Set Of Admissible States Research Articles

The signaling dimension of two-dimensional and polytopic systems

The signaling dimension of any given physical system represents its classical simulation cost, that is, the minimum dimension of a classical system capable of reproducing all the input/output correlations of the given system. The signaling dimension landscape is vastly unexplored; the only non-trivial systems whose signaling dimension is known -- other than quantum systems -- are the octahedron and the composition of two squares.}{Building on previous results by Matsumoto, Kimura, and Frenkel, our first result consists of deriving bounds on the signaling dimension of any system as a function of its Minkowski measure of asymmetry. We use such bounds to prove that the signaling dimension of any two-dimensional system (i.e. with two-dimensional set of admissible states, such as polygons and the real qubit) is two if and only if such a set is centrally symmetric, and three otherwise, thus conclusively settling the problem of the signaling dimension for such systems.} {Guided by the relevance of symmetries in the two dimensional case, we propose a branch and bound division-free algorithm for the exact computation of the symmetries of any given polytope, in polynomial time in the number of vertices and in factorial time in the dimension of the space. Our second result then consist of providing an algorithm for the exact computation of the signaling dimension of any given system, that outperforms previous proposals by exploiting the aforementioned bounds to improve its pruning techniques and incorporating as a subroutine the aforementioned symmetries-finding algorithm. We apply our algorithm to compute the exact value of the signaling dimension for all rational Platonic, Archimedean, and Catalan solids, and for the class of hyper-octahedral systems up to dimension five.

High-order accurate positivity-preserving and well-balanced discontinuous Galerkin schemes for ten-moment Gaussian closure equations with source terms

This paper proposes novel high-order accurate discontinuous Galerkin (DG) schemes for the one- and two-dimensional ten-moment Gaussian closure equations with source terms defined by a known potential function. Our DG schemes exhibit the desirable capability of being well-balanced (WB) for a known hydrostatic equilibrium state while simultaneously preserving positive density and positive-definite anisotropic pressure tensor. The well-balancedness is built on carefully modifying the solution states in the Harten–Lax–van Leer–contact (HLLC) flux, and appropriate reformulation and discretization of the source terms. Our novel modification technique overcomes the difficulties posed by the anisotropic effects, maintains the high-order accuracy, and ensures that the modified solution state remains within the physically admissible state set. We provide the rigorous positivity-preserving analyses of our WB DG schemes, based on several key properties of the admissible state set, the HLLC flux and the HLLC solver, as well as the geometric quasilinearization (GQL) approach in Wu and Shu (2023) [52], which was originally applied to analyze the admissible state set and the physical-constraints-preserving schemes for the relativistic magnetohydrodynamic equations in Wu and Tang (2017) [54], to address the difficulties arising from the nonlinear constraints on the pressure tensor. Moreover, the proposed WB DG schemes satisfy the weak positivity for the cell averages, implying the use of a simple scaling limiter to enforce the physical admissibility of the DG solution polynomials at certain points of interest. Extensive numerical experiments are conducted to validate the preservation of equilibrium states, accuracy in capturing small perturbations to such states, robustness in solving problems involving low density or low pressure, and high resolution for both smooth and discontinuous solutions.

Reliability model of the security subsystem countering to the impact of typed cyber-physical attacks

The article's main contribution is the description of the process of the security subsystem countering the impact of typed cyber-physical attacks as a model of end states in continuous time. The input parameters of the model are the flow intensities of typed cyber-physical attacks, the flow intensities of possible cyber-immune reactions, and the set of probabilities of neutralization of cyber-physical attacks. The set of admissible states of the info-communication system is described taking into account possible variants of the development of the modeled process. The initial parameters of the model are the probabilities of the studied system in the appropriate states at a particular moment. The dynamics of the info-communication system's life cycle are embodied in the form of a matrix of transient probabilities. The mentioned matrix connects the initial parameters in the form of a system of Chapman's equations. The article presents a computationally efficient concept based on Gershgorin's theorems to solve such a system of equations with given initiating values. Based on the presented scientific results, the article proposes the concept of calculating the time to failure as an indicator of the reliability of the info-communication system operating under the probable impact of typical cyber-physical attacks. The adequacy of the model and concepts presented in the article is proved by comparing a statically representative amount of empirical and simulated data. We emphasize that the main contribution of the research is the description of the process of the security subsystem countering the impact of typed cyber-physical attacks as a model of end states in continuous time. Based on the created model, the concept of computationally efficient solution of Chapman's equation system based on Gershgorin's theorems and calculating time to failure as an indicator of the reliability of the info-communication system operating under the probable impact of typed cyber-physical attacks are formalized. These models and concepts are the highlights of the research.

