Approximating the Steady-State Periodic Solutions of Contractive Systems

“…Such systems have convergence properties that can be defined using a generalization of Equation ( 6) (a generalization containing an expression of the form C∥p 1 (0) − p 2 (0)∥ instead of the constant C) [65,66]. Perturbation and approximation bounds for both regular [67,68] and singular [69] perturbations of contrac-tive systems have been derived. Whereas this perturbation theory developed within the domain of differential equations independently of Markov chain theory, there appear to be possibilities for cross-fertilization.…”

Section: Discussionmentioning

confidence: 99%

The Arsenal of Perturbation Bounds for Finite Continuous-Time Markov Chains: A Perspective

Mitrophanov

2024

Mathematics

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Perturbation bounds are powerful tools for investigating the phenomenon of insensitivity to perturbations, also referred to as stability, for stochastic and deterministic systems. This perspective article presents a focused account of some of the main concepts and results in inequality-based perturbation theory for finite state-space, time-homogeneous, continuous-time Markov chains. The diversity of perturbation bounds and the logical relationships between them highlight the essential stability properties and factors for this class of stochastic processes. We discuss the linear time dependence of general perturbation bounds for Markov chains, as well as time-independent (i.e., time-uniform) perturbation bounds for chains whose stationary distribution is unique. Moreover, we prove some new results characterizing the absolute and relative tightness of time-uniform perturbation bounds. Specifically, we show that, in some of them, an equality is achieved. Furthermore, we analytically compare two types of time-uniform bounds known from the literature. Possibilities for generalizing Markov-chain stability results, as well as connections with stability analysis for other systems and processes, are also discussed.

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“…Entrainment in nonlinear systems is non-trivial to prove. A typical proof is based on contraction theory [ 6 , 7 ], yet this type of proof provides no information on the attractive periodic solution, except for its period (see [ 39 ] for some related considerations). Nevertheless, we show here that determining the gain of entrainment can be cast as an optimal control problem .…”

Section: Introductionmentioning

confidence: 99%

Maximizing average throughput in oscillatory biochemical synthesis systems: an optimal control approach

2021

Self Cite

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A dynamical system entrains to a periodic input if its state converges globally to an attractor with the same period. In particular, for a constant input, the state converges to a unique equilibrium point for any initial condition. We consider the problem of maximizing a weighted average of the system’s output along the periodic attractor. The gain of entrainment is the benefit achieved by using a non-constant periodic input relative to a constant input with the same time average. Such a problem amounts to optimal allocation of resources in a periodic manner. We formulate this problem as a periodic optimal control problem, which can be analysed by means of the Pontryagin maximum principle or solved numerically via powerful software packages. We then apply our framework to a class of nonlinear occupancy models that appear frequently in biological synthesis systems and other applications. We show that, perhaps surprisingly, constant inputs are optimal for various architectures. This suggests that the presence of non-constant periodic signals, which frequently appear in biological occupancy systems, is a signature of an underlying time-varying objective functional being optimized.

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Approximating the Steady-State Periodic Solutions of Contractive Systems

Cited by 3 publications

References 44 publications

Revisiting totally positive differential systems: A tutorial and new results

Revisiting totally positive differential systems: A tutorial and new results

The Arsenal of Perturbation Bounds for Finite Continuous-Time Markov Chains: A Perspective

Maximizing average throughput in oscillatory biochemical synthesis systems: an optimal control approach

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