Theory Of Dissipative Systems Research Articles

Asymptotic Behavior of a Chemostat Model with Constant Recycle Sludge Concentration.

In this work, we study a several species aerobic chemostat model with constant recycle sludge concentration in continuous culture. We reduce the number of parameters by considering a dimensionless model. First, the existence of a global positive uniform attractor for the model with different removal rates is proved using the theory of dissipative dynamical systems. Hence, we investigate the asymptotic behavior of the model under small perturbations using methods of singular perturbation theory and we prove that, in the case of two species in competition, the unique equilibrium which is positive is globally asymptotically stable. Finally, we establish the link between the open problem of the chemostat with different removal rates and monotone functional responses, and our model when two species compete on the same nutrient. We give some numerical simulations to illustrate the results.

Dissipative open systems theory as a foundation for the thermodynamics of linear systems.

In this paper, we advocate the use of open dynamical systems, i.e. systems sharing input and output variables with their environment, and the dissipativity theory initiated by Jan Willems as models of thermodynamical systems, at the microscopic and macroscopic level alike. We take linear systems as a study case, where we show how to derive a global Lyapunov function to analyse networks of interconnected systems. We define a suitable notion of dynamic non-equilibrium temperature that allows us to derive a discrete Fourier law ruling the exchange of heat between lumped, discrete-space systems, enriched with the Maxwell-Cattaneo correction. We complete these results by a brief recall of the steps that allow complete derivation of the dissipation and fluctuation in macroscopic systems (i.e. at the level of probability distributions) from lossless and deterministic systems.This article is part of the themed issue 'Horizons of cybernetical physics'.

A Dissipative Systems Theory for FDTD With Application to Stability Analysis and Subgridding

This paper establishes a far-reaching connection between the Finite-Difference Time-Domain method (FDTD) and the theory of dissipative systems. The FDTD equations for a rectangular region are written as a dynamical system having the magnetic and electric fields on the boundary as inputs and outputs. Suitable expressions for the energy stored in the region and the energy absorbed from the boundaries are introduced, and used to show that the FDTD system is dissipative under a generalized Courant-Friedrichs-Lewy condition. Based on the concept of dissipation, a powerful theoretical framework to investigate the stability of FDTD methods is devised. The new method makes FDTD stability proofs simpler, more intuitive, and modular. Stability conditions can indeed be given on the individual components (e.g. boundary conditions, meshes, embedded models) instead of the whole coupled setup. As an example of application, we derive a new subgridding method with material traverse, arbitrary grid refinement, and guaranteed stability. The method is easy to implement and has a straightforward stability proof. Numerical results confirm its stability, low reflections, and ability to handle material traverse.

Synthesis of Distributed Robust H-Infinity Controllers for Interconnected Discrete Time Systems

This paper presents an algorithm for the synthesis of robust distributed controllers for interconnected linear discrete-time systems. For a network of interconnected uncertain linear time-invariant systems, the distributed controller achieves robust stability and a guaranteed level of robust performance in a well-defined H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> sense. The setting of this paper is in discrete time. Based on the theory of dissipative dynamical systems, conditions for the analysis of robust stability and robust performance of networks are derived in terms of feasibility tests of linear matrix inequalities. From these conditions, computationally tractable synthesis conditions are derived. An iterative D-K type of synthesis algorithm is proposed that yields a robust distributed controller. Convergence properties of the algorithm are inferred.

Observer‐Based Adaptive L2 Disturbance Attenuation Control of Semi‐Active Suspension with MR Damper

AbstractA novel nonlinear observer‐based adaptive disturbance attenuation control strategy was proposed for a quarter semi‐active suspension system with a magneto‐rheological (MR) damper in light of the intrinsic nonlinearity, parameter uncertainty, state immeasurability and road randomness. Adaptive adjusting parameters were adopted to avoid the curve fitting and identification of the system parameters by a great deal of experimental data for shortening the development cycle of the control system. Based on the reduced‐order observer, the system states including the immeasurable virtual state of MR damper and inconveniently measured states of suspension system were estimated for the realistic frame of the proposed controller in practice. The dissipative system theory was utilized to reduce the influence of the road disturbance on the system control performance. Simulation results in the bump road and B‐class road indicate that, whether there are perturbations of the system parameters or not, the proposed control scheme always ensures a better performance on the suspension travel, ride comfort and handling stability in comparison with other existing methods.