Sections of Hamiltonian Systems

A section of a Hamiltonian system is a hypersurface in the phase space of the system, usually representing a set of one-sided constraints (e.g. a boundary, an obstacle or a set of admissible states). In this paper we give local classification results for all typical singularities of sections of regular (non-singular) Hamiltonian systems, a problem equivalent to the classification of typical singularities of Hamiltonian systems with one-sided constraints. In particular we give a complete list of exact normal forms with functional invariants, and we show how these are related/obtained by the symplectic classification of mappings with prescribed (Whitney-type) singularities, naturally defined on the reduced phase space of the Hamiltonian system.

Application of the mini-batch adaptive method of random search (MAMRS) in problems of optimal in mean control of the trajectory pencils

The article discusses the application of one of the new methods of constrained optimization to solve the problem of finding optimal control of a pencil of trajectories of nonlinear deterministic systems emanating from a given set of initial states. The structure of a feedback system is proposed, which contains a model of a control object, a measurement system model, a state observer that generates an estimate of the state vector from incoming measurements, and a regulator. The quality of control is assessed by the value of the average value of the functional determined on individual trajectories. Unknown control laws for the object model and the state observer are found in the form of expansions in terms of orthonormal systems of basic functions defined on the set of admissible states of the dynamic system. The problem of controlling a pencil of trajectories is reduced to the problem of parametric optimization, which is solved using a mini-batch adaptive random search method. A step-by-step algorithm for solving the problem is proposed, which is demonstrated by solving the problem of tracking various coordinates of a dynamical system according to the measurement results. The influence of the mini-batch size on the achieved tracking accuracy is investigated.

A Provably Positive Discontinuous Galerkin Method for Multidimensional Ideal Magnetohydrodynamics

The density and pressure are positive physical quantities in magnetohydrodynamics (MHD). Design of provably positivity-preserving (PP) numerical schemes for ideal compressible MHD is highly desirable, but remains a challenge especially in the multidimensional cases. In this paper, we first develop uniformly high-order discontinuous Galerkin (DG) schemes which provably preserve the positivity of density and pressure for multidimensional ideal MHD. The schemes are constructed by using the locally divergence-free DG schemes for the symmetrizable ideal MHD equations as the base schemes, a PP limiter to enforce the positivity of the DG solutions, and the strong stability preserving methods for time discretization. The significant innovation is that we discover and rigorously prove the PP property of the proposed DG schemes by using a novel equivalent form of the admissible state set and very technical estimates. Several two-dimensional numerical examples further confirm the PP property, and demonstrate the accuracy, effectiveness and robustness of the proposed PP methods.

Positivity-Preserving Analysis of Numerical Schemes for Ideal Magnetohydrodynamics

Numerical schemes provably preserving the positivity of density and pressure are highly desirable for MHD, but the rigorous positivity-preserving (PP) analysis remains challenging. The difficulties mainly arise from the intrinsic complexity of the MHD equations as well as the indeterminate relation between the PP property and the divergence-free condition on magnetic field. We present the first rigorous PP analysis of conservative schemes with Lax-Friedrichs (LF) flux for ideal MHD. The significant innovation is the discovery of theoretical connection between PP property and a discrete divergence-free (DDF) condition. This connection is established through the generalized LF splitting properties, which are alternatives of the usually-expected LF splitting property that does not hold for ideal MHD. The generalized LF splitting properties involve a number of admissible states strongly coupled by DDF condition, making their derivation very difficult. We derive these properties via a novel equivalent form of the admissible state set and an important inequality skillfully constructed by technical estimates. Rigorous PP analysis is presented for finite volume and discontinuous Galerkin schemes with LF flux on uniform Cartesian meshes. In 1D case, PP property is proved for the first-order scheme with proper numerical viscosity, and also for arbitrarily high-order schemes under conditions accessible by a PP limiter. In 2D case, we show that the DDF condition is necessary and crucial for achieving PP property. It is observed that even slightly violating the proposed DDF condition may cause failure to preserve the positivity of pressure. We prove that the 2D LF type scheme with proper numerical viscosity preserves the positivity and DDF condition. Sufficient conditions are derived for 2D PP high-order schemes, and extension to 3D is discussed. Numerical examples confirm the theoretical findings.

