Theory Of Dissipative Systems Research Articles

On a 3D crack tracking algorithm and its variational nature

Abstract The crack propagation problem for linear elastic fracture mechanics has been studied by several authors exploiting its analogy with standard dissipative systems theory (see e.g. Nguyen (2000), Mielke (2005) and Francfort and Marigo (1998)). In recent publications Salvadori and Carini (2011) and Salvadori and Fantoni (2013) minimum theorems were derived in terms of crack tip “quasi static velocity” for two- and three-dimensional fracture mechanics. They were reminiscent of Ceradini's theorem Ceradini (1965, 1966) in plasticity. Such an incremental picture naturally leads to explicit methods for integration in time, with well know drawbacks in terms of accuracy and stability. The present work introduces an implicit Newton–Raphson based crack tracking algorithm which is endowed with a variational formulation.

Analysis and simulation for an isotropic phase-field model describing grain growth

A phase-field system of coupled Allen--Cahn type PDEs describing grain growth is analyzed and simulated. In the periodic setting, we prove the existence and uniqueness of global weak solutions to the problem. Then we investigate the long-time behavior of the solutions within the theory of infinite-dimensional dissipative dynamical systems. Namely, the problem possesses a global attractor as well as an exponential attractor, which entails that the global attractor has finite fractal dimension.Moreover, we show that each trajectory converges to a single equilibrium. A time-adaptive numerical scheme based on trigonometric interpolation is presented.It allows to track the approximated long-time behavior accurately and leads to a convergence rate. The scheme exhibits a physically consistent discrete free energy dissipation.

Attractors for the Magnetic Bénard Problem

The influence of the magnetic field on the heat convection in a viscous and resistive fluid is an important topic for researches in various industrial situations, which include steel processing, the production of semi-conductors, and so on. A mathematical formulation of the heat convection phenomena under the presence of the magnetic field is provided by the magnetic Benard equations, which is a coupled system of the magnetohydrodynamic equations and the thermohydraulics equations. We refer to [1]–[3] for more details. We deal here with the asymptotic behavior of solutions to the magnetic Benard system. In particular, we are concerned with the attractors, which characterize the long-time dynamics of the original system and possess the finite dimensional structure. Indeed, the theory of infinite dimensional dissipative dynamical systems developed so far and the analysis of the asymptotic behavior of solutions to the nonlinear evolution equations of these types generally show the finite dimensional features of the problems. Our intension is then to estimate the fractal dimension of the attractors in terms of the various physical constants; that is, the (magnetic) Prandtl number, Rayleigh number to name a few. We believe that such characteristics may reveal the degree and the extent of the influence of magnetic factors and may be used as controlling parameters.

Preface

The theory of infinite dimensional and stochastic dynamical systems is a rapidly expanding and vibrant field of mathematics. In the recent three decades it has been highlighted as a core knowledge and an advancing thrust in the qualitative study of complex systems and processes described by evolutionary partial differential equations in many different settings, stochastic differential equations, functional differential equations and lattice differential equations. The central research topics include the invariant and attracting sets, stability and bifurcation of patterns and waves, asymptotic theory of dissipative systems and reduction of dimensions, and more and more problems of nonlocal systems, ill-posed systems, multicomponent and network dynamics, random dynamics and chaotic dynamics.For more information please click the “Full Text” above

Three Dimensional Fracture Growth as a Standard Dissipative System: Some General Theorems and Numerical Simulations

The crack propagation problem for linear elastic fracture mechanics has been studied by several authors exploiting its analogy with standard dissipative systems theory (see e.g. Nguyen (1994); Bourdin et al. (2008); Salvadori (2008)). In recent publications (Salvadori and Carini (2011); Salvadori and Fantoni (2013)) minimum theorems were derived in terms of crack tip “quasi static velocity” for two-dimensional fracture mechanics. They were reminiscent of Ceradini's theorem (Ceradini (1966)) in plasticity. Following the cornerstone work of Rice (Rice (1989)) on weight function theories, Leblond and coworkers (Leblond (1999); Leblond et al. (1999)) proposed asymptotic expansions for Stress Intensity Factors (SIFs) in three dimensions. As formerly in 2D, expansions can be given a Colonnetti's decomposition interpretation (Colonnetti (1950)). In view of the expression of the expansions proposed in (Leblond (1999); Leblond et al. (1999)) however, symmetry of Ceradini's theorem operators was not evident and the extension of outcomes proposed in (Salvadori and Fantoni (2013)) not straightforward. Following a different path of reasoning, minimum theorems have been finally derived. Moving from well established theorems in plasticity, algorithms for crack advancing have been formulated (Salvadori and Fantoni (2014,a); Salvadori and Fantoni (2014,b)). Their performance is here presented within a set of classical benchmarks.