Nonequilibrium Response of Nanosystems Coupled to Driven Quantum Baths.

Commonly, nanosystems are characterized by their response to time-dependent external fields in the presence of inevitable environmental fluctuations. The direct impact of the external driving on the environment is generally neglected. While this approach is satisfactory for macroscopic systems, on the nanoscale, an interaction of external fields with the environment is often unavoidable on principle. We extend the standard linear response theory of quantum dissipative systems to strongly driven baths. Significant modifications are found for two paradigm examples. First, we evaluate the polarizability of a molecule immersed in a strongly polarizable medium that responds to terahertz radiation. We find an increase of the molecular polarizability by about 30%. Second, we determine the response of a semiconductor quantum dot in close proximity to a metallic nanoparticle. Both are placed in a polarizable medium and exposed to electromagnetic irradiation. We show that the response of the quantum dot is qualitatively modified by the driven nanoparticle, including the generation of an additional channel of stimulated emission.

Transience and multifractal analysis

We study dimension theory for dissipative dynamical systems, proving a conditional variational principle for the quotients of Birkhoff averages restricted to the recurrent part of the system. On the other hand, we show that when the whole system is considered (and not just its recurrent part) the conditional variational principle does not necessarily hold. Moreover, we exhibit an example of a topologically transitive map having discontinuous Lyapunov spectrum. The mechanism producing all these pathological features on the multifractal spectra is transience, that is, the non-recurrent part of the dynamics.

Energetics of an rf SQUID Coupled to Two Thermal Reservoirs.

We study energetics of a Josephson tunnel junction connecting a superconducting loop pierced by an external magnetic flux (an rf SQUID) and coupled to two independent thermal reservoirs of different temperature. In the framework of the theory of quantum dissipative systems, we analyze energy currents in stationary states. The stationary energy flow can be periodically modulated by the external magnetic flux exemplifying the rf SQUID as a quantum heat interferometer. We also consider the transient regime and identify three distinct regimes: monotonic decay, damped oscillations and pulse-type behavior of energy currents. The first two regimes can be controlled by the external magnetic flux while the last regime is robust against its variation.

Experimental evidence that evolution by niche construction affects dissipative ecosystem dynamics

Evolution by niche construction occurs when organism-mediated modification of the environment causes an evolutionary response. Physicists have postulated that evolution in general, and evolution mediated via feedbacks between organisms and their environment in particular (i.e. evolution by niche construction), could increase the capacity of biological systems to dissipate free energy in an open thermodynamic system, and help them maintain a state far from thermodynamic equilibrium. Here, we propose using the bacterium Pseudomonas fluorescens (strain SBW25) as a model system to experimentally test theories in both evolutionary biology (e.g. niche construction) and physics (e.g. dissipative systems theory). P. fluorescens rapidly and predictably evolves multiple strategies for exploiting oxygen in unmixed culture flasks. This evolutionary dynamic is mediated by feedbacks between the modification of the oxygen gradient by P. fluorescens and the ecological and evolutionary responses of Pseudomonas to modified environmental conditions. To confirm this, we experimentally manipulated two aspects of the system that influence the strength of the feedback between P. fluorescens and oxygen gradients in the system. First, we inhibited the metabolism of the strain used to inoculate the cultures, and, second, we disturbed the formation of mats at the air–liquid interface. Overall, we found convincing experimental evidence of evolution by niche construction, and conclude that this study system is amenable to experimental investigations of both niche construction and dissipative systems theory.

