Negative Definite Function Research Articles

On eigenvalues of certain special matrices

Let g : R → [ 0 , ∞ ) be a conditionally negative definite function and f : [ 0 , ∞ ) → [ 0 , ∞ ) be a Bernstein function. We prove that the function h = f ∘ g is conditionally negative definite and that for distinct real numbers p 1 , p 2 , … , p n the inertia of the matrix [ h ( p i − p j ) ] is ( 1 , 0 , n − 1 ) if f is non-linear and g ( x ) = 0 only for x = 0. We also present a new and simple proof to show that the matrix [ log ⁡ ( 1 − p i p j ) ] is negative definite for n distinct positive real numbers p i < 1 , ∀ i . These results supplement and unify earlier results proved by a number of authors including Dyn, Goodman and Michelli, Bhatia and Jain, Garg and Aujla, Garg and Agarwal for operator monotone functions.

Simulating space-time random fields with nonseparable Gneiting-type covariance functions

Two algorithms are proposed to simulate space-time Gaussian random fields with a covariance function belonging to an extended Gneiting class, the definition of which depends on a completely monotone function associated with the spatial structure and a conditionally negative definite function associated with the temporal structure. In both cases, the simulated random field is constructed as a weighted sum of cosine waves, with a Gaussian spatial frequency vector and a uniform phase. The difference lies in the way to handle the temporal component. The first algorithm relies on a spectral decomposition in order to simulate a temporal frequency conditional upon the spatial one, while in the second algorithm the temporal frequency is replaced by an intrinsic random field whose variogram is proportional to the conditionally negative definite function associated with the temporal structure. Both algorithms are scalable as their computational cost is proportional to the number of space-time locations that may be irregular in space and time. They are illustrated and validated through synthetic examples.

Counterexample to a Lyapunov Condition for Uniform Asymptotic Partial Stability

Partial stability characterizes dynamical systems for which only a part of the state variables exhibits a stable behavior. In his book on partial stability, Vorotnikov proposed a sufficient condition to establish this property through a Lyapunov-like function whose total derivative is upper-bounded by a negative definite function involving only the sub-state of interest. In this note, we show with a simple two-dimensional system that this statement is wrong in general. More precisely, we show that the convergence rate of the relevant state variables may not be uniform in the initial state. We also discuss the impact of this lack of uniformity on the connected issue of robustness with respect to exogenous disturbances.

Distance multivariance: New dependence measures for random vectors

We introduce two new measures for the dependence of $n\ge2$ random variables: distance multivariance and total distance multivariance. Both measures are based on the weighted $L^{2}$-distance of quantities related to the characteristic functions of the underlying random variables. These extend distance covariance (introduced by Székely, Rizzo and Bakirov) from pairs of random variables to $n$-tuplets of random variables. We show that total distance multivariance can be used to detect the independence of $n$ random variables and has a simple finite-sample representation in terms of distance matrices of the sample points, where distance is measured by a continuous negative definite function. Under some mild moment conditions, this leads to a test for independence of multiple random vectors which is consistent against all alternatives.

Application of the Lyapunov Algorithm to Optimize the Control Strategy of Low-Voltage and High-Current Synchronous DC Generator Systems

In the present study, a novel multiple three-phase low-voltage and high-current permanent magnet synchronous generation system is proposed, which has only half-turn coils per phase. The proposed system is composed of a generator and two confluence plates with 108 rectifier modules. The output can reach up to 10,000 A continuous DC power supply, which is suitable for the outdoors and non-commercial power supply. The application of the Lyapunov algorithm in the synchronous rectification control was optimized. A current sharing loop control was added to the closed-loop control to ensure a stable output voltage and the output current sharing of each rectifier module. Since the two control variables solved by the Lyapunov algorithm were coupled and the negative definite function of the Lyapunov algorithm could not be guaranteed in this system, a simple decoupling method was used to decouple the control variables. Compared to the conventional control, the proposed strategy highly improved the dynamic performance of the system. The effectiveness of the proposed strategy was verified by the simulation. The 5 V/10,000 A hardware experiment platform was built, which proved the feasibility and validity of the proposed strategy for a high-power generation system.

