Chaplygin’s nonholonomic systems are familiar mechanical systems subject to unintegrable linear constraints, which can be reduced into holonomic nonconservative systems in a subspace of the original state space. The ...Chaplygin’s nonholonomic systems are familiar mechanical systems subject to unintegrable linear constraints, which can be reduced into holonomic nonconservative systems in a subspace of the original state space. The inverse problem of the calculus of variations or Lagrangian inverse problem for such systems is analyzed by making use of a reduction of the systems into new ones with time reparametrization symmetry and a genotopic transformation related with a conformal transformation. It is evident that the Lagrangian inverse problem does not have a direct universality. By meaning of a reduction of Chaplygin’s nonholonomic systems into holonomic, regular, analytic, nonconservative, first-order systems, the systems admit a Birkhoffian representation in a star-shaped neighborhood of a regular point of their variables, which is universal due to the Cauchy-Kovalevski theorem and the converse of the Poincaré lemma.展开更多
For symbolic reachability analysis of rectangular hybrid systems, the basic issue is finding a formal structure to represent and manipulate its infinite state spaces. Firstly, this structure must be closed to the reac...For symbolic reachability analysis of rectangular hybrid systems, the basic issue is finding a formal structure to represent and manipulate its infinite state spaces. Firstly, this structure must be closed to the reachability operation which means that reachable states from states expressed by this structure can be presented by it too. Secondly, the operation of finding reachable states with this structure should take as less computation as possible. To this end, a constraint system called rectangular zone is formalized, which is a conjunction of fixed amount of inequalities that compare fixed types of linear expressions with two variables to rational numbers. It is proved that the rectangular zone is closed to those reachability operations-intersection, elapsing of time and edge transition. Since the number of inequalities and the linear expression of each inequality is fixed in rectangular zones, so to obtain reachable rectangular zones, it just needs to change the rational numbers to which these linear expressions need to compare. To represent rectangular zones and unions of rectangular zones, a data structure called three dimensional constraint matrix(TDCM) and a BDD-like structure rectangular hybrid diagram(RHD) are introduced.展开更多
基金supported by the National Natural Science Foundation of China (Grant Nos. 10932002, 10872084, and 10472040)the Outstanding Young Talents Training Fund of Liaoning Province of China (Grant No. 3040005)+2 种基金the Research Program of Higher Education of Liaoning Prov- ince, China (Grant No. 2008S098)the Program of Supporting Elitists of Higher Education of Liaoning Province, China (Grant No. 2008RC20)the Program of Constructing Liaoning Provincial Key Laboratory, China (Grant No. 2008403009)
文摘Chaplygin’s nonholonomic systems are familiar mechanical systems subject to unintegrable linear constraints, which can be reduced into holonomic nonconservative systems in a subspace of the original state space. The inverse problem of the calculus of variations or Lagrangian inverse problem for such systems is analyzed by making use of a reduction of the systems into new ones with time reparametrization symmetry and a genotopic transformation related with a conformal transformation. It is evident that the Lagrangian inverse problem does not have a direct universality. By meaning of a reduction of Chaplygin’s nonholonomic systems into holonomic, regular, analytic, nonconservative, first-order systems, the systems admit a Birkhoffian representation in a star-shaped neighborhood of a regular point of their variables, which is universal due to the Cauchy-Kovalevski theorem and the converse of the Poincaré lemma.
基金supported by the National Natural Science Foundation of China(Grant Nos.61373043&61003079)the Fundamental Research Funds for the Central Universities(Grant No.JB140316)
文摘For symbolic reachability analysis of rectangular hybrid systems, the basic issue is finding a formal structure to represent and manipulate its infinite state spaces. Firstly, this structure must be closed to the reachability operation which means that reachable states from states expressed by this structure can be presented by it too. Secondly, the operation of finding reachable states with this structure should take as less computation as possible. To this end, a constraint system called rectangular zone is formalized, which is a conjunction of fixed amount of inequalities that compare fixed types of linear expressions with two variables to rational numbers. It is proved that the rectangular zone is closed to those reachability operations-intersection, elapsing of time and edge transition. Since the number of inequalities and the linear expression of each inequality is fixed in rectangular zones, so to obtain reachable rectangular zones, it just needs to change the rational numbers to which these linear expressions need to compare. To represent rectangular zones and unions of rectangular zones, a data structure called three dimensional constraint matrix(TDCM) and a BDD-like structure rectangular hybrid diagram(RHD) are introduced.