The topology and geometry of self-adjoint and elliptic boundary conditions for Dirac and Laplace operators

The theory of self-adjoint extensions of first- and second-order elliptic differential operators on manifolds with boundary is studied via its most representative instances: Dirac and Laplace operators. The theory is developed by exploiting the geometrical structures attached to them and, by using an adapted Cayley transform on each case, the space [Formula: see text] of such extensions is shown to have a canonical group composition law structure. The obtained results are compared with von Neumann's theorem characterizing the self-adjoint extensions of densely defined symmetric operators on Hilbert spaces. The 1D case is thoroughly investigated. The geometry of the submanifold of elliptic self-adjoint extensions [Formula: see text] is studied and it is shown that it is a Lagrangian submanifold of the universal Grassmannian Gr. The topology of [Formula: see text] is also explored and it is shown that there is a canonical cycle whose dual is the Maslov class of the manifold. Such cycle, called the Cayley surface, plays a relevant role in the study of the phenomena of topology change. Self-adjoint extensions of Laplace operators are discussed in the path integral formalism, identifying a class of them for which both treatments leads to the same results. A theory of dissipative quantum systems is proposed based on this theory and a unitarization theorem for such class of dissipative systems is proved. The theory of self-adjoint extensions with symmetry of Dirac operators is also discussed and a reduction theorem for the self-adjoint elliptic Grassmannian is obtained. Finally, an interpretation of spontaneous symmetry breaking is offered from the point of view of the theory of self-adjoint extensions.

Global dynamics of a nonlocal population model with age structure in a bounded domain: A non-monotone case

We study the global dynamics of a nonlocal population model with age structure in a bounded domain. We mainly concern with the case where the birth rate decreases as the mature population size become large. The analysis is rather subtle and it is inadequate to apply the powerful theory of monotone dynamical systems. By using the method of super-sub solutions, combined with the careful analysis of the kernel function in the nonlocal term, we prove nonexistence, existence and uniqueness of positive steady states of the model. Moreover, due to the mature individuals do not diffuse, the solution semiflow to the model is not compact. To overcome the difficulty of non-compactness in describing the global asymptotic stability of the unique positive steady state, we first establish an appropriate comparison principle. With the help of the comparison principle, we can employ the theory of dissipative systems to obtain the global asymptotic stability of the unique positive steady state. The main results are illustrated with the nonlocal Nicholson’s blowflies equation and the nonlocal Mackey-Glass equation.

Reconfigurable distributed model predictive control

An approach to reconfigurable distributed model predictive control based on reconfigurable controller dissipativity properties is developed. The dissipativity properties of the controllers are updated online to reconfigure themselves for changes in the process network topology, which may be due to changing product specifications, feedstock type or scheduled or unscheduled maintenance; allowing for more flexible and agile manufacturing processes. The use of dissipative systems theory allows for the interaction effects between individual processes to be taken into account in control design to achieve high levels of plant-wide performance. Plant-wide performance and stability bounds are developed based on dissipative systems theory, which in turn are translated into the dissipative trajectory conditions on each local controller. This approach is enabled by the use of dynamic supply rates in quadratic difference form parameterised as linear functions of the process network structure.

Dissipativity-based Analysis of Controller Networks with Reduced Rate Communication

Distributed control is a computationally efficient and high performance approach for plantwide process systems, where the interactions between process units can be dealt with using the information exchanged among controllers. In this work we develop an approach to analyze distributed control systems where the controllers communicate at a rate lower than the sampling rate for the local process units. This is motivated by the observation that many chemical processes have plant-wide dynamics on a time-scale slower than that of the individual processes’ dynamics. Based on the dissipative systems theory, this approach can be used to determine the impact of the reduced communication rate on the stability and control performance of the plant-wide system.

Modified plasma–fluid equations from nonstandard Lagrangians with applications to nuclear fusion

Nonstandard Lagrangian dynamics have gained great interest recently, in particular within the theory of nonlinear differential equations and dissipative dynamical systems. In this paper, we address their implications in plasma–fluid dynamics. The mathematical settings are constructed starting from the modified Vlasov–Boltzmann transport equation, which is derived from modified Euler–Lagrange equations of motion. Far from giving a self-consistent nonstandard Lagrangian theory of plasma–fluid dynamics, in this paper we have just introduced the basic settings and discussed some illustrative examples that such a modified theory should have in plasma–fluid theory and nuclear fusion.