A theory of robust omega-regular software synthesis

A key property for systems subject to uncertainty in their operating environment is robustness : ensuring that unmodeled but bounded disturbances have only a proportionally bounded effect upon the behaviors of the system. Inspired by ideas from robust control and dissipative systems theory, we present a formal definition of robustness as well as algorithmic tools for the design of optimally robust controllers for ω-regular properties on discrete transition systems. Formally, we define metric automata —automata equipped with a metric on states—and strategies on metric automata which guarantee robustness for ω-regular properties. We present fixed-point algorithms to construct optimally robust strategies in polynomial time. In contrast to strategies computed by classical graph theoretic approaches, the strategies computed by our algorithm ensure that the behaviors of the controlled system gracefully degrade under the action of disturbances; the degree of degradation is parameterized by the magnitude of the disturbance. We show an application of our theory to the design of controllers that tolerate infinitely many transient errors provided they occur infrequently enough.

Multi-Rate Dissipative Control of Large-Scale Systems

An approach to distributed multi-rate control for large-scale systems, and in particular 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, which facilitate less conservative stability and performance conditions as compared to classical types of supply rates.

Finance Mechanism of Industrial Risk Preventing Based on Knowledge Sharing: A Dissipative Theory Perspective

This paper analyzed financial mechanism in handling industrial risks in the light of dissipative system theory that internal and external fluctuations could influence system stability. Internal fluctuations from system factors and external fluctuations from environment could lead a system to diverge its path, and in industrial system an upward path is viewed as opportunity and a downward path as risk. This paper classified risk preventing finance into three types: (1) risks from fluctuations inside industries, including node fluctuations and link fluctuations; (2) risks from fluctuations from adjacent environment; (3) risks from macro-environment fluctuation. This paper demonstrates effects of industrial risk management finance by Lotka-Volterra stochastic differential dynamics equation modeling and numerical simulation method.

Minimum theorems in 3D incremental linear elastic fracture mechanics

The crack propagation problem for linear elastic fracture mechanics has been studied by several authors exploiting its analogy with standard dissipative systems theory (see e.g. Nguyen in Appl Mech Rev 47, 1994, Stability and nonlinear solid mechanics. Wiley, New York, 2000; Mielke in Handbook of differential equations, evolutionary equations. Elsevier, Amsterdam, 2005; Bourdin et al. in The variational approach to fracture. Springer, Berlin, 2008). In a recent publication (Salvadori and Carini in Int J Solids Struct 48:1362–1369, 2011) minimum theorems were derived in terms of crack tip “quasi static velocity” for two-dimensional fracture mechanics. They were reminiscent of Ceradini’s theorem (Ceradini in Rendiconti Istituto Lombardo di Scienze e Lettere A99, 1965, Meccanica 1:77–82, 1966) in plasticity. Following the cornerstone work of Rice (1989) on weight function theories, Leblond et al. (Leblond in Int J Solids Struct 36:79–103, 1999; Leblond et al. in Int J Solids Struct 36:105–142, 1999) proposed asymptotic expansions for stress intensity factors in three dimensions—see also Lazarus (J Mech Phys Solids 59:121–144, 2011). As formerly in 2D, expansions can be given a Colonnetti’s decomposition (Colonnetti in Rend Accad Lincei 5, 1918, Quart Appl Math 7:353–362, 1950) interpretation. In view of the expression of the expansions proposed in Leblond (Int J Solids Struct 36:79–103, 1999), Leblond et al. (Int J Solids Struct 36:105–142, 1999) however, symmetry of Ceradini’s theorem operators was not evident and the extension of outcomes proposed in Salvadori and Carini (Int J Solids Struct 48:1362–1369, 2011) not straightforward. Following a different path of reasoning, minimum theorems have been finally derived.

Geometric Construction of Mixed Potentials

This paper presents an approach to construct mixed potentials, to be used as storage functions in the context of dissipative systems theory. The objective is to obtain, through a locally-defined geometric decomposition of a given drift vector field, a potential similar to the thermodynamically-defined availability function proposed in the literature. The mixed potential is obtained by homotopy integration of a differential one-form for the drift vector field which decomposes the system into an exact part (generated by a potential) and an anti-exact part (which is not directly generated by a potential). The key element in the proposed approach is the computation of an integrating factor for the anti-exact part of the dynamics.

The New Creatures of Difference: A Look at the Concept of Repetition Within Dissipative Systems Theory

Repetition is a major theme in contemporary thought, aligned more with difference and monstrosity than similarity and banality. This paper examines the logic of repetition through its deployment as a critique of the Western philosophical tradition’s understanding of time as temporal succession and examines the development of new concepts of repetition in dissipative systems theory and evolutionary thinking.