Strong Markov processes and negative definite functions associated with non-Archimedean elliptic pseudo-differential operators

In this article we prove that the heat kernel attached to the non-Archimedean elliptic pseudodifferential operators determine a Feller semigroup and a uniformly stochastically continuous $$C_{0}$$-transition function of some strong Markov processes $${\mathfrak {X}}$$ with state space $${\mathbb {Q}}_{p}^{n}.$$ We explicitly write the Feller semigroup and the Markov transition function associated with the heat kernel. Also, we show that the symbols of these pseudo-differential operators are a negative definite function and moreover, that this symbols can be represented as a combination of a constant $$c\ge 0,$$ a continuous homomorphism $$l: {\mathbb {Q}}_{p}^{n}\rightarrow {\mathbb {R}}$$ and a non-negative, continuous quadratic form $$q: {\mathbb {Q}}_{p}^{n}\rightarrow {\mathbb {R}}.$$

Robust synchronization of chaotic systems with novel fuzzy rule-based controllers

This paper proposes a set of novel fuzzy controllers with adjustable control forces for the robust synchronization of different chaotic systems that include uncertain terms or unexpected disturbances. The proposed fuzzy rule-based control system is developed through Lyapunov stability theory and the fuzzy logic concept to form a multi-input multi-output (MIMO) system. This system provides flexible control inputs based on the error dynamics of the system, and the inputs satisfy the synchronization condition of being asymptotically stable. In this case, the error derivatives always have signs opposite those of the error states, which yields negative definite first-order Lyapunov function derivatives. Furthermore, the proposed system appropriately adjusts the control inputs through the constructed fuzzy if-then rules. This method is robust to both unmodeled terms and unexpected interference signals during the synchronization process. The simulation results demonstrate the feasibility and effectiveness of the proposed fuzzy rule-based control system.

Conditionally Negative Definite Functions

Let $$f:[0,\infty )\rightarrow [0,\infty )$$ be an operator monotone function and $$g: \mathbb {R}\rightarrow [0,\infty )$$ be a conditionally negative definite(in short cnd) function. We obtain that $$f\circ g:\mathbb {R}\rightarrow [0,\infty )$$ is also conditionally negative definite. This generalizes and subsumes several existing results. A versatile direct connection between cnd functions and functions having Weierstrass factorization is established and consequently a reasonable account for cnd functions is presented.

$\alpha $-positive/$\alpha $-negative definite functions on groups

In this paper, we introduce the notions of an $\alpha $-positive/$\alpha $-negative definite function on a (discrete) group. We first construct the Naimark-GNS type representation associated to an $\alpha $-positive definite function and prove the Schoenberg type theorem for a matricially bounded $\alpha $-negative definite function. Using a $J$-representation on a Krein space $(\mathcal{K} ,J)$ associated to a nonnegative normalized $\alpha $-negative definite function, we also construct a $J$-cocycle associated to a $J$-representation. Using a $J$-cocycle, we show that there exist two sequences of $\alpha $-positive definite functions and proper $(\alpha ,J)$-actions on a Krein space $(\mathcal{K} ,J)$ corresponding to a proper matricially bounded $\alpha $-negative definite function.

Bernstein-gamma functions and exponential functionals of Lévy processes

In this work we analyse the solution to the recurrence equation \[ M_\Psi (z+1)=\frac{-z} {\Psi (-z)}M_\Psi (z), \quad M_\Psi (1)=1, \] defined on a subset of the imaginary line and where $-\Psi $ is any continuous negative definite function. Using the analytic Wiener-Hopf method we solve this equation as a product of functions that extend the gamma function and are in bijection with the Bernstein functions. We call these functions Bernstein-gamma functions. We establish universal Stirling type asymptotic in terms of the constituting Bernstein function. This allows the full understanding of the decay of $\vert M_\Psi (z)\vert $ along imaginary lines and an access to quantities important for many studies in probability and analysis. This functional equation is a central object in several recent studies ranging from analysis and spectral theory to probability theory. As an application of the results above, we study from a global perspective the exponential functionals of Levy processes whose Mellin transform satisfies the equation above. Although these variables have been intensively studied, our new approach based on a combination of probabilistic and analytical techniques enables us to derive comprehensive properties and strengthen several results on the law of these random variables for some classes of Levy processes that could be found in the literature. These encompass smoothness for its density, regularity and analytical properties, large and small asymptotic behaviour, including asymptotic expansions and bounds. We furnish a thorough study of the weak convergence of exponential functionals on a finite time horizon when the latter expands to infinity. We deliver intertwining relation between members of the class of positive self-similar semigroups.