A nonlocal diffusion population model with age structure and Dirichlet boundary condition

In this paper, we study the global dynamics of a population modelwith age structure. The model is given by a nonlocalreaction-diffusion equation carrying a maturation time delay,together with the homogeneous Dirichlet boundary condition. Thenon-locality arises from spatial movements of the immatureindividuals. We are mainly concerned with the case when the birthrate decays as the mature population size becomes large. The analysis is rathersubtle and it is inadequate to apply the powerful theory of monotonedynamical systems. By using the method of super-sub solutions,combined with the careful analysis of the kernel function in thenonlocal term, we prove nonexistence, existence and uniqueness ofthe positive steady states of the model. By establishing anappropriate comparison principle and applying the theory ofdissipative systems, we obtain some sufficient conditions for theglobal asymptotic stability of the trivial solution and the uniquepositive steady state.

Global existence and optimal decay rates for the Timoshenko system: The case of equal wave speeds

We first show the global existence and optimal decay rates of solutions to the classical Timoshenko system in the framework of Besov spaces. Due to the non-symmetric dissipation, the general theory for dissipative hyperbolic systems (see [31]) cannot be applied to the Timoshenko system directly. In the case of equal wave speeds, we construct global solutions to the Cauchy problem pertaining to data in the spatially Besov spaces. Furthermore, the dissipative structure enables us to give a new decay framework which pays less attention on the traditional spectral analysis. Consequently, the optimal decay estimates of solution and its derivatives of fractional order are shown by time-weighted energy approaches in terms of low-frequency and high-frequency decompositions. As a by-product, the usual decay estimate of L1(R)-L2(R) type is also shown.

Multi-rate dissipativity based control of process networks

An approach to multi-rate distributed control design for process networks is presented, where the local measurements, local control and controller communication are allowed to operate at different sampling rates. Dissipative systems theory is used to facilitate stability and performance analysis of the process network, based upon dynamic supply rates which have been lifted into a global sampling rate. Quadratic difference forms are used as supply rates and storage functions, leading to less conservative stability and performance conditions as compared to classical types of supply rates. These theoretical results are illustrated by a case study of a heat exchanger network.

Dynamics of a fractional hydrodynamical equation for the Heisenberg paramagnet

In the present work, we study the global solvability and large time dynamics for a fractional generalization of the hydrodynamical equation modeling the soft micromagnetic materials. Introducing a cancellation property, we prove the existence of weak solutions and establish a uniqueness criterion. A maximal principle is obtained and the global existence and uniqueness of smooth solutions are proved by some a priori estimates. Finally, we analyze the asymptotic behavior of the solutions within the theory of infinite dimensional dissipative dynamical systems. We prove that the problem generates a strongly continuous semigroup on a suitable phase space and show the existence of a maximal global attractor A in this phase space. Moreover, in absence of external force, global attractor A converges exponentially to a single equilibrium.

Study on Construction of Student Work Groups in China’s Colleges Based on Dissipative Structure Theory

This paper aims to study the student work groups in China’s colleges on the basis of the dissipative structure theory. It attempts to analyze the features of development from the perspective of a dissipative system theory, take such measures as reducing entropy and increasing negative entropy to ensure the stability of system of the student work groups, and propose approaches to developing sustainable student work groups. The specific approaches include clarifying roles and responsibilities of workers, developing scientific mechanisms for assessment and rewards and penalty, designing rational mechanisms for personnel selection, building platforms for enhancing the overall quality of student work groups, improving their research capabilities, and promoting the process of being specialized and expertized.

Decentralized nonlinear control of process networks based on dissipativity—A Hamilton–Jacobi equation approach

This paper presents an approach of decentralized nonlinear control for process networks based on the theory of dissipative systems. First, the dissipativity analysis is performed to determine the candidate dissipative functions for each subsystem such that desired global network performance criteria are achieved. Then, a nonlinear control law is designed for each subsystem to locally shape the candidate dissipative function. We show that the dissipativity shaping of each subsystem is equivalent to the robust nonlinear H∞ control problem for an equivalent system. The solution to such problem can be obtained by solving a linear partial differential Hamilton–Jacobi equation, for which many approximate solution methods have been developed. In particular, our results are related to the expansion approach. The performance of the proposed algorithm is demonstrated via a benchmark example of a reactor–separator network.