Let u be a solution to second order elliptic equations in Dini domains, a direct and elementary proof of the doubling property for u 2 over balls centered at points in the domain is presented. Moreover, under the zero...Let u be a solution to second order elliptic equations in Dini domains, a direct and elementary proof of the doubling property for u 2 over balls centered at points in the domain is presented. Moreover, under the zero Dirichlet boundary condition, the unique continuation at the boundary for Dini domains has been proved.展开更多
A numerical method for computing Nash equilibrium strategies(NES)of the spacecraft time-optimal orbit pursuitevasion game(TOOPEG)with continuous thrust reachable domain(RD)analysis is proposed.Through theoretical deri...A numerical method for computing Nash equilibrium strategies(NES)of the spacecraft time-optimal orbit pursuitevasion game(TOOPEG)with continuous thrust reachable domain(RD)analysis is proposed.Through theoretical derivation and Monte Carlo validation,the equivalence among the minimum time of the TOOPEG problem with NES,the minimum time of a virtual single spacecraft for a time-optimal approach to the origin,and the minimum time required for the envelope of the pursuer's RD to enclose that of the evader is established.First,the necessary conditions for NES are derived using Pontryagin's maximum principle(PMP),converting the original bilateral optimal control problem into a 7-dimensional two-point boundary value problem(TPBVP).Then,the TOOPEG is transformed into a virtual single-spacecraft time-optimal approach problem,with the above necessary conditions.By exploiting the evolutionary characteristics of the continuous-thrust RD,the problem is further reduced to a 3-dimensional nonlinear differential equation.An improved Broyden quasi-Newton iterative(IBQNI)algorithm is employed to obtain high-precision numerical solutions,and an iterative initial value construction method based on a linearized orbit dynamic model is proposed.Furthermore,a set of criteria is developed to assess the relative spatial configuration between the RD of different spacecraft.Numerical simulations demonstrate that the proposed method achieves excellent convergence and remarkable computational efficiency.展开更多
文摘Let u be a solution to second order elliptic equations in Dini domains, a direct and elementary proof of the doubling property for u 2 over balls centered at points in the domain is presented. Moreover, under the zero Dirichlet boundary condition, the unique continuation at the boundary for Dini domains has been proved.
基金supported by the National Natural Science Foundation of China(Grant No.12572054)。
文摘A numerical method for computing Nash equilibrium strategies(NES)of the spacecraft time-optimal orbit pursuitevasion game(TOOPEG)with continuous thrust reachable domain(RD)analysis is proposed.Through theoretical derivation and Monte Carlo validation,the equivalence among the minimum time of the TOOPEG problem with NES,the minimum time of a virtual single spacecraft for a time-optimal approach to the origin,and the minimum time required for the envelope of the pursuer's RD to enclose that of the evader is established.First,the necessary conditions for NES are derived using Pontryagin's maximum principle(PMP),converting the original bilateral optimal control problem into a 7-dimensional two-point boundary value problem(TPBVP).Then,the TOOPEG is transformed into a virtual single-spacecraft time-optimal approach problem,with the above necessary conditions.By exploiting the evolutionary characteristics of the continuous-thrust RD,the problem is further reduced to a 3-dimensional nonlinear differential equation.An improved Broyden quasi-Newton iterative(IBQNI)algorithm is employed to obtain high-precision numerical solutions,and an iterative initial value construction method based on a linearized orbit dynamic model is proposed.Furthermore,a set of criteria is developed to assess the relative spatial configuration between the RD of different spacecraft.Numerical simulations demonstrate that the proposed method achieves excellent convergence and remarkable computational efficiency.
