62 research outputs found
Mean Square Summability of Solution of Stochastic Difference Second-Kind Volterra Equation with Small Nonlinearity
Stochastic difference second-kind Volterra equation with continuous time and small nonlinearity is considered. Via the general method of Lyapunov functionals construction, sufficient conditions for uniform mean square summability of solution of the considered equation are obtained
Constructive Methods of Invariant Manifolds for Kinetic Problems
We present the Constructive Methods of Invariant Manifolds for model reduction in physical and chemical kinetics, developed during last two decades. The physical problem of reduced description is studied in a most general form as a problem of constructing the slow invariant manifold. The invariance conditions are formulated as the differential equation for a manifold immersed in the phase space (the invariance equation). The equation of motion for immersed manifolds is obtained (the film extension of the dynamics). Invariant manifolds are fixed points for this equation, and slow invariant manifolds are Lyapunov stable fixed points, thus slowness is presented as stability. A collection of methods for construction of slow invariant manifolds is presented, in particular, the Newton method subject to incomplete linearization is the analogue of KAM methods for dissipative systems.
The systematic use of thermodynamics structures and of the quasi--chemical representation allow to construct approximations which are in concordance with physical restrictions.
We systematically consider a discrete analogue of the slow (stable) positively invariant manifolds for dissipative systems, invariant grids. Dynamic and static postprocessing procedures give us the opportunity to estimate the accuracy of obtained approximations, and to improve this accuracy significantly.
The following examples of applications are presented: Nonperturbative deviation of physically consistent hydrodynamics from the Boltzmann equation and from the reversible dynamics, for Knudsen numbers Kn~1; construction of the moment equations for nonequilibrium media and their dynamical correction (instead of extension of list of variables) to gain more accuracy in description of highly nonequilibrium flows; determination of molecules dimension (as diameters of equivalent hard spheres) from experimental viscosity data; invariant grids for a two-dimensional catalytic reaction and a four-dimensional oxidation reaction (six species, two balances); universal continuous media description of dilute polymeric solution; the limits of macroscopic description for polymer molecules, etc
Sum-of-Squares approach to feedback control of laminar wake flows
A novel nonlinear feedback control design methodology for incompressible
fluid flows aiming at the optimisation of long-time averages of flow quantities
is presented. It applies to reduced-order finite-dimensional models of fluid
flows, expressed as a set of first-order nonlinear ordinary differential
equations with the right-hand side being a polynomial function in the state
variables and in the controls. The key idea, first discussed in Chernyshenko et
al. 2014, Philos. T. Roy. Soc. 372(2020), is that the difficulties of treating
and optimising long-time averages of a cost are relaxed by using the
upper/lower bounds of such averages as the objective function. In this setting,
control design reduces to finding a feedback controller that optimises the
bound, subject to a polynomial inequality constraint involving the cost
function, the nonlinear system, the controller itself and a tunable polynomial
function. A numerically tractable approach to the solution of such optimisation
problems, based on Sum-of-Squares techniques and semidefinite programming, is
proposed.
To showcase the methodology, the mitigation of the fluctuation kinetic energy
in the unsteady wake behind a circular cylinder in the laminar regime at
Re=100, via controlled angular motions of the surface, is numerically
investigated. A compact reduced-order model that resolves the long-term
behaviour of the fluid flow and the effects of actuation, is derived using
Proper Orthogonal Decomposition and Galerkin projection. In a full-information
setting, feedback controllers are then designed to reduce the long-time average
of the kinetic energy associated with the limit cycle. These controllers are
then implemented in direct numerical simulations of the actuated flow. Control
performance, energy efficiency, and physical control mechanisms identified are
analysed. Key elements, implications and future work are discussed
Normal Form of Equivariant Maps and Singular Symplectic Reduction in Infinite Dimensions with Applications to Gauge Field Theory
Inspired by problems in gauge field theory, this thesis is concerned with various
aspects of infinite-dimensional differential geometry.
In the first part, a local normal form theorem for smooth equivariant maps
between tame Fréchet manifolds is established. Moreover, an elliptic version of
this theorem is obtained. The proof these normal form results is inspired by
the Lyapunov–Schmidt reduction for dynamical systems and by the Kuranishi
method for moduli spaces, and uses a slice theorem for Fréchet manifolds as
the main technical tool. As a consequence of this equivariant normal form
theorem, the abstract moduli space obtained by factorizing a level set of the
equivariant map with respect to the group action carries the structure of a
Kuranishi space, i.e., such moduli spaces are locally modeled on the quotient
by a compact group of the zero set of a smooth map.
In the second part of the thesis, the theory of singular symplectic reduction
is developed in the infinite-dimensional Fréchet setting. By refining the above
construction, a normal form for momentum maps similar to the classical
Marle–Guillemin–Sternberg normal form is established. Analogous to the
reasoning in finite dimensions, this normal form result is then used to show
that the reduced phase space decomposes into smooth manifolds each carrying
a natural symplectic structure.
Finally,the singular symplectic reduction scheme is further investigated in the
situation where the original phase space is an infinite-dimensional cotangent
bundle. The fibered structure of the cotangent bundle yields a refinement of
the usual orbit-momentum type strata into so-called seams. Using a suitable
normal form theorem, it is shown that these seams are manifolds. Taking
the harmonic oscillator as an example, the influence of the singular seams on
dynamics is illustrated.
The general results stated above are applied to various gauge theory models.
The moduli spaces of anti-self-dual connections in four dimensions and of
Yang–Mills connections in two dimensions is studied. Moreover, the stratified
structure of the reduced phase space of the Yang–Mills–Higgs theory is
investigated in a Hamiltonian formulation after a (3 + 1)-splitting
Active Brownian Heat Engines
When do non-equilibrium forms of disordered energy qualify as heat?
\textcolor{blue}{We address this question in the context of cyclically
operating heat engines in contact with a non-equilibrium energy reservoir that
defies the zeroth law of thermodynamics. To consistently address the latter as
a heat bath requires the existence of a precise mapping to an equivalent cycle
with an equilibrium bath at a time-dependent effective temperature. We identify
the most general setup for which this can generically be ascertained and
thoroughly discuss an analytically tractable, experimentally relevant
scenario}: a Brownian particle confined in a \textcolor{blue}{periodically}
modulated harmonic potential and coupled to some non-equilibrium bath of
variable activity. We deduce formal limitations for its thermodynamic
performance, including maximum efficiency, efficiency at maximum power, and
maximum efficiency at fixed power. They can guide the design of new
micro-machines and clarify how much these can outperform passive-bath designs,
which has been a debated issue for recent experimental realizations. To
illustrate the general principles for practical quasi-static and finite-rate
protocols, we further analyze a specific realization of such an active heat
engine based on the paradigmatic Active Brownian Particle (ABP) model. This
reveals some non-intuitive features of the explicitly computed dynamical
effective temperature, illustrates various conceptual and practical limitations
of the effective-equilibrium mapping, and clarifies the operational relevance
of various coarse-grained measures of dissipation.Comment: 26 pages, 12 figure
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