Dissipativity-based decentralized control of interconnected nonlinear chemical processes

This paper presents an approach, based on dissipative systems theory, to the analysis and control design of interconnected nonlinear processes. The objective is to design distributed feedback controllers to achieve plant-wide stability. Extensions of classical results on the stability of large-scale interconnected systems lead to input–output dissipativity constraints for each subsystems, encoded as supply rates from input to output interconnecting ports. For each subsystem, a parameterized nonlinear feedback controller is designed using nonlinear dissipative inequalities to ensure that the aforementioned dissipativity constraints are met in closed-loop. One focus of this paper is the design of domination-based nonlinear feedback controllers to meet the above interconnection constraints. This paper also presents new results on the construction of storage functions for control affine systems, as a generalization of physics-based approaches to dissipative systems theory. Control of interconnected chemical reactors with a recycle stream is presented throughout the paper to demonstrate the proposed construction.

Special issue on robust stability and control of large-scale nonlinear systems

In a broad sense, large-scale systems theory is concerned with the analysis and design of often very high-dimensional systems that admit some kind of partition into smaller subsystems. Research in linear large-scale systems analysis and synthesis can be traced back to at least the 1970s. Recently, interest in this subject has been revived by new developments in nonlinear systems and control, which include but are not limited to dissipative systems theory, input-to-state stability, and various Lyapunov and smallgain design schemes. Systems and control theory itself has seen many important advances since the 1970s, including, e.g., concepts like control Lyapunov functions, effective implementations of nonlinear model predictive control, or entire emerging areas like hybrid systems and cooperative multi-agent systems. Naturally, these developments also leap into the area of large-scale systems. It is hence to be expected that the interest in this area will continue to increase, ultimately also due to, e.g., current research efforts in distributed model predictive control, next generation smart electricity grids, incorporating green and renewable energy sources, or bioand neuro-engineering. Many questions are associated with the theory of large-scale systems. A quite central one is about sufficient conditions on interconnections of a number of stable systems such that the composite system is also stable. Another one is how individual

Communication: Transition state theory for dissipative systems without a dividing surface

Transition state theory is a central cornerstone in reaction dynamics. Its key step is the identification of a dividing surface that is crossed only once by all reactive trajectories. This assumption is often badly violated, especially when the reactive system is coupled to an environment. The calculations made in this way then overestimate the reaction rate and the results depend critically on the choice of the dividing surface. In this Communication, we study the phase space of a stochastically driven system close to an energetic barrier in order to identify the geometric structure unambiguously determining the reactive trajectories, which is then incorporated in a simple rate formula for reactions in condensed phase that is both independent of the dividing surface and exact.

Nonlinear Robust Adaptive Control for Static Var Compensator Based on Backstepping and Dissipative System Theory

Based on adaptive backstepping nonlinear control scheme and dissipative system theory, the robust adaptive nonlinear controller is designed for single machine infinite bus (SMIB) power system with static Var compensator (SVC), which conquers the obstacle of relative degree one of passivity feedback scheme. During the procedure of designing controller and parameter estimator, the uncertainties of disturbances and the damping coefficient are taken into account; therefore, the designed controller has the ability of robustness and adaptive estimation. The globally uniformly boundness of all the states in the system and the asymptotical stability of the system errors are verified by the theories. Simulation experiments show that not only the transient stability of the SVC power system is guaranteed, but also the robustness performance is significantly improved even with disturbances.

Finite Dimensional Reduction and Convergence to Equilibrium for Incompressible Smectic-A Liquid Crystal Flows

We consider a hydrodynamic system that models smectic-A liquid crystal flow. The model consists of the Navier–Stokes equation for the fluid velocity coupled with a fourth-order equation for the layer variable, endowed with periodic boundary conditions. We analyze the long-time behavior of the solutions within the theory of infinite-dimensional dissipative dynamical systems. We first prove that in two dimensions, the problem possesses a global attractor $\mathcal{A}$ in a certain phase space. Then we establish the existence of an exponential attractor $\mathcal{M}$, which entails that the global attractor $\mathcal{A}$ has finite fractal dimension. Moreover, we show that each trajectory converges to a single equilibrium by means of a suitable Łojasiewicz–Simon inequality. Corresponding results in three dimensions are also discussed.

Attractor structures of riftogenesis as an attribute of the cenozoic stage in evolution of the lithosphere within the Baikal rift system

