) ] (11)

Equation 11 is about the concave function of E M . Finding the first-order partial derivative of Eq. 11 with respect to E M and setting it equal to zero, the manufacturer’s KS effort E M can be obtained as E M = α U M N ′ ( x ) ( 1 − φ M ) η M (12)

It can be seen from Eqs 8, 12 that the degree of KS effort is positively correlated with the government subsidy coefficient and the influence coefficient of knowledge stock, while it is negatively correlated with the effort cost coefficient of KS. Substitute Eq. 8 into Eqs 7, 12 into Eq. 11, and get ρ U S N ( x ) = γ P ( μ + ε x ) + U S N ′ ( x ) [ α 2 U M N ′ ( x ) ( 1 − φ M ) η M + β 2 U S N ′ ( x ) 2 ( 1 − φ S ) η S − δ x ] (13) ρ U M N ( x ) = ( 1 − γ ) P ( μ + ε x ) + U M N ′ ( x ) [ α 2 U M N ′ ( x ) 2 ( 1 − φ M ) η M + β 2 U S N ′ ( x ) ( 1 − φ S ) η S − δ x ] (14)

Suppose the linear expressions of U S N ( x ) and U M N ( x ) with respect to x are U S N ( x ) = a 1 N x + b 1 N (15) U M N ( x ) = a 2 N x + b 2 N (16) Where, a 1 N , b 1 N , a 2 N and b 2 N are all constants. Substitute Eqs 15, 16 into Eqs 13, 14, as shown below { ρ a 1 N = γ P ε − U S N ′ ( x ) δ ρ a 2 N = ( 1 − γ ) P ε − U M N ′ ( x ) δ ρ b 1 N = γ P μ + U S N ′ ( x ) [ α 2 U M N ′ ( x ) ( 1 − φ M ) η M + β 2 U S N ′ ( x ) 2 ( 1 − φ S ) η S ] ρ b 2 N = ( 1 − γ ) P μ + U M N ′ ( x ) [ α 2 U M N ′ ( x ) 2 ( 1 − φ M ) η M + β 2 U S N ′ ( x ) ( 1 − φ S ) η S ] (17)

In this study, the undetermined coefficient method was used to solve the equation, as shown below { a 1 N ∗ = γ P ε ρ + δ a 2 N ∗ = ( 1 − γ ) P ε ρ + δ b 1 N ∗ = γ P μ ρ + γ ( P ε ) 2 ρ ( ρ + δ ) 2 [ α 2 ( 1 − γ ) ( 1 − φ M ) η M + β 2 γ 2 ( 1 − φ S ) η S ] b 2 N ∗ = ( 1 − γ ) P μ ρ + ( 1 − γ ) ( P ε ) 2 ρ ( ρ + δ ) 2 [ α 2 ( 1 − γ ) 2 ( 1 − φ M ) η M + β 2 γ ( 1 − φ S ) η S ]

a 1 N ∗ , b 1 N ∗ , a 2 N ∗ and b 2 N ∗ are substituted into Eqs 15, 16. The expressions of U S N ( x ) and U M N ( x ) are as follows: { U S N ( x ) = a 1 N ∗ x + b 1 N ∗ U M N ( x ) = a 2 N ∗ x + b 2 N ∗ (18)

Equation 18 and its first derivative U S N ′ ( x ) and U M N ′ ( x ) are substituted into Eqs 8, 12. The equilibrium solution E S N ∗ of the supplier’s KS effort and the equilibrium solution E M N ∗ of the manufacturer’s KS effort can be obtained as { E S N ∗ = β γ P ε ( 1 − φ S ) ( ρ + δ ) η S E M N ∗ = α ( 1 − γ ) P ε ( 1 − φ M ) ( ρ + δ ) η M (19)

Equation 19 is substituted into x ˙ ( t ) = α E M ( t ) + β E S ( t ) − δ x ( t ) . According to the initial condition x ( 0 ) = x 0 > 0 , the optimal trajectory of the GSC knowledge stock can be obtained as x N ∗ ( t ) = l 1 − ( l 1 − x 0 ) e − δ t , where, l 1 = β 2 γ P ε ( 1 − φ S ) δ ( ρ + δ ) η S + α 2 ( 1 − γ ) P ε ( 1 − φ M ) δ ( ρ + δ ) η M . Substituting Eq. 18 into Eqs 5, 9 can obtain the optimal function of supplier and manufacturer’s income. The certificate is complete.

