The current work proposes a new and constructive proof for the Caratheodory's theorem on existence and uniqueness of trajectories of dynamical systems. The key concern is the numerical uncertainty, i.e., the discr...The current work proposes a new and constructive proof for the Caratheodory's theorem on existence and uniqueness of trajectories of dynamical systems. The key concern is the numerical uncertainty, i.e., the discrepancy between mathematical proofs, algorithms, and their implementations, which may affect the correct functioning of a control system. Due to growing demands on security and compliance with specifications,correctness of the control system functioning is becoming ever more important. Since in both dynamical systems and many control design approaches, one of the central notions is the system trajectory, it is important to address existence and uniqueness of system trajectories in a way which incorporates numerical uncertainty. Constructive analysis is a particular approach to formalizing numerical uncertainty and is used as the basis of the current work. The major difficulties of guaranteeing existence and uniqueness of system trajectories arise in the case of systems and controllers which possess discontinuities in time,since classical solutions to initial value problems do not exist.This issue is addressed in Caratheodory's theorem. A particular constructive variant of the theorem is proven which covers a large class of problems found in practice.展开更多
文摘The current work proposes a new and constructive proof for the Caratheodory's theorem on existence and uniqueness of trajectories of dynamical systems. The key concern is the numerical uncertainty, i.e., the discrepancy between mathematical proofs, algorithms, and their implementations, which may affect the correct functioning of a control system. Due to growing demands on security and compliance with specifications,correctness of the control system functioning is becoming ever more important. Since in both dynamical systems and many control design approaches, one of the central notions is the system trajectory, it is important to address existence and uniqueness of system trajectories in a way which incorporates numerical uncertainty. Constructive analysis is a particular approach to formalizing numerical uncertainty and is used as the basis of the current work. The major difficulties of guaranteeing existence and uniqueness of system trajectories arise in the case of systems and controllers which possess discontinuities in time,since classical solutions to initial value problems do not exist.This issue is addressed in Caratheodory's theorem. A particular constructive variant of the theorem is proven which covers a large class of problems found in practice.
基金Supported by the National Natural Science Foundation of China(11071053)the Natural Science Foundation of Hebei Province(A2014207010)+1 种基金the Key Project of Science and Research of Hebei Educational Department(ZH2012080)the Key Project of Science and Research of Hebei University of Economics and Business(2013KYZ01)
