scholarly journals A generalization of the Freidlin–Wentcell theorem on averaging of Hamiltonian systems

2020 ◽  
pp. 1-35
Author(s):  
Yichun Zhu
Keyword(s):  
Probability Theory ◽  
Weak Convergence ◽  
Hamiltonian Systems ◽  
Random Perturbations ◽  
Identity Matrix ◽  
Compact Sets ◽  
Original Result ◽  
Drift Term ◽  
State Dependent ◽  
Auxiliary Process

In this paper, we generalize the classical Freidlin-Wentzell’s theorem for random perturbations of Hamiltonian systems. In (Probability Theory and Related Fields 128 (2004) 441–466), M.Freidlin and M.Weber generalized the original result in the sense that the coefficient for the noise term is no longer the identity matrix but a state-dependent matrix and taking the drift term into consideration. In this paper, We generalize the result by adding a state-dependent matrix that converges uniformly to 0 on any compact sets as ϵ tends to 0 to a state-dependent noise and considering the drift term which contains two parts, the state-dependent mapping and a state-dependent mapping that converges uniformly to 0 on any compact sets as ϵ tends to 0. In the proof, we adapt a new way to prove the weak convergence inside the edge by constructing an auxiliary process and modify the proof in (Probability Theory and Related Fields 128 (2004) 441–466) when proving gluing condition.

"Robustifying passive closed-loop Port-Hamiltonian systems using Observer Based Control"

2018 ◽  
Author(s):  
Matías Nacusse ◽  
Alejandro Donaire ◽  
Sergio Junco
Keyword(s):  
Control Systems ◽  
Hamiltonian Systems ◽  
Closed Loop ◽  
Design Procedure ◽  
Ph Control ◽  
Bond Graph ◽  
Control Law ◽  
Second Stage ◽  
State Dependent ◽  
Passivity Based Control

"This paper contributes a passivity-based approach to obtain a control law that robustifies Port-Hamiltonian (pH) control systems under external and state-dependent disturbances using disturbance observers (DO). A twostage design procedure is used to define the Disturbance Observed Based Control (DOBC) scheme. In the first stage a passivity based control law, called Interconnection and Damping assignment (IDA-PBC) is designed in the Bond Graph (BG) domain via BG prototyping, using an undisturbed model of the physical system. This stage is not the main issue of this paper and therefore the IDA-PBC law will be assumed to be known. The second stage, the main result of this paper, consists in the design of the DO and its integration with the IDA-PBC control law. The DO is derived in the BG domain via the integration of the residual signal computed from a Diagnostic Bond Graph (DBG). The methodology is developed through examples in the BG domain and formalized and extended in the pH framework."

scholarly journals Weak Convergence of Topological Measures

2021 ◽  
Author(s):  
Svetlana V. Butler
Keyword(s):  
Probability Theory ◽  
Weak Convergence ◽  
Metric Spaces ◽  
Measure Theory ◽  
Convergent Subsequence ◽  
Borel Measures ◽  
Nonlinear Functionals ◽  
Topological Measures ◽  
Standard Techniques ◽  
Uniformly Bounded

AbstractTopological measures and deficient topological measures are defined on open and closed subsets of a topological space, generalize regular Borel measures, and correspond to (nonlinear in general) functionals that are linear on singly generated subalgebras or singly generated cones of functions. They lack subadditivity, and many standard techniques of measure theory and functional analysis do not apply to them. Nevertheless, we show that many classical results of probability theory hold for topological and deficient topological measures. In particular, we prove a version of Aleksandrov’s theorem for equivalent definitions of weak convergence of deficient topological measures. We also prove a version of Prokhorov’s theorem which relates the existence of a weakly convergent subsequence in any sequence in a family of topological measures to the characteristics of being a uniformly bounded in variation and uniformly tight family. We define Prokhorov and Kantorovich–Rubenstein metrics and show that convergence in either of them implies weak convergence of (deficient) topological measures on metric spaces. We also generalize many known results about various dense and nowhere dense subsets of deficient topological measures. The present paper constitutes a necessary step to further research in probability theory and its applications in the context of (deficient) topological measures and corresponding nonlinear functionals.

A method of approximating Markov jump processes

1988 ◽  
Vol 20 (01) ◽  
pp. 33-58
Author(s):  
Keith N. Crank ◽  
Prem S. Puri
Keyword(s):  
Weak Convergence ◽  
Branching Process ◽  
Rates Of Convergence ◽  
Jump Processes ◽  
Markov Jump ◽  
Markov Jump Processes ◽  
State Dependent ◽  
Convergence Results ◽  
Fixed Times ◽  
Special Case

We present a method of approximating Markov jump processes which was used by Fuhrmann [7] in a special case. We generalize the method and prove weak convergence results under mild assumptions. In addition we obtain bounds on the rates of convergence of the probabilities at arbitrary fixed times. The technique is demonstrated using a state-dependent branching process as an example.

Quantifier elimination for neocompact sets

1998 ◽  
Vol 63 (4) ◽  
pp. 1442-1472 ◽  
Author(s):  
H. Jerome Keisler
Keyword(s):  
Probability Theory ◽  
Existence Theorems ◽  
Quantifier Elimination ◽  
General Setting ◽  
Random Variable ◽  
Compact Sets ◽  
The Family ◽  
Countably Compact ◽  
Universal Quantifiers

AbstractWe shall prove quantifier elimination theorems for neocompact formulas, which define neocompact sets and are built from atomic formulas using finite disjunctions, infinite conjunctions, existential quantifiers, and bounded universal quantifiers. The neocompact sets were first introduced to provide an easy alternative to nonstandard methods of proving existence theorems in probability theory, where they behave like compact sets. The quantifier elimination theorems in this paper can be applied in a general setting to show that the family of neocompact sets is countably compact. To provide the necessary setting we introduce the notion of a law structure. This notion was motivated by the probability law of a random variable. However, in this paper we discuss a variety of model theoretic examples of the notion in the light of our quantifier elimination results.

A method of approximating Markov jump processes

1988 ◽  
Vol 20 (1) ◽  
pp. 33-58
Author(s):  
Keith N. Crank ◽  
Prem S. Puri
Keyword(s):  
Weak Convergence ◽  
Branching Process ◽  
Rates Of Convergence ◽  
Jump Processes ◽  
Markov Jump ◽  
Markov Jump Processes ◽  
State Dependent ◽  
Convergence Results ◽  
Fixed Times ◽  
Special Case

We present a method of approximating Markov jump processes which was used by Fuhrmann [7] in a special case. We generalize the method and prove weak convergence results under mild assumptions. In addition we obtain bounds on the rates of convergence of the probabilities at arbitrary fixed times. The technique is demonstrated using a state-dependent branching process as an example.

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