ON THE STABILITY OF DYNAMIC SYSTEMS WITH CERTAIN SWITCHINGS, WHICH CONSISTS OF LINEAR SUBSYSTEMS WITHOUT DELAY

This work is devoted to the further development of the study of the stability of dynamic systems with switchings. There are many different classes of dynamical systems described by switched equations. The authors of the work divide systems with switches into two classes. Namely, on systems with definite and indefinite switchings. In this paper, the system with certain switching, namely a system composed of differential and difference sub-systems with the condition of decreasing Lyapunov function. One of the most versatile methods of studying the stability of the zero equilibrium state is the second Lyapunov method, or the method of Lyapunov functions. When using it, a positive definite function is selected that satisfies certain properties on the solutions of the system. If a system of differential equations is considered, then the condition of non-positiveness (negative definiteness) of the total derivative due to the system is imposed. If a difference system of equations is considered, then the first difference is considered by virtue of the system. For more general dynamical systems (in particular, for systems with switchings), the condition is imposed that the Lyapunov function does not increase (decrease) along the solutions of the system. Since the paper considers a system consisting of differential and difference subsystems, the condition of non-increase (decrease of the Lyapunov function) is used.For a specific type of subsystems (linear), the conditions for not increasing (decreasing) are specified. The basic idea of using the second Lyapunov method for systems of this type is to construct a sequence of Lyapunov functions, in which the level surfaces of the next Lyapunov function at the switching points are either «stitched» or «contain the level surface of the previous function».

Evenly positive definite function of Hilbert space. Self-adjoint operators which are connected by algebraic relationship

A generalization of P. A. Minlos, V. V. Sazonov’s theorem is proved in a case of bounded evenly positive definite function given in Hilbert space. The integral representation is obtained for a family of bounded commutative self-adjoint operators which are connected by algebraic relationship.

Explicit solutions of the kinetic and potential matching conditions of the energy shaping method

<p style='text-indent:20px;'>In the context of underactuated Hamiltonian systems defined by simple Hamiltonian functions, the matching conditions of the energy shaping method split into two decoupled subsets of equations: the <i>kinetic</i> and <i>potential</i> equations. The unknown of the kinetic equation is a metric on the configuration space of the system, while the unknown of the potential equation are the same metric and a positive-definite function around some critical point of the Hamiltonian function. In this paper, assuming that a solution of the kinetic equation is given, we find conditions (in the <inline-formula><tex-math id="M1">\begin{document}$ C^{\infty} $\end{document}</tex-math></inline-formula> category) for the existence of positive-definite solutions of the potential equation and, moreover, we present a procedure to construct, up to quadratures, some of these solutions. In order to illustrate such a procedure, we consider the subclass of systems with one degree of underactuation, where we find in addition a concrete formula for the general solution of the kinetic equation. As a byproduct, new global and local expressions of the matching conditions are presented in the paper.</p>

Positive definite functions and cut-off for discrete groups

We consider the sequence of powers of a positive definite function on a discrete group. Taking inspiration from random walks on compact quantum groups, we give several examples of situations where a cut-off phenomenon occurs for this sequence, including free groups and infinite Coxeter groups. We also give examples of absence of cut-off using free groups again.

Interconnection and Damping Assignment Passivity–Based Control of an Underactuated 2–DOF Gyroscope

In this paper we present interconnection and damping assignment passivity-based control (IDA-PBC) applied to a 2 degrees of freedom (DOFs) underactuated gyroscope. First, the equations of motion of the complete system (3-DOF) are presented in both Lagrangian and Hamiltonian formalisms. Moreover, the conditions to reduce the system from a 3-DOF to a 2- DOF gyroscope, by using Routh’s equations of motion, are shown. Next, the solutions of the partial differential equations involved in getting the proper controller are presented using a reduction method to handle them as ordinary differential equations. Besides, since the gyroscope has no potential energy, it presents the inconvenience that neither the desired potential energy function nor the desired Hamiltonian function has an isolated minimum, both being only positive semidefinite functions; however, by focusing on an open-loop nonholonomic constraint, it is possible to get the Hamiltonian of the closed-loop system as a positive definite function. Then, the Lyapunov direct method is used, in order to assure stability. Finally, by invoking LaSalle’s theorem, we arrive at the asymptotic stability of the desired equilibrium point. Experiments with an underactuated gyroscopic mechanical system show the effectiveness of the proposed scheme.

The Point Value Maximization Problem for Positive Definite Functions Supported in a Given Subset of a Locally Compact Group

The century-old extremal problem, solved by Carathéodory and Fejér, concerns a non-negative trigonometric polynomial $T(t) = a_0 + \sum\nolimits_{k = 1}^n {a_k} \cos (2\pi kt) + b_k\sin (2\pi kt){\ge}0$, normalized by a0=1, where the quantity to be maximized is the coefficient a1 of cos (2π t). Carathéodory and Fejér found that for any given degree n, the maximum is 2 cos(π/n+2). In the complex exponential form, the coefficient sequence (ck) ⊂ ℂ will be supported in [−n, n] and normalized by c0=1. Reformulating, non-negativity of T translates to positive definiteness of the sequence (ck), and the extremal problem becomes a maximization problem for the value at 1 of a normalized positive definite function c: ℤ → ℂ, supported in [−n, n]. Boas and Kac, Arestov, Berdysheva and Berens, Kolountzakis and Révész and, recently, Krenedits and Révész investigated the problem in increasing generality, reaching analogous results for all locally compact abelian groups. We prove an extension to all the known results in not necessarily commutative locally compact groups.

Ideals of the Fourier algebra, supports and harmonic operators

We examine the common null spaces of families of Herz–Schur multipliers and apply our results to study jointly harmonic operators and their relation with jointly harmonic functionals. We show how an annihilation formula obtained in [1] can be used to give a short proof as well as a generalisation of a result of Neufang and Runde concerning harmonic operators with respect to a normalised positive definite function. We compare the two notions of support of an operator that have been studied in the literature and show how one can be expressed in terms of the other.

Semi-isotropic design of a Stewart platform through constrained optimization

The isotropic property at home configuration for a semi-regular Stewart platform manipulator with equal leg lengths has been found to be nonachievable by kinematic design optimization study. In this context, a new approach of design optimization has been formulated here for achieving semi-isotropicity through variable transformation that has rendered the constrained optimization over a finite workspace to infinite workspace in the transformed domain. The proposed minimization methodology of a positive definite function for all the angular and translational motion has exhibited strong convergence to zero for values of the design parameters that can be worked out as a closed-form solution only for the cases of linear translation. Finally, the variations of the condition number over the permissible range of all single-degree-of-freedom motions have been carried out. The absence of any other minima in the entire workspace has clearly established the home position as globally optimized.

Generalized Gaussian processes and relations with random matrices and positive definite functions on permutation groups

The main purpose of this paper is an explicit construction of generalized Gaussian process with function tb(V) = bH(V), where H(V) = n - h(V), h(V) is the number of singletons in a pair-partition V ∈ 𝒫2(2n). This gives another proof of Theorem of A. Buchholtz15 that tb is positive definite function on the set of all pair-partitions. Here there are some new combinatorial formulas presented. Connections with free additive convolutions probability measure on ℝ are also done. There are new positive definite functions on permutations presented. What is more, it is proven that the function H is norm (on the group S(∞) = ⋃S(n)).