In accordance with the principle of actualism, theresults of studying the modern geodynamics and tectonophysics of the Baikal rift system (BRS) have beenapplied for development and broadening of ideasabout the Cenozoic Stage of the lithosphere in theregion. Identification of three attractor structures ofriftogenesis (ASR) has allowed the theory of moderngeodynamics and tectonophysics of the BRS to bedeveloped: the regional energy sources for geodynamical and seismic reconstructions have been includedinto the pattern, and their nature and effects have beendescribed in the first approximation. Within theframework of the theory of dissipative dynamical systems of selforganization and nonlinear media, theASRs have been considered as an attribute of Cenozoic geodynamics of the BRS lithosphere: they formednonlinearity and instability of geodynamical and tectonophysical processes in this period of time. Threestages of volcanic activation of the BRS lithosphere areexplained by sequential origination and developmentof three ASRs: firstly, in the South Baikal Depression(Late Cretaceous–Paleocene), then in the KhubsugulDepression (Eocene–Oligocene), and finally in theMuya Depression (postOligocene). The start of theCenozoic riftogenic stage of the Baikal Depressionabout 70 Ma BP is explained by appearance of theASR is the South Baikal Depression. Two stages of rifting, an increase in the rate of rift processes in the BRS,and propagation of riftogenesis southwestwards andnortheastwards from the South Baikal Depression areattributed to the appearance and gradual twolateralevolution of the ASRs in the Khubsugul (southwesternflank) and Muya (northeastern flank) depressions.In the hierarcy of natural systems, intracontinentalrift systems, which consist of deeplaid fault zones andzones functioning in a relatively narrow thermodynamical regime, are classified as mesosystems [1].N.A. Logachev et al. [2, 3] found the main features ofthe Cenozoic riftogenesis dynamics of the intracontinental BRS: evolution of the process southwestwardsand northeastwards from the South Baikal Depression, which is the historical core of the BRS, tookplace during two stages, namely, slow and rapid rifting.Such systems that change in space and time are identified as dynamic [4], and it follows from the theorythat dissipative dynamical systems in geological–geophysical media should have spatiotemporal attractors;in the lithosphere of a rift system, these attractors areto be classified as attractors of riftogenesis (spacedomains and time periods where normal faultrelatedearthquakes, which are typical for this mode, dominate). It has been found that temporal attractors of riftogenesis in the BRS lithosphere correspond to assemblage structure within the framework of the evolutional scenario with bifurcation of triple equilibrium[5]. The detailed study and generalization of theseparameters of seismic sources enabled the identification of three ASRs in the lithosphere of the region:South Baikal, Khubsugul, and Muya depressions [6](Fig. 1). ASRs are the main peculiarity of regionallithosphere since they form nonlinearity and instability of modern geodynamical and tectonophysical processes in the BRS lithosphere.In the present paper, the ASRs are considered as anattribute of Cenozoic nonlinear geodynamics of theBRS lithosphere. Their inclusion in the history ofregional Cenozoic geodynamics allows us to supplement the available description [2, 3, 7] with modernconcepts about the origin and evolution of riftogenesisin the BRS lithosphere within the framework of theselforganization theory for dissipative systems andnonlinear media. We lean upon the actualism principle here; it is not understood as a statement about similarity between present and past geological–geophysical processes, but as a statement that systems preserve

Passivity-based Visual Motion Observer with Panoramic Camera for Pose Control

This paper considers the vision-based estimation and pose control with a panoramic camera via passivity approach. First, a hyperbolic projection of a panoramic camera is presented. Next, using standard body-attached coordinate frames (the world frame, mirror frame, camera frame and object frame), we represent the body velocity of the relative rigid body motion (position and orientation). After that, we propose a visual motion observer to estimate the relative rigid body motion from the measured camera data. We show that the estimation error system with a panoramic camera has the passivity which allows us to prove stability in the sense of Lyapunov. The visual motion error system which consists of the estimation error system and the pose control error system preserves the passivity. After that, stability and L2-gain performance analysis for the closed-loop system are discussed via Lyapunov method and dissipative systems theory, respectively. Finally, simulation and experimental results are shown in order to confirm the proposed method.

Attractor structures of riftogenesis in the lithosphere of Baikal Rift System

The study results of modern geodynamics and tectonophysics of the lithosphere of Baikal Rift System (BRS) are generalized. By the data on radii of dislocations, three areas of maximal strain-strength anisotropy of the medium are distinguished, while analysis of seismic moments of earthquakes has showed that in these parts of the lithosphere mostly dip-slip fault-causing quakes of various energy classes take place; i.e., riftogenesis processes dominate. Within the framework of the theory of nonlinear dissipative dynamical systems, these areas are classified as attractor structures of riftogenesis (ASR). ASRs are located in the central part and in the flanks of the BRS, and they form nonlinearity and instability of modern geodynamical and tectonophysical processes in the lithosphere, which are manifested in seismicity of the Baikal Region and Mongolia.

A dissipativity-based approach to traffic signal control for an over-saturated intersection

In this paper, using the energy concept, the control of traffic flow by traffic lights on an over-saturated single intersection is addressed. By regarding the flow entry of vehicles as the energy stored in the system, we show that the dissipative system theory can be applied. As a result, a state feedback control law is found in terms of the solution of an associated dissipation equation that renders the system non-accumulative, i.e., to get out of the over-saturation situation. The results of the simulations indicate that our control strategy guarantees a high degree of control benefit.