In order to obtain more information that is conducive to LC technology innovation, manufacturers encourage suppliers to actively share knowledge and share the cost of supplier sharing. The cost sharing ratio is θ ( t ) ( 0 < θ ( t ) < 1 ) . In the process of the game, a Stackellberg master-slave game with manufacturers as the leading and supplier as the subordinate is gradually formed. The manufacturer first determines its own KS effort E M Y ( t ) and cost sharing ratio θ ( t ) , and the supplier determines its own effort E S Y ( t ) according to the manufacturer’s cost sharing ratio θ ( t ) . It can be determined that the supplier’s revenue function P S Y and the manufacturer’s revenue function P M Y is: P S Y = m a x E S Y ∫ 0 ∞ e − ρ t [ γ D ( x ) − ( 1 − φ S − θ ) C S ( E S ) ] d t (20) P M Y = m a x E M Y ∫ 0 ∞ e − ρ t [ ( 1 − γ ) D ( x ) − ( 1 − φ M ) C M ( E M ) − θ C S ( E S ) ] d t (21)

In order to determine the equilibrium strategy of the supplier and the manufacturer, the HJB equation is used to solve it. The solution result is shown in Proposition 2.

Proposition 2

The equilibrium result of decentralized decision-making under cost sharing is as follows:

(1) The optimal trajectory of the knowledge stock x Y ∗ ( t ) in the GSC is

Proof: Using the inverse induction method to solve, from Eq. 20, we can see that the supplier’s optimal profit function P S Y ∗ ( x , t ) at time t is P S Y ∗ ( x , t ) = e − ρ t U S Y ( x ) (22) where, U S Y ( x ) = m a x E S ∫ t ∞ e − ρ ( r − t ) [ γ D ( x ) − ( 1 − φ S − θ ) C S ( E S ) ] d r . The supplier’s optimal revenue function satisfies the following HJB equation ρ U S Y ( x ) = m a x E S [ γ P ( μ + ε x ) − ( 1 − φ S − θ ) 1 2 η S E S 2 + U S Y ′ ( x ) ( α E M + β E S − δ x ) ] (23)

It can be seen that Eq. 23 is about the concave function of E S . Find the first-order partial derivative of E S and make it equal to zero, and the following equation is obtained E S = β U S Y ′ ( x ) ( 1 − φ S − θ ) η S (24)

In the same way, the manufacturer’s optimal profit function P M Y ∗ ( x , t ) at time t is P M Y ∗ ( x , t ) = e − ρ t U M Y ( x ) (25) where, U M Y ( x ) = max E M ∫ t ∞ e − ρ ( r − t ) [ ( 1 − γ ) D ( x ) − ( 1 − φ M ) C M ( E M ) − θ C S ( E S ) ] d r . At this time, the manufacturer’s optimal profit function satisfies the HJB equation, as shown below ρ U M Y ( x ) = m a x E M , θ { ( 1 − γ ) P ( μ + ε x ) − 1 2 ( 1 − φ M ) η M E M 2 − θ β 2 [ U S Y ′ ( x ) ] 2 2 ( 1 − φ S − θ ) 2 η S + U M Y ′ ( x ) ( α E M + β 2 U S Y ′ ( x ) ( 1 − φ S − θ ) η S − δ x ) } (26)

According to the related theory of Hessian matrix (Wang D. et al., 2019), Eq. 26 is a concave function, and the maximum value is obtained when the partial derivative is equal to zero. In order to solve E M and θ , let the first partial derivative of E M and θ in Eq. 26 be zero, the following equation can be obtained E M = α U M Y ′ ( x ) ( 1 − φ M ) η M ,   θ = ( 1 − φ S ) ( 2 U M Y ′ ( x ) − U S Y ′ ( x ) ) 2 U M Y ′ ( x ) + U M Y ′ ( x ) (27)

The manufacturer’s share of the supplier’s KS cost ratio θ is meaningful only when U M Y ′ ( x ) > U S Y ′ ( x ) 2 . Otherwise, the manufacturer will not share the supplier’s innovation costs.

Substituting E S , E M and θ in Eqs 24, 27 into Eqs 23, 26, the following equations can be obtained ρ U S Y ( x ) = γ P ( μ + ε x ) + U S Y ′ ( x ) ( α 2 U M Y ′ ( x ) ( 1 − φ M ) η M − δ x ) + β 2 U S Y ′ ( x ) [ 2 U M Y ′ ( x ) + U S Y ′ ( x ) ] 4 ( 1 − φ S ) η S (28) ρ U M Y ( x ) = ( 1 − γ ) P ( μ + ε x ) + U M Y ′ ( x ) [ α 2 U M Y ′ ( x ) 2 ( 1 − φ M ) η M − δ x ] + β 2 [ 2 U M Y ′ ( x ) + U S Y ′ ( x ) ] 2 8 ( 1 − φ S ) η S (29)