Admissible states and physical-constraints-preserving schemes for relativistic magnetohydrodynamic equations

This paper first studies the admissible state set [Formula: see text] of relativistic magnetohydrodynamics (RMHD). It paves a way for developing physical-constraints-preserving (PCP) schemes for the RMHD equations with the solutions in [Formula: see text]. To overcome the difficulties arising from the extremely strong nonlinearities and no explicit formulas of the primitive variables and the flux vectors with respect to the conservative vector, two equivalent forms of [Formula: see text] with explicit constraints on the conservative vector are skillfully discovered. The first is derived by analyzing roots of several polynomials and transferring successively them, and further used to prove the convexity of [Formula: see text] with the aid of semi-positive definiteness of the second fundamental form of a hypersurface. While the second is derived based on the convexity, and then used to show the orthogonal invariance of [Formula: see text]. The Lax–Friedrichs (LxF) splitting property does not hold generally for the nonzero magnetic field, but by a constructive inequality and pivotal techniques, we discover the generalized LxF splitting properties, combining the convex combination of some LxF splitting terms with a discrete divergence-free condition of the magnetic field. Based on the above analyses, several 1D and 2D PCP schemes are then studied. In the 1D case, a first-order accurate LxF-type scheme is first proved to be PCP under the Courant–Friedrichs–Lewy (CFL) condition, and then the high-order accurate PCP schemes are proposed via a PCP limiter. In the 2D case, the discrete divergence-free condition and PCP property are analyzed for a first-order accurate LxF-type scheme, and two sufficient conditions are derived for high-order accurate PCP schemes. Our analysis reveals in theory for the first time that the discrete divergence-free condition is closely connected with the PCP property. Several numerical examples demonstrate the theoretical findings and the performance of numerical schemes.

Design of provably physical-constraint-preserving methods for general relativistic hydrodynamics

The paper develops high-order physical-constraint-preserving (PCP) methods for general relativistic hydrodynamic (GRHD) equations, equipped with a general equation of state. Here the physical constraints, describing the admissible states of GRHD, are referred to the subluminal constraint on the fluid velocity and the positivity of the density, pressure and specific internal energy. Preserving these constraints is very important for robust computations, otherwise violating one of them will lead to the ill-posed problem and numerical instability. To overcome the difficulties arising from the inherent strong nonlinearity contained in the constraints, we derive an equivalent definition of the admissible states. Using this definition, we prove the convexity, scaling invariance and Lax-Friedrichs (LxF) splitting property of the admissible state set $\mathcal G$, and discover the dependence of $\mathcal G$ on the spacetime metric. Unfortunately, such dependence yields the non-equivalence of $\mathcal G$ at different points in curved spacetime, and invalidates the convexity of $\mathcal G$ in analyzing PCP schemes. This obstacle is effectively overcame by introducing a new formulation of the GRHD equations. Based on this formulation and the above theories, a first-order LxF scheme is designed on general unstructured mesh and rigorously proved to be PCP under a CFL condition. With two types of PCP limiting procedures, we design high-order, {\em provably} (not probably) PCP methods under discretization on the proposed new formulation. These high-order methods include the PCP finite difference, finite volume and discontinuous Galerkin methods.

An admissibility and asymptotic preserving scheme for systems of conservation laws with source term on 2D unstructured meshes with high-order MOOD reconstruction

The aim of this work is to design an explicit finite volume scheme with high-order MOOD reconstruction for specific systems of conservation laws with stiff source terms which degenerate into diffusion equations. We propose a general framework to design an asymptotic preserving scheme that is stable and consistent under a classical hyperbolic CFL condition in both hyperbolic and diffusive regimes for any 2D unstructured mesh. Moreover, the developed scheme also preserves the set of admissible states, which is mandatory to conserve physical solutions in stiff configurations. This construction is achieved by using a non-linear scheme as a target scheme for the limit diffusion equation, which gives the form of the global scheme for the full system. The high-order polynomial reconstructions allow to improve the accuracy of the scheme without getting a full high-order scheme. Numerical results are provided to validate the scheme in every regime.

PHYSICAL-CONSTRAINT-PRESERVING CENTRAL DISCONTINUOUS GALERKIN METHODS FOR SPECIAL RELATIVISTIC HYDRODYNAMICS WITH A GENERAL EQUATION OF STATE

ABSTRACT The ideal gas equation of state (EOS) with a constant adiabatic index is a poor approximation for most relativistic astrophysical flows, although it is commonly used in relativistic hydrodynamics (RHD). This paper develops high-order accurate, physical-constraints-preserving (PCP), central, discontinuous Galerkin (DG) methods for the one- and two-dimensional special RHD equations with a general EOS. It is built on our theoretical analysis of the admissible states for RHD and the PCP limiting procedure that enforce the admissibility of central DG solutions. The convexity, scaling invariance, orthogonal invariance, and Lax–Friedrichs splitting property of the admissible state set are first proved with the aid of its equivalent form. Then, the high-order central DG methods with the PCP limiting procedure and strong stability-preserving time discretization are proved, to preserve the positivity of the density, pressure, specific internal energy, and the bound of the fluid velocity, maintain high-order accuracy, and be L 1-stable. The accuracy, robustness, and effectiveness of the proposed methods are demonstrated by several 1D and 2D numerical examples involving large Lorentz factor, strong discontinuities, or low density/pressure, etc.

