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Random Vectors and Random Functions
Published in Richard M. Golden, Statistical Machine Learning, 2020
A probability measure P on a measurable space (Ω,F) is a countably additive measure P:F→[0,1] such that P(Ω) = 1.
Stochastic differential games and inverse optimal control and stopper policies
Published in International Journal of Control, 2019
Tanmay Rajpurohit, Wassim M. Haddad, Wei Sun
We define a complete probability space as , where Ω denotes the sample space, denotes a σ-algebra, and defines a probability measure on the σ-algebra ; that is, is a non-negative countably additive set function on such that (Arnold, 1974). Furthermore, we assume that w(·) is a standard d-dimensional Wiener process defined by , where is the classical Wiener measure (Øksendal, 1995, p. 10), with a continuous-time filtration generated by the Wiener process w(t) up to time t. We denote a stochastic dynamical system by generating a adapted to the stochastic process on satisfying , 0 ≤ τ < t, such that , t ≥ 0, for all Borel sets contained in the Borel σ-algebra . Here we use the notation x(t) to represent the stochastic process x(t, ω) omitting its dependence on ω.
An adaptive control architecture for leader–follower multiagent systems with stochastic disturbances and sensor and actuator attacks
Published in International Journal of Control, 2019
We define a complete probability space as , where Ω denotes the sample space, denotes a σ-algebra, and defines a probability measure on the σ-algebra ; that is, is a nonnegative countably additive set function on such that (Arnold, 1974). Furthermore, we assume that w(·) is a standard d-dimensional Wiener process defined by , where is the classical Wiener measure (Øksendal, 1995, p. 10), with a continuous-time filtration generated by the Wiener process w(t) up to time t. We denote a stochastic dynamical system by generating a filtration adapted to the stochastic process on satisfying , 0 ≤ τ < t, such that , t ≥ 0, for all Borel sets contained in the Borel σ-algebra . Here, we use the notation x(t) to represent the stochastic process x(t, ω) omitting its dependence on ω.
Dissipativity, inverse optimal control, and stability margins for nonlinear discrete-time stochastic feedback regulators
Published in International Journal of Control, 2023
Wassim M. Haddad, Manuel Lanchares
Throughout the paper, denotes a complete probability space, where Ω denotes the sample space, denotes a σ-algebra of subsets of Ω, and defines a probability measure on the σ-algebra ; that is, is a nonnegative countably additive set function on such that . Given , we denote the set of equivalence classes of measurable -valued random vectors on by , where the equivalence relation is the one induced by -almost-sure equality. In particular, elements of take finite values -almost surely (a.s.). Hence, depending on the context, will denote either the set of real vectors or the subspace of comprising -valued random vectors that are constant almost surely. Furthermore, and denote the subsets of comprising random vectors whose elements are integrable and square-integrable, respectively, on .