Applications of Borovkov's Renovation Theory to Non-Stationary Stochastic Recursive Sequences and Their Control

1997 ◽  
Vol 29 (2) ◽  
pp. 388-413 ◽  
Author(s):  
Eitan Altman ◽  
Arie Hordijk
Keyword(s):  
Stochastic Processes ◽  
Finite Time ◽  
Probability Space ◽  
Initial State ◽  
Control Policies ◽  
Recursive Sequences ◽  
Stationary Stochastic Processes ◽  
Initial States ◽  
Natural Criterion ◽  
The Stability

We investigate in this paper the stability of non-stationary stochastic processes, arising typically in applications of control. The setting is known as stochastic recursive sequences, which allows us to construct on one probability space stochastic processes that correspond to different initial states and even different control policies. It does not require any Markovian assumptions. A natural criterion for stability for such processes is that the influence of the initial state disappears after some finite time; in other words, starting from different initial states, the process will couple after some finite time to the same limiting (not necessarily stationary nor ergodic) stochastic process. We investigate this as well as other types of coupling, and present conditions for them to occur uniformly in some class of control policies. We then use the coupling results to establish new theoretical aspects in the theory of non-Markovian control.

Applications of Borovkov's Renovation Theory to Non-Stationary Stochastic Recursive Sequences and Their Control

1997 ◽  
Vol 29 (02) ◽  
pp. 388-413 ◽  
Author(s):  
Eitan Altman ◽  
Arie Hordijk
Keyword(s):  
Stochastic Processes ◽  
Finite Time ◽  
Probability Space ◽  
Initial State ◽  
Control Policies ◽  
Recursive Sequences ◽  
Stationary Stochastic Processes ◽  
Initial States ◽  
Natural Criterion ◽  
The Stability

We investigate in this paper the stability of non-stationary stochastic processes, arising typically in applications of control. The setting is known as stochastic recursive sequences, which allows us to construct on one probability space stochastic processes that correspond to different initial states and even different control policies. It does not require any Markovian assumptions. A natural criterion for stability for such processes is that the influence of the initial state disappears after some finite time; in other words, starting from different initial states, the process will couple after some finite time to the same limiting (not necessarily stationary nor ergodic) stochastic process. We investigate this as well as other types of coupling, and present conditions for them to occur uniformly in some class of control policies. We then use the coupling results to establish new theoretical aspects in the theory of non-Markovian control.

scholarly journals Discrete-time inverse linear quadratic optimal control over finite time-horizon under noisy output measurements

2021 ◽  
Author(s):  
Han Zhang ◽  
Yibei Li ◽  
Xiaoming Hu
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Discrete Time ◽  
Finite Time ◽  
Time Horizon ◽  
Model Structure ◽  
Initial State ◽  
Quadratic Optimal Control ◽  
Initial States ◽  
Finite Time Horizon

AbstractIn this paper, the problem of inverse quadratic optimal control over finite time-horizon for discrete-time linear systems is considered. Our goal is to recover the corresponding quadratic objective function using noisy observations. First, the identifiability of the model structure for the inverse optimal control problem is analyzed under relative degree assumption and we show the model structure is strictly globally identifiable. Next, we study the inverse optimal control problem whose initial state distribution and the observation noise distribution are unknown, yet the exact observations on the initial states are available. We formulate the problem as a risk minimization problem and approximate the problem using empirical average. It is further shown that the solution to the approximated problem is statistically consistent under the assumption of relative degrees. We then study the case where the exact observations on the initial states are not available, yet the observation noises are known to be white Gaussian distributed and the distribution of the initial state is also Gaussian (with unknown mean and covariance). EM-algorithm is used to estimate the parameters in the objective function. The effectiveness of our results are demonstrated by numerical examples.

scholarly journals THE STABILITY OF QUANTUM MARKOV FILTERS

2009 ◽  
Vol 12 (01) ◽  
pp. 153-172 ◽  
Author(s):  
RAMON VAN HANDEL
Keyword(s):  
Differential Equation ◽  
Stochastic Model ◽  
Stochastic Differential Equation ◽  
Initial State ◽  
Rank Condition ◽  
Quantum Stochastic Differential Equation ◽  
Absolutely Continuous ◽  
Finite Dimensional ◽  
Initial States ◽  
The Stability

When are quantum filters asymptotically independent of the initial state? We show that this is the case for absolutely continuous initial states when the quantum stochastic model satisfies an observability condition. When the initial system is finite dimensional, this condition can be verified explicitly in terms of a rank condition on the coefficients of the associated quantum stochastic differential equation.

Effects of Initial Conditions and Circuit Parameters on the SBT-Memristor-Based Chaotic Circuit

2019 ◽  
Vol 29 (12) ◽  
pp. 1950171 ◽  
Author(s):  
Gang Dou ◽  
Huaying Duan ◽  
Wenyan Yang ◽  
Hai Yang ◽  
Mei Guo ◽  
...  
Keyword(s):  
Chaotic System ◽  
Initial Conditions ◽  
Chaotic Circuit ◽  
Mathematical Methods ◽  
Initial State ◽  
Dynamical Characteristics ◽  
Dynamical Behaviors ◽  
Initial States ◽  
The Stability ◽  
Stable Points

In the paper, a fourth-order SBT-memristor-based chaotic system described by the flux-controlled model is investigated. The stability of the chaotic system is analyzed, and the effects of initial conditions and circuit parameters on the SBT-memristor-based chaotic circuit are discussed by mathematical methods of Lyapunov exponents spectra, bifurcation diagrams, phase orbits and Poincaré maps. Through simulations, it is observed that the dynamical characteristics vary with initial states and circuit parameters. Complex dynamical behaviors such as stable points, period cycles and chaos can be found in the SBT-memristor-based system. It is also found that the system exhibits multistability, which is closely dependent on the initial state of the SBT memristor. This study provides insightful guidance for the design and analysis of memristor-based circuits towards potential real applications.