Regional Stability Analysis of Discrete-Time Dynamic Output Feedback Under Aperiodic Sampling and Input Saturation

This technical note addresses the problem of regional stability analysis of a continuous-time plant controlled by a discrete-time dynamic output feedback control law with saturation constraints under aperiodic sampling. To cope with the aperiodic sampling problem, the proposed approach is based on an impulsive system modeling and the use of looped functionals. A generalized sector-based relation is applied to tackle the control saturation effects. From these ingredients, conditions to ensure regional asymptotic stability of the closed-loop system under aperiodic sampling are derived. Based on these conditions, LMI-based optimization problems are proposed to compute estimates of the region of attraction of the closed-loop system or to maximize the bound on the maximal admissible interval between two successive sampling instants, for which the regional stability can be ensured with respect to a given set of admissible states. A numerical example illustrates the application of the results.

An admissibility and asymptotic-preserving scheme for systems of conservation laws with source term on 2D unstructured meshes

The objective of this work is to design explicit finite volumes schemes for specific systems of conservations laws with stiff source terms, which degenerate into diffusion equations. We propose a general framework to design an asymptotic preserving scheme, that is stable and consistent under a classical hyperbolic CFL condition in both hyperbolic and diffusive regime, for any two-dimensional unstructured mesh. Moreover, the scheme developed also preserves the set of admissible states, which is mandatory to keep physical solutions in stiff configurations. This construction is achieved by using a non-linear scheme as a target scheme for the diffusive equation, which gives the form of the global scheme for the complete system of conservation laws. Numerical results are provided to validate the scheme in both regimes.

High-order accurate physical-constraints-preserving finite difference WENO schemes for special relativistic hydrodynamics

The paper develops high-order accurate physical-constraints-preserving finite difference WENO schemes for special relativistic hydrodynamical (RHD) equations, built on the local Lax–Friedrichs splitting, the WENO reconstruction, the physical-constraints-preserving flux limiter, and the high-order strong stability preserving time discretization. They are extensions of the positivity-preserving finite difference WENO schemes for the non-relativistic Euler equations [20]. However, developing physical-constraints-preserving methods for the RHD system becomes much more difficult than the non-relativistic case because of the strongly coupling between the RHD equations, no explicit formulas of the primitive variables and the flux vectors with respect to the conservative vector, and one more physical constraint for the fluid velocity in addition to the positivity of the rest-mass density and the pressure. The key is to prove the convexity and other properties of the admissible state set and discover a concave function with respect to the conservative vector instead of the pressure which is an important ingredient to enforce the positivity-preserving property for the non-relativistic case.Several one- and two-dimensional numerical examples are used to demonstrate accuracy, robustness, and effectiveness of the proposed physical-constraints-preserving schemes in solving RHD problems with large Lorentz factor, or strong discontinuities, or low rest-mass density or pressure etc.

Tracking and rejection of periodic signals for discrete‐time linear systems subject to control saturation

This work addresses the tracking and rejection problem of periodic signals for discrete-time linear systems subject to control saturation. To ensure the periodic tracking/rejection an internal model-based controller is considered. Based on this control structure, conditions allowing the simultaneous synthesis of a stabilising state feedback gain and an anti-windup gain are proposed. Provided that references and disturbances belong to the interior of a certain admissible set, these gains guarantee that the trajectories of the closed-loop system starting in an ellipsoidal set contract to the linearity region of the closed-loop system, where the presence of the internal model ensures the periodic tracking/rejection. To compute the gains, an LMI-based optimisation problem aiming at the maximisation of the set of admissible states and/or the set of admissible references/disturbances is proposed.