Making conditionally negative definite radial basis function interpolation well-conditioned by adding cardinal basis functions

Abstract A class of basis functions so called well-conditioned RBF (WRBFs) has been introduced. This basis has been manipulated by adding cardinal functions to the conditionally negative definite RBFs of order 1, such as Multiquadric functions 1 + ( ∊ r ) 2 (MQ) and log ( 1 + ( ∊ r ) 2 ) (LOG). The condition number of the interpolation matrix arising from this basis is of O ( N ) , where N is the number of center nodes. This order is independent of shape parameter and therefore applying this basis functions would recover the ill–posed linear system associated with the order 1 conditionally negative definite RBFs interpolation.

Global Stability Results for Switched Systems Based on Weak Lyapunov Functions

In this paper we study the stability of nonlinear and time-varying switched systems under restricted switching. We approach the problem by decomposing the system dynamics into a nominal-like part and a perturbation-like one. Most stability results for perturbed systems are based on the use of strong Lyapunov functions, i.e. functions of time and state whose total time derivative along the nominal system trajectories is bounded by a negative definite function of the state. However, switched systems under restricted switching may not admit strong Lyapunov functions, even when asymptotic stability is uniform over the set of switching signals considered. The main contribution of the current paper consists in providing stability results that are based on the stability of the nominal-like part of the system and require only a weak Lyapunov function. These results may have wider applicability than results based on strong Lyapunov functions. The results provided follow two lines. First, we give very general global uniform asymptotic stability results under reasonable boundedness conditions on the functions that define the dynamics of the nominal-like and the perturbation-like parts of the system. Second, we provide input-to-state stability (ISS) results for the case when the nominal-like part is switched linear-time-varying. We provide two types of ISS results: standard ISS that involves the essential supremum norm of the input and a modified ISS that involves a power-type norm.

Global finite-time stabilization for a class of high-order nonlinear systems with multiple unknown control directions

This paper considers the problem of the global finite-time stabilization for the high-order nonlinear systems with unknown control directions. Due to the uncertainty of control directions, the paper analyzes all possible conditions of the directions. The Lyapunov-based logic switching rule ensures that we can find the correct control directions. The adaptive switching controller with a switching parameter which is to be tuned online guarantees that the derivative of Lyapunov function is less than a negative definite function and the closed-loop system is globally finite-time stable. The effectiveness of the proposed method is illustrated by an example.

Bank-to-Turn Guidance Law with Terminal Impact-Angle Constraint

AbstractTo avoid the mathematical singularity of polar converting in bank-to-turn (BTT) guidance and to provide the specified terminal impact angle, a BTT guidance law with impact-angle constraint (BTT-GL/IA) is presented. Considering the influence of the impact-angle control and target maneuver to the miss-distance, a new form zero-effort miss-distance with impact angle (ZEM/IA) is introduced. Referring to the Lyapunov-like approach used to drive the guidance law in the prior literature, a Lyapunov function is constructed in terms of ZEM/IA. Based on the Lyapunov stability theorem, a negative definite function is designed, and the BTT-GL/IA is derived. Next, BTT-GL/IA is implemented in another form with the variables as the line-of-sight angle and its rate, which are easily measured by the seeker. When using BTT-GL/IA, the guidance law tends to maintain smooth and continuously-varied body acceleration and roll-angle commands while satisfying the impact position and angle constraints. The problem of a mat...

