In this paper,we discuss the closed finite-to-one mapping theorems on generalized metric spaces and their applications.It is proved that point-G_δ properties,■-snf-countability and csf-countability are invariants an...In this paper,we discuss the closed finite-to-one mapping theorems on generalized metric spaces and their applications.It is proved that point-G_δ properties,■-snf-countability and csf-countability are invariants and inverse invariants under closed finite-to-one mappings.By the relationships between the weak first-countabilities,we obtain the closed finite-to-one mapping theorems of weak quasi-first-countability,quasi-first-countability,snf-countability,gfcountability and sof-countability.Furthermore,these results are applied to the study of symmetric products of topological spaces.展开更多
Within the affine connection framework of Lagrangia, n control systems,based on the results of Sussmann on controllability of general affine control systems defined on a finite-dimensional manifold, a computable suffi...Within the affine connection framework of Lagrangia, n control systems,based on the results of Sussmann on controllability of general affine control systems defined on a finite-dimensional manifold, a computable sufficient condition of configuration controllability for the simple mechanical control systems was extended to the case of systems with strictly dissipative energy terms of linear isotropic nature, and a sufficient condition of equilibrium controllability for the systems was also given, where Lagrangian is kinetic energy minus potential energy. Lie bracketting of vector fields in controllable Lie algebra, and the symmetric product associated with Levi-Civita connection show virtues in the discussion. Liouville vector field simplified the computation of controllable Lie algebra for the systems, although the terms of potential energy complicated the study of configuration controllability.展开更多
Within the affine connection framework of Lagrangian control systems, basedon the results of Sussmann on small-time locally controllability of single-input affine nonlinearcontrol systems, the controllability results ...Within the affine connection framework of Lagrangian control systems, basedon the results of Sussmann on small-time locally controllability of single-input affine nonlinearcontrol systems, the controllability results for mechanical control systems with single-input areextended to the case of the systems with isotropic damping, where the Lagrangian is the kineticenergy associated with a Riemannian metric. A sufficient condition of negative small-time locallycontrollability for the system is obtained. Then,it is demonstrated that such systems are small-timelocally configuration controllable if and only if the dimension of the configuration manifold isone. Finally, two examples are given to illustrate the results. Lie bracketting of vector fields andthe symmetric product show the advantages in the discussion.展开更多
基金Supported by the National Natural Science Foundation of China(11801254,11471153)
文摘In this paper,we discuss the closed finite-to-one mapping theorems on generalized metric spaces and their applications.It is proved that point-G_δ properties,■-snf-countability and csf-countability are invariants and inverse invariants under closed finite-to-one mappings.By the relationships between the weak first-countabilities,we obtain the closed finite-to-one mapping theorems of weak quasi-first-countability,quasi-first-countability,snf-countability,gfcountability and sof-countability.Furthermore,these results are applied to the study of symmetric products of topological spaces.
文摘Within the affine connection framework of Lagrangia, n control systems,based on the results of Sussmann on controllability of general affine control systems defined on a finite-dimensional manifold, a computable sufficient condition of configuration controllability for the simple mechanical control systems was extended to the case of systems with strictly dissipative energy terms of linear isotropic nature, and a sufficient condition of equilibrium controllability for the systems was also given, where Lagrangian is kinetic energy minus potential energy. Lie bracketting of vector fields in controllable Lie algebra, and the symmetric product associated with Levi-Civita connection show virtues in the discussion. Liouville vector field simplified the computation of controllable Lie algebra for the systems, although the terms of potential energy complicated the study of configuration controllability.
文摘Within the affine connection framework of Lagrangian control systems, basedon the results of Sussmann on small-time locally controllability of single-input affine nonlinearcontrol systems, the controllability results for mechanical control systems with single-input areextended to the case of the systems with isotropic damping, where the Lagrangian is the kineticenergy associated with a Riemannian metric. A sufficient condition of negative small-time locallycontrollability for the system is obtained. Then,it is demonstrated that such systems are small-timelocally configuration controllable if and only if the dimension of the configuration manifold isone. Finally, two examples are given to illustrate the results. Lie bracketting of vector fields andthe symmetric product show the advantages in the discussion.
