203 research outputs found
Optimal control formulation of transition path problems for Markov Jump Processes
Among various rare events, the effective computation of transition paths
connecting metastable states in a stochastic model is an important problem.
This paper proposes a stochastic optimal control formulation for transition
path problems in an infinite time horizon for Markov jump processes on polish
space. An unbounded terminal cost at a stopping time and a controlled
transition rate for the jump process regulate the transition from one
metastable state to another. The running cost is taken as an entropy form of
the control velocity, in contrast to the quadratic form for diffusion
processes. Using the Girsanov transformation for Markov jump processes, the
optimal control problem in both finite time and infinite time horizon with
stopping time fit into one framework: the optimal change of measures in the
C\`adl\`ag path space via minimizing their relative entropy. We prove that the
committor function, solved from the backward equation with appropriate boundary
conditions, yields an explicit formula for the optimal path measure and the
associated optimal control for the transition path problem. The unbounded
terminal cost leads to a singular transition rate (unbounded control velocity),
for which, the Gamma convergence technique is applied to pass the limit for a
regularized optimal path measure. The limiting path measure is proved to solve
a Martingale problem with an optimally controlled transition rate and the
associated optimal control is given by Doob-h transformation. The resulting
optimally controlled process can realize the transitions almost surely.Comment: 31 page
On the Value of Linear Quadratic Zero-sum Difference Games with Multiplicative Randomness: Existence and Achievability
We consider a wireless networked control system (WNCS) with multiple
controllers and multiple attackers. The dynamic interaction between the
controllers and the attackers is modeled as a linear quadratic (LQ) zero-sum
difference game with multiplicative randomness induced by the multiple-input
and multiple-output (MIMO) wireless fading channels of the controllers and
attackers. We focus on analyzing the existence and achievability of the value
of the zero-sum game. We first establish a general sufficient and necessary
condition for the existence of the game value. This condition relies on the
solvability of a modified game algebraic Riccati equation (MGARE) under an
implicit concavity constraint, which is generally difficult to verify due to
the intermittent controllability or almost sure uncontrollability caused by the
multiplicative randomness. We then introduce a new positive semidefinite (PSD)
kernel decomposition method induced by multiplicative randomness, through which
we obtain a closed-form tight verifiable sufficient condition. Under the
existence condition, we finally construct a saddle-point policy that is able to
achieve the game value in a certain class of admissible policies. We
demonstrate that the proposed saddle-point policy is backward compatible to the
existing strictly feedback stabilizing saddle-point policy.Comment: 32 pages, 3 figure
Almost Sure Stability and Stabilization for Hybrid Stochastic Systems with Time-Varying Delays
The problems of almost sure (a.s.) stability and a.s. stabilization are investigated for hybrid stochastic systems (HSSs) with time-varying delays. The different time-varying delays in the drift part and in the diffusion part are considered. Based on nonnegative semimartingale convergence theorem, Hölder’s inequality, Doob’s martingale inequality, and Chebyshev’s inequality, some sufficient conditions are proposed to guarantee that the underlying nonlinear hybrid stochastic delay systems (HSDSs) are almost surely (a.s.) stable. With these conditions, a.s. stabilization problem for a class of nonlinear HSDSs is addressed through designing linear state feedback controllers, which are obtained in terms of the solutions to a set of linear matrix inequalities (LMIs). Two numerical simulation examples are given to show the usefulness of the results derived
Interest rate models with Markov chains
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Particle systems with a singular mean-field self-excitation. Application to neuronal networks
We discuss the construction and approximation of solutions to a nonlinear
McKean-Vlasov equation driven by a singular self-excitatory interaction of the
mean-field type. Such an equation is intended to describe an infinite
population of neurons which interact with one another. Each time a proportion
of neurons 'spike', the whole network instantaneously receives an excitatory
kick. The instantaneous nature of the excitation makes the system singular and
prevents the application of standard results from the literature. Making use of
the Skorohod M1 topology, we prove that, for the right notion of a 'physical'
solution, the nonlinear equation can be approximated either by a finite
particle system or by a delayed equation. As a by-product, we obtain the
existence of 'synchronized' solutions, for which a macroscopic proportion of
neurons may spike at the same time
Reduced realizations and model reduction for switched linear systems:a time-varying approach
In the last decades, switched systems gained much interest as a modeling framework in many applications. Due to a large number of subsystems and their high-dimensional dynamics, such systems result in high complexity and challenges. This motivates to find suitable reduction methods that produce simplified models which can be used in simulation and optimization instead of the original (large) system. In general, the study aims to find a reduced model for a given switched system with a fixed switching signal and known mode sequence. This thesis concerns first the reduced realization of switched systems with known mode sequence which has the same input-output behavior as original switched systems. It is conjectured that the proposed reduced system has the smallest order for almost all switching time duration. Secondly, a model reduction method is proposed for switched systems with known switching signals which provide a good model with suitable thresholds for the given switched system. The quantitative information for each mode is carried out by defining suitable Gramians and, these Gramians are exploited at the midpoint of the given switching time duration. Finally, balanced truncation leads to a modewise reduction. Later, a model reduction method for switched differential-algebraic equations in continuous time is proposed. Thereto, a switched linear system with jumps and impulses is constructed which has the identical input-output behavior as original systems. Finally, a model reduction approach for singular linear switched systems in discrete time is studied. The choice of initial/final values of the reachability and observability Gramians are also investigated
International Conference on Mathematical Analysis and Applications in Science and Engineering – Book of Extended Abstracts
The present volume on Mathematical Analysis and Applications in Science and Engineering - Book of
Extended Abstracts of the ICMASC’2022 collects the extended abstracts of the talks presented at the
International Conference on Mathematical Analysis and Applications in Science and Engineering –
ICMA2SC'22 that took place at the beautiful city of Porto, Portugal, in June 27th-June 29th 2022 (3 days).
Its aim was to bring together researchers in every discipline of applied mathematics, science, engineering,
industry, and technology, to discuss the development of new mathematical models, theories, and
applications that contribute to the advancement of scientific knowledge and practice. Authors proposed
research in topics including partial and ordinary differential equations, integer and fractional order
equations, linear algebra, numerical analysis, operations research, discrete mathematics, optimization,
control, probability, computational mathematics, amongst others.
The conference was designed to maximize the involvement of all participants and will present the state-of-
the-art research and the latest achievements.info:eu-repo/semantics/publishedVersio
Large Scale Stochastic Dynamics
In focus are interacting stochastic systems with many components, ranging from stochastic partial differential equations to discrete systems as interacting particles on a lattice moving through random jumps.
More specifically one wants to understand the large scale behavior, large in spatial extent but also over long time spans, as entailed by the characterization of stationary measures, effective macroscopic evolution laws, transport of conserved fields, homogenization, self-similar structure and scaling, critical dynamics, aging, dynamical phase transitions, large deviations, to mention only a few key items
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