A theory of subtree matching and tree kernels based on the edit distance concept

Edit distances provide us with an established method to capture structural features of data, and a distance between data objects represents their dissimilarity. In contrast, kernels form a category of similarity functions, and a positive definite kernel enables us to leverage abundant techniques of multivariate analysis. This paper aims to fill the gap between distances and kernels. In the literature, we have several formulas that convert a negative definite distance function into a positive definite kernel. Edit distance functions, however, are not necessarily negative definite, and our first contribution is to introduce an alternative method to derive positive definite kernels from edit distance functions that are not necessarily negative definite. The method is equipped with an easy-to-check and strong sufficient condition for positive definiteness, and the condition turns out to be tightly related with the triangle inequality. In fact, to our knowledge, all of the edit distance functions in the literature that support the triangle inequality meet the condition for positive definiteness. Secondly, we apply this method to four well-known edit distance functions for trees to introduce four novel kernels and show that three of them are positive definite. Thirdly, we develop a theory of subtree matching to study these kernels. Our kernels count matchings between subtrees of the input trees with weights determined according to individual matchings. Although the number of such matchings is an exponential function of the size of the input trees (the number of vertices), our theory enables us to develop dynamic-programming-based algorithms, whose asymptotic computational complexities fall between a quadratic function and a cubic function of the size.

A New Approach to Stable Optimal Control of Complex Nonlinear Dynamical Systems

This paper gives a simple approach to designing a controller that minimizes a user-specified control cost for a mechanical system while ensuring that the control is stable. For a user-given Lyapunov function, the method ensures that its time rate of change is negative and equals a user specified negative definite function. Thus a closed-form, optimal, nonlinear controller is obtained that minimizes a desired control cost at each instant of time and is guaranteed to be Lyapunov stable. The complete nonlinear dynamical system is handled with no approximations/linearizations, and no a priori structure is imposed on the nature of the controller. The methodology is developed here for systems modeled by second-order, nonautonomous, nonlinear, differential equations. The approach relies on some recent fundamental results in analytical dynamics and uses ideas from the theory of constrained motion.

Robust adaptive backstepping control of uncertain Lorenz system

In this paper, a novel robust adaptive control method is proposed for controlling the Lorenz chaotic attractor. A new backstepping controller for the Lorenz system based on the Lyapunov stability theorem is proposed to overcome the singularity problem that appeared in using the typical backstepping control method. By exploiting the property of the system, the resulting controller is shown to be singularity free and the closed loop system is globally stable. Due to unavailability of system states measurement in practice, the controller is selected such that only one system state is needed. To overcome the problem of parameter uncertainty, an additional term to Lyapunov function is added and an identification scheme is adopted to have a negative definite Lyapunov function derivative. The simulation results demonstrate the effectiveness of the proposed controllers and approaches.

Construction of non-Gaussian random fields with any given correlation structure

A positive definite function can be thought of as the covariance function of a Gaussian random field, according to the celebrated Kolmogorov existence theorem. A question of great theoretical and practical interest is: how could one construct a non-Gaussian random field with the given positive definite function as its covariance function? In this paper we demonstrate a novel and simple method for constructing many such non-Gaussian random fields, with the corresponding finite-dimensional distributions identified. Also, we show how to construct a non-Gaussian random field with a given negative definite function as its variogram.

Non‐local Dirichlet forms generated by pseudodifferential operators on compact abelian groups

AbstractLet Gi, 1 ≤ i ≤ n, be compact abelian groups and let Γi , 1 ≤ i ≤ n, be countable dual groups. We consider G = G1 ⊕ G2 ⊕ … ⊕ Gn and Γ = Γ1 ⊕ Γ2 ⊕ … ⊕ Γn . For 1 ≤ j ≤ n, let aj be a negative definite function on Γj and a (γ) = . For φ ∈ S (G), the set of all generalized trigonometrical polynomials on G, we define , where (γ) = aj (γj) (γ), 1 ≤ j ≤ n. Then is a Dirichlet form with the domain on L2 (G). (© 2008 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Feller Processes Generated by Pseudo-Differential Operators: On the Hausdorff Dimension of Their Sample Paths1

Let {Xt}t≥0 be a Feller process generated by a pseudo-differential operator whose symbol satisfiesC∈∝n|q(C,ξ)|≤c(1=ψ)(ξ)) for some fixed continuous negative definite function ψ(ξ). The Hausdorff dimension of the set {Xt:t∈E}, E ⊂ [0, 1] is any analytic set, is a.s. bounded above by βψ dim E. βψ is the Blumenthal−Getoor upper index of the Levy Process associated with ψ(ξ).