According to the characteristics of Eqs 28, 29, it is assumed that the linear expressions of U S Y ( x ) and U M Y ( x ) with respect to x are U S Y ( x ) = a 1 Y x + b 1 Y ,     U M Y ( x ) = a 2 Y x + b 2 Y (30) where, a 1 Y , b 1 Y , a 2 Y and b 2 Y are all constants. Substituting Eq. 30 into Eqs 28, 29, the equations for a 1 Y , b 1 Y , a 2 Y and b 2 Y can be obtained. Using the method of undetermined coefficients to solve the equations, the following equations can be obtained { a 1 Y ∗ = γ P ε ρ + δ a 2 Y ∗ = ( 1 − γ ) P ε ρ + δ b 1 Y ∗ = γ P μ ρ + γ ( P ε ) 2 ρ ( ρ + δ ) 2 [ α 2 ( 1 − γ ) ( 1 − φ M ) η M + β 2 ( 2 − γ ) 4 ( 1 − φ S ) η S ] b 2 Y ∗ = ( 1 − γ ) P μ ρ + ( P ε ) 2 2 ρ ( ρ + δ ) 2 [ α 2 ( 1 − γ ) 2 ( 1 − φ M ) η M + β 2 ( 2 − γ ) 2 4 ( 1 − φ S ) η S ]

Substituting a 1 Y ∗ , b 1 Y ∗ , a 2 Y ∗ and b 2 Y ∗ into Eq. 30, U S Y ( x ) and U M Y ( x ) can be determined as shown below U S Y ( x ) = a 1 Y ∗ x + b 1 Y ∗ ,   U M Y ( x ) = a 2 Y ∗ x + b 2 Y ∗ (31)

Equations 31, U S Y ′ ( x ) and U M Y ′ ( x ) are substituted into Eqs 24, 27, the supplier’s KS effort E S Y ∗ and manufacturer’s KS effort E M Y ∗ are shown as follows E S Y ∗ = β P ε ( 2 − γ ) 2 ( ρ + δ ) ( 1 − φ S ) η S ,   E M Y ∗ = α ( 1 − γ ) P ε ( 1 − φ M ) ( ρ + δ ) η M (32)

Equation 32 is substituted into x ˙ ( t ) = α E M ( t ) + β E S ( t ) − δ x ( t ) . According to the initial condition x ( 0 ) = x 0 > 0 , the optimal trajectory of the knowledge stock of the GSC can be obtained as x Y ∗ ( t ) = l 2 − ( l 2 − x 0 ) e − δ t , where l 2 = β 2 P ε ( 2 − γ ) 2 δ ( ρ + δ ) ( 1 − φ S ) η S + α 2 ( 1 − γ ) P ε δ ( 1 − φ M ) ( ρ + δ ) η M . Equation 32 is substituted into Eqs 22, 25 respectively, the optimal function of supplier and manufacturer’s income can be obtained. The certificate is complete.

Collaborative sharing is knowledge cooperation between suppliers and manufacturers aiming at the overall technology innovation of the GSC and maximizing the benefits of emission reduction. The collaborative sharing of LC technologies between suppliers and manufacturers in the GSC can improve the level of KS, realize resource integration and complement each other’s advantages. In addition, it can also reduce the information asymmetry between suppliers and manufacturers and promote LC technology innovation. In the case of collaborative sharing, it is assumed that both parties make decisions with the goal of maximizing the overall benefits of the GSC. The decision variables include the supplier’s KS effort E S and the manufacturer’s KS effort E M . It can be determined that the GSC emission reduction benefit P C under collaborative sharing can be expressed as: P C = max E S C , E M C ∫ 0 ∞ e − ρ t [ D ( x ) − ( 1 − φ S ) C S ( E S ) − ( 1 − φ M ) C M ( E M ) ] d t

Using the HJB equation, the equilibrium strategy of the decision problem can be obtained, as shown in Proposition 3.

Proposition 3

The equilibrium results in centralized decision-making are as follows:

(1) The optimal trajectory of the knowledge stock x C ∗ ( t ) in the GSC is

Proof: From Eq. 33, the optimal carbon emission reduction revenue function P C ∗ ( x , t ) of the GSC at time t can be obtained as P C ∗ ( x , t ) = max E S C , E M C ∫ t ∞ e − ρ r [ D ( x ) − ( 1 − φ S ) C S ( E S ) − ( 1 − φ M ) C M ( E M ) ] d r

Let U C ( x ) = max E S C , E M C ∫ t ∞ e − ρ ( r − t ) [ D ( x ) − ( 1 − φ S ) C S ( E S ) − ( 1 − φ M ) C M ( E M ) ] d r , then the optimal benefit function of carbon emission reduction in the GSC at time t can be expressed as P C ∗ ( x , t ) = e − ρ t U C ( x )