Stability of current Antarctica grounding lines

2021 ◽  
Author(s):  
Urruty Benoît ◽  
Gagliardini Olivier ◽  
Gillet-Chaulet Fabien
Keyword(s):  
Ice Sheet ◽  
Tipping Points ◽  
Ice Shelf ◽  
Initial State ◽  
Flow Models ◽  
Grounding Line ◽  
Huge Impact ◽  
Initial States ◽  
The Stability ◽  
Ice Model

<p>Global warming has a huge impact on the different climatic components. In Antarctica, small changes on the ice-sheet or the ocean may drive the continent to some large instabilities. At a certain threshold, a tipping point might be crossed and the ice-sheet might retreat faster and irreversibly. The TiPACCs (Tipping Points in Antarctic Climate Components) project aims to a better understanding of the tipping points of Antarctica, both in the ocean and in the glaciers.  3 different ice-flow models (Elmer/Ice, PISM, and Ua) are used in the project.  This study is focusing on the Elmer/Ice model to determine and characterize tipping points for its grounding lines. In this presentation, the most famous instability of Antarctica, the Marine Ice-Sheet Instability (MISI), will be investigated. The goal is to define the stability regime of the current Antarctic ice-sheet. For this purpose, multiple initial states have been created.  The Elmer/Ice model uses the inverse method as it has been done in InitMIP-Antarctica (Seroussi et al. 2019) to define initial states. A common initial state for the three TiPACCs models has been defined by the use of common datasets and parameters. The melt at the base of the ice-shelf is defined by the PICO parametrization (Reese, 2016) which permits to define the melting per basins with a box model. Then, perturbations of basalt melt are be applied by modifying the ocean far-field temperature and salinity. The stability of the current grounding line is evaluated by calculating the grounding line migration for the different ice-shelf. The experiments are driven by a small but numerically significant perturbation to observe a retreat of the grounding line. If the grounding line is moving backward when removing the perturbation, then we can conclude that it is stable. Otherwise, if the grounding line is continuing its retreat then it is unstable.</p><p> </p>

scholarly journals All adapted topologies are equal

2020 ◽  
Vol 178 (3-4) ◽  
pp. 1125-1172
Author(s):  
Julio Backhoff-Veraguas ◽  
Daniel Bartl ◽  
Mathias Beiglböck ◽  
Manu Eder
Keyword(s):  
Stochastic Processes ◽  
Weak Topology ◽  
Equilibrium Problems ◽  
Diffusion Processes ◽  
Temporal Structure ◽  
Wasserstein Distance ◽  
Optimal Stopping Problems ◽  
Reward Functions ◽  
Nested Distance ◽  
The Stability

Abstract A number of researchers have introduced topological structures on the set of laws of stochastic processes. A unifying goal of these authors is to strengthen the usual weak topology in order to adequately capture the temporal structure of stochastic processes. Aldous defines an extended weak topology based on the weak convergence of prediction processes. In the economic literature, Hellwig introduced the information topology to study the stability of equilibrium problems. Bion–Nadal and Talay introduce a version of the Wasserstein distance between the laws of diffusion processes. Pflug and Pichler consider the nested distance (and the weak nested topology) to obtain continuity of stochastic multistage programming problems. These distances can be seen as a symmetrization of Lassalle’s causal transport problem, but there are also further natural ways to derive a topology from causal transport. Our main result is that all of these seemingly independent approaches define the same topology in finite discrete time. Moreover we show that this ‘weak adapted topology’ is characterized as the coarsest topology that guarantees continuity of optimal stopping problems for continuous bounded reward functions.

scholarly journals Stimulated emission of relic gravitons and their super-Poissonian statistics

2019 ◽  
Vol 34 (23) ◽  
pp. 1950185 ◽  
Author(s):  
Massimo Giovannini
Keyword(s):  
Coherent State ◽  
Second Order ◽  
Stimulated Emission ◽  
Initial State ◽  
Einstein Distribution ◽  
Hubble Radius ◽  
Initial States ◽  
Bose Einstein ◽  
Final Degree ◽  
Poissonian Statistics

The degree of second-order coherence of the relic gravitons produced from the vacuum is super-Poissonian and larger than in the case of a chaotic source characterized by a Bose–Einstein distribution. If the initial state does not minimize the tensor Hamiltonian and has a dispersion smaller than its averaged multiplicity, the overall statistics is by definition sub-Poissonian. Depending on the nature of the sub-Poissonian initial state, the final degree of second-order coherence of the quanta produced by stimulated emission may diminish (possibly even below the characteristic value of a chaotic source) but it always remains larger than one (i.e. super-Poissonian). When the initial statistics is Poissonian (like in the case of a coherent state or for a mixed state weighted by a Poisson distribution) the degree of second-order coherence of the produced gravitons is still super-Poissonian. Even though the quantum origin of the relic gravitons inside the Hubble radius can be effectively disambiguated by looking at the corresponding Hanbury Brown–Twiss correlations, the final distributions caused by different initial states maintain their super-Poissonian character which cannot be altered.

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