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This meshless collocation method is applicable not only to the Helmholtz equation with Dirichlet boundary condition but also mixed boundary conditions. It can calculate not only the high wavenumber problems, but also the variable wave number problems.

In this paper, the authors developed a meshless collocation method by using barycentric Lagrange interpolation basis function based on the Chebyshev nodes to deduce the scheme for solving the three-dimensional Helmholtz equation. First, the spatial variables and their partial derivatives are treated by interpolation basis functions, and the collocation method is established for solving second order differential equations. Then the differential matrix is employed to simplify the differential equations which is on a given test node. Finally, numerical experiments show the accuracy and effectiveness of the proposed method.

The numerical experiments show the advantages of the present method, such as less number of collocation nodes needed, shorter calculation time, higher precision, smaller error and higher efficiency. What is more, the numerical solutions agree well with the exact solutions.

Compared with finite element method, finite difference method and other traditional numerical methods based on grid solution, meshless method can reduce or eliminate the dependence on grid and make the numerical implementation more flexible.

The Helmholtz equation has a wide application background in many fields, such as physics, mechanics, engineering and so on.

This meshless method is first time applied for solving the 3D Helmholtz equation. What is more the present work not only gives the relationship of interpolation nodes but also the test nodes.

This paper aims to study the application of reinforcement learning (RL) in the control of an output-constrained flapping-wing micro aerial vehicle (FWMAV) with system uncertainty.

A six-degrees-of-freedom hummingbird model is used without consideration of the inertial effects of the wings. A RL algorithm based on actor–critic framework is applied, which consists of an actor network with unknown policy gradient and a critic network with unknown value function. Considering the good performance of neural network (NN) in fitting nonlinearity and its optimum characteristics, an actor–critic NN optimization algorithm is designed, in which the actor and critic NNs are used to generate a policy and approximate the cost functions, respectively. In addition, to ensure the safe and stable flight of the FWMAV, a barrier Lyapunov function is used to make the flight states constrained in predefined regions. Based on the Lyapunov stability theory, the stability of the system is analyzed, and finally, the feasibility of RL in the control of a FWMAV is verified through simulation.

The proposed RL control scheme works well in ensuring the trajectory tracking of the FWMAV in the presence of output constraint and system uncertainty.

A novel RL algorithm based on actor–critic framework is applied to the control of a FWMAV with system uncertainty. For the stable and safe flight of the FWMAV, the output constraint problem is considered and solved by barrier Lyapunov function-based control.

The relationship between industrial policy and exploratory innovation is imperfect.

The authors use Chinese high-tech enterprise identification policy (HTEP) as a natural experimental group to test policy impacts, spillover effects and mechanisms of action.

First, HTEP promotes exploratory innovation. In addition, HTEP has a greater impact on non-exploratory innovation. Second, HTEP has spillover effects in two phases: HTEP (2008) and the 2016 policy reform. HTEP affects exploratory innovation in nearby non-high-tech firms, and the policy effect decreases monotonically with increasing distance from the treatment group. Third, HTEP affects innovation capacity through financing constraints, technical personnel flow and knowledge flow, which explains not only policy effects but also spillover effects. Fourth, the analysis of policy heterogeneity shows that the 2016 policy reforms reinforce the positive effect of HTEP (2008). By deducting the effects of other policies, the HTEP effect is found to be less volatile. In terms of the continuity of policy identification, continuous uninterrupted identification has a crucial impact on the improvement of firms’ innovation capacity compared to repeated certification and certification expiration. Finally, HTEP has a crowding-out effect in state-owned enterprises and large firms’ innovation.

This paper contributes to the existing literature in several ways. First, the authors enrich the literature on industrial policy through exploratory innovation research. While previous studies have focused on R&D investment and patents (Dai and Wang, 2019), exploratory innovation helps firms break away from the inherent knowledge mindset and achieve sustainable innovation. Second, few studies have explored the characteristics of industrial policies. In this paper, the authors subdivide the sample into repeated certification, continuous certification and certification expiration according to high-tech enterprise identification. In addition, the authors compare the differences in policy implementation effects between the 2016 policy reform and the 2008 policy to provide new directions for business managers and policy makers. Third, innovation factors guided by industrial policies may cluster in specific regions, which in turn manifest externalities. This is when the policy spillover effect is worth considering. This paper fills a gap in the industrial policy literature by examining the spillover effects. Finally, this paper also explores the mechanisms of policy effects from three perspectives: financing constraints, technician mobility and knowledge mobility, which can affect not only the innovation of beneficiary firms directly but also indirectly the innovation of neighboring non-beneficiary firms.