Magnetic states at short distances

The magnetic interactions between a fermion and an antifermion of opposite electric or color charges in the 1S0 and 3P0 states are very attractive and singular near the origin and may allow the formation of new bound and resonance states at short distances. In the two body Dirac equations formulated in constraint dynamics, the short-distance attraction for these states for point particles leads to a quasipotential that behaves near the origin as -\alpha^2/r^2. Both 1S0 and 3P0 states admit two types of eigenstates with drastically different behaviors for the radial wave function u=r\psi. One type of states, with u growing as r^{1+\sqrt(1-4*\alpha^2)/2} at small r, will be called usual states. The other type of states with u growing as r^{(1-\sqrt(1-4*\alpha^2))/2} will be called peculiar states. Both of the usual and peculiar eigenstates have admissible behaviors at short distances. The usual bound 1S0 states possess attributes the same as those one usually encounters in QED and QCD. In contrast, the peculiar bound 1S0 states, yet to be observed, have distinctly different bound state properties and scattering phase shifts. For the 3P0 states, the usual solutions lead to the standard bound state energies and no resonance, but resonances have been found for the peculiar states whose energies depend on the description of the internal structure of the charges, the mass of the constituent, and the coupling constant. The existence of both usual and peculiar eigenstates in the same system leads to the non-self-adjoint property of the mass operator and two non-orthogonal complete sets. The mass operator can be made self-adjoint with a single complete set of admissible states by introducing a new peculiarity quantum number and an enlarged Hilbert space.

Analysis of quantum walks with time-varying coin on d-dimensional lattices

In this paper, we present a study of discrete time quantum walks whose underlying graph is a d-dimensional lattice. The dynamical behavior of these systems is of current interest because of their applications in quantum information theory as tools to design quantum algorithms. We assume that, at each step of the walk evolution, the coin transformation is allowed to change so that we can use it as a control variable to drive the evolution in a desired manner. We give an exact description of the possible evolutions and of the set of possible states that can be achieved with such a system. In particular, we show that it is possible to go from a state where there is probability 1 for the walker to be found in a vertex to a state where all the vertices have equal probability. We also prove a number of properties of the set of admissible states in terms of the number of steps needed to obtain them. We provide explicit algorithms for state transfer in low dimensional cases as well as results that allow to reduce algorithms on two-dimensional lattices to algorithms on the one-dimensional lattice, the cycle.

On the Tracking Problem for Linear Systems subject to Control Saturation

This paper addresses the problem of tracking constant references for linear systems subject to control saturation. Considering an unitary output feedback loop, containing an integral action, conditions in LMI form are proposed to compute a state feedback and an integrator anti-windup gain. These conditions ensure that the trajectories of the closed-loop system are bounded in an invariant ellipsoidal set, provided that the initial conditions are taken in this set and the references and the disturbances belong to a certain admissible set. Based on these conditions, optimization problems aiming at the maximization of the invariant set of admissible states and/or the maximization of the set of admissible references/disturbances are proposed.

Constrained model predictive control: Stability and optimality

Model predictive control is a form of control in which the current control action is obtained by solving, at each sampling instant, a finite horizon open-loop optimal control problem, using the current state of the plant as the initial state; the optimization yields an optimal control sequence and the first control in this sequence is applied to the plant. An important advantage of this type of control is its ability to cope with hard constraints on controls and states. It has, therefore, been widely applied in petro-chemical and related industries where satisfaction of constraints is particularly important because efficiency demands operating points on or close to the boundary of the set of admissible states and controls. In this review, we focus on model predictive control of constrained systems, both linear and nonlinear and discuss only briefly model predictive control of unconstrained nonlinear and/or time-varying systems. We concentrate our attention on research dealing with stability and optimality; in these areas the subject has developed, in our opinion, to a stage where it has achieved sufficient maturity to warrant the active interest of researchers in nonlinear control. We distill from an extensive literature essential principles that ensure stability and use these to present a concise characterization of most of the model predictive controllers that have been proposed in the literature. In some cases the finite horizon optimal control problem solved on-line is exactly equivalent to the same problem with an infinite horizon; in other cases it is equivalent to a modified infinite horizon optimal control problem. In both situations, known advantages of infinite horizon optimal control accrue.

A reduced-order framework applied to linear systems with constrained controls

A major issue in the control of dynamical systems is the integration of both technological constraints and some dynamic performance requirements in the design of the control system. The authors show in this work that it is possible to solve a class of constrained control problems of linear systems by using a reduced-order system obtained by the projection of the trajectories of the original system onto a subspace associated with the undesirable open-loop eigenvalues. The class of regulation schemes considered uses full state feedback to guarantee that any trajectory emanating from a given polyhedral set of admissible initial states remains in that set. This set of admissible states is said to be positively invariant with respect to the closed-loop system. The authors also address the important issues of numerical stability and complexity of the computations.