At this time, the decision-making problem of collaborative KS satisfies the following HJB equation ρ U C ( x ) = m a x E S C , E M C [ D ( x ) − ( 1 − φ S ) C S ( E S ) − ( 1 − φ M ) C M ( E M ) + U C ′ ( x ) x ˙ ] (33)

Expand the Eq. 33 as follows ρ U C ( x ) = m a x E S C , E M C [ P ( μ + ε x ) − ( 1 − φ S ) 1 2 η S E S 2 − ( 1 − φ M ) 1 2 η M E M 2 + U C ' ( x ) ( α E M + β E S − δ x ) ] (34)

The Hessian matrix of Eq. 34 for E S and E M is H = [ − ( 1 − φ S ) η S 0 0 − ( 1 − φ M ) η M ]

Since d e t ( H ) = ( 1 − φ S ) ( 1 − φ M ) η S η M > 0 and − ( 1 − φ S ) η S < 0 , it can be determined that the Hessian matrix is negative definite, so Eq. 34 is a concave function. The maximum value of E S and E M can be obtained, and the maximum value is obtained when the partial derivative is equal to zero. In order to find the first-order partial derivative of E S and E M in Eq. 34 and set it equal to zero, the following equation can be obtained E S = β U C ′ ( x ) ( 1 − φ S ) η S ,   E M = α U C ′ ( x ) ( 1 − φ M ) η M (35)

Substituting Eq. 35 into Eq. 34 can get: ρ U C ( x ) = P ( μ + ε x ) − [ β U C ' ( x ) ] 2 2 ( 1 − φ S ) η S − [ α U C ' ( x ) ] 2 2 ( 1 − φ M ) η M + U C ' ( x ) ( α 2 U C ' ( x ) ( 1 − φ M ) η M + β 2 U C ' ( x ) ( 1 − φ S ) η S − δ x ) (36)

The optimal value function of Eq. 36 with respect to x is the solution of the HJB equation, so the linear expression of U C ( x ) with respect to x is U C ( x ) = a 1 C x + b 1 C (37)

Equation 37 is substituted into Eq. 36, the constraint equations about a 1 C and b 1 C can be obtained. Use the method of undetermined coefficients to get a 1 C ∗ = P ε ρ + δ , b 1 C ∗ = P μ ρ + ( P ε ) 2 2 ρ ( ρ + δ ) 2 [ β 2 ( 1 − φ S ) η S + α 2 ( 1 − φ M ) η M ] .

Substituting a 1 C ∗ and b 1 C ∗ into Eq. 37, the expression of U C ( x ) can be determined as U C ( x ) = a 1 C ∗ x + b 1 C ∗ (38)

Substituting the first derivative of Eq. 38 into Eq. 35, the equilibrium solution E S C ∗ of the supplier’s KS effort and the equilibrium solution E M C ∗ of the manufacturer’s KS effort can be determined as E S C ∗ = β P ε ( 1 − φ S ) ( ρ + δ ) η S ,   E M C ∗ = α P ε ( 1 − φ M ) ( ρ + δ ) η M (39)

Substitute Eq. 39 into x ˙ ( t ) = α E M ( t ) + β E S ( t ) − δ x ( t ) . According to the initial condition x ( 0 ) = x 0 > 0 , the optimal trajectory of the knowledge stock of the GSC can be obtained x C ∗ ( t ) = l 3 − ( l 3 − x 0 ) e − δ t , where l 3 = β 2 P ε δ ( ρ + δ ) ( 1 − φ S ) η S + α 2 P ε δ ( 1 − φ M ) ( ρ + δ ) η M . Substituting Eq. 38 into Eq. 36 can obtain the optimal value function P C ∗ ( x , t ) of the carbon emission reduction benefit of the GSC. The certificate is complete.

In the case of decentralized decision-making without cost sharing and decentralized decision-making with cost sharing, a comparative analysis of the degree of KS effort between suppliers and manufacturers can be obtained: { E M Y ∗ − E M N ∗ = 0 E S Y ∗ − E S N ∗ = ( 2 − 3 γ ) β P ε 2 ( 1 − φ S ) ( ρ + δ ) η S E S Y ∗ − E S N ∗ E N Y ∗ = ( 2 − 3 γ ) 2 γ > 0

Corollary 1

When the supplier’s carbon emission reduction revenue distribution ratio is 0 < γ < 2 3 , E M Y ∗ = E M N ∗ , E S Y > E S N ∗ . The cost-sharing mechanism does not change the degree of KS efforts of manufacturers. However, the degree of KS efforts of suppliers has increased, and the degree of improvement is related to the supplier’s carbon emission reduction revenue distribution ratio γ . The introduction of a cost-sharing mechanism can increase the willingness of suppliers to share knowledge, and promote the game between the two to change from the Nash non-cooperative game to the Stackelberg master-slave game.