A theory of hegemon-provoked instability, with an application to NATO and the Ukraine-Russia war

This paper investigates the robust stability and stabilization analysis of interval fractional-order systems with time-varying delay. The stability problem of such systems is solved first, and then using the proposed results, a stabilization theorem is also included, where sufficient conditions are obtained for designing a stabilizing controller with a predetermined order, which can be chosen to be as low as possible. Utilizing efficient lemmas, the stability and stabilization theorems are proposed in LMI form, which is more suitable to check due to various existing efficient convex optimization parsers and solvers. Finally, some numerical examples have shown the effectiveness of our results.

This book on stability theory and robustness will interest researchers and advanced graduate students in the area of feedback control engineering, circuits, and systems. It will also appeal to mathematicians who are involved in applications of functional analysis to engineering problems. The book provides a methodology for the rigorous treatment of such inherently feedback aspects of dynamical system design as robustness and sensitivity, just as many researchers are beginning to realize that this type of methodology is mandatory if modern systems theory is to be used to design complicated multivariable and large-scale systems. The main objective of the book is to provide a clear mathematical formulation of the issues that arise in designing feedback systems that are robust against the destabilizing effects of unknown-but-bounded uncertainty in component dynamics. It is the first study to identify formal methods for the quantitative analysis of multiloop feedback system robustness. The view that is presents of nonlinear, multiloop feedback system stability theory is unique, lucid, and conceptually appealing. Lyapunov and input-output stability theories are unified in a new and simple geometrical perspective based on the topological separation of spaces. This perspective greatly facilitates visualization of the underlying conceptual issues in stability and robustness theory and serves to motivate specific results concerning the robustness of feedback systems. Potentially, this methodology may be applied to nonlinear feedback design, validation of modeling approximations, hierarchical control system design, and stability margin analysis for multiloop feedback systems. This book is the third publication in The MIT Press Series in Signal Processing, Optimization, and Control, edited by Alan S. Willsky.

Introduction. 1.1 Linear Time-Invariant Systems. 1.2 Nonlinear Systems. 1.3 Equilibrium Points. 1.4 First-Order Autonomous Nonlinear Systems. 1.5 Second-Order Systems: Phase-Plane Analysis. 1.6 Phase-Plane Analysis of Linear Time-Invariant Systems. 1.7 Phase-Plane Analysis of Nonlinear Systems. 1.8 Higher-Order Systems. 1.9 Examples of Nonlinear Systems. 1.10 Exercises. Mathematical Preliminaries. 2.1 Sets. 2.2 Metric Spaces. 2.3 Vector Spaces. 2.4 Matrices. 2.5 Basic Topology. 2.6 Sequences. 2.7 Functions. 2.8 Differentiability. 2.9 Lipschitz Continuity. 2.10 Contraction Mapping. 2.11 Solution of Differential Equations. 2.12 Exercises. Lyapunov Stability I: Autonomous Systems. 3.1 Definitions. 3.2 Positive Definite Functions. 3.3 Stability Theorems. 3.4 Examples. 3.5 Asymptotic Stability in the Large. 3.6 Positive Definite Functions Revisited. 3.7 Construction of Lyapunov Functions. 3.8 The Invariance Principle. 3.9 Region of Attraction. 3.10 Analysis of Linear Time-Invariant Systems. 3.11 Instability. 3.12 Exercises. Lyapunov Stability II: Nonautonomous Systems. 4.1 Definitions. 4.2 Positive Definite Functions. 4.3 Stability Theorems. 4.4 Proof of the Stability Theorems. 4.5 Analysis of Linear Time-Varying Systems. 4.6 Perturbation Analysis. 4.7 Converse Theorems. 4.8 Discrete-Time Systems. 4.9 Discretization. 4.10 Stability of Discrete-Time Systems. 4.11 Exercises. Feedback Systems. 5.1 Basic Feedback Stabilization. 5.2 Integrator Backstepping. 5.3 Backstepping: More General Cases. 5.4 Examples. 5.5 Exercises. Input-Output Stability. 6.1 Function Spaces. 6.2 Input-Output Stability. 6.3 Linear Time-Invariant Systems. 6.4 Lp Gains for LTI Systems. 6.5 Closed Loop Input-Output Stability. 6.6 The Small Gain Theorem. 6.7 Loop Transformations. 6.8 The Circle Criterion. 6.9 Exercises. Input-to-State Stability. 7.1 Motivation. 7.2 Definitions. 7.3 Input-to-State Stability (ISS) Theorems. 7.4 Input-to-State Stability Revisited. 7.5 Cascade Connected Systems. 7.6 Exercises. Passivity. 8.1 Power and Energy: Passive Systems. 8.2 Definitions. 8.3 Interconnections of Passivity Systems. 8.4 Stability of Feedback Interconnections. 8.5 Passivity of Linear Time-Invariant Systems. 8.6 Strictly Positive Real Rational Functions. Exercises. Dissipativity. 9.1 Dissipative Systems. 9.2 Differentiable Storage Functions. 9.3 QSR Dissipativity. 9.4 Examples. 9.5 Available Storage. 9.6 Algebraic Condition for Dissipativity. 9.7 Stability of Dissipative Systems. 9.8 Feedback Interconnections. 9.9 Nonlinear L2 Gain. 9.10 Some Remarks about Control Design. 9.11 Nonlinear L2-Gain Control. 9.12 Exercises. Feedback Linearization. 10.1 Mathematical Tools. 10.2 Input-State Linearization. 10.3 Examples. 10.4 Conditions for Input-State Linearization. 10.5 Input-Output Linearization. 10.6 The Zero Dynamics. 10.7 Conditions for Input-Output Linearization. 10.8 Exercises. Nonlinear Observers. 11.1 Observers for Linear Time-Invariant Systems. 11.2 Nonlinear Observability. 11.3 Observers with Linear Error Dynamics. 11.4 Lipschitz Systems. 11.5 Nonlinear Separation Principle. Proofs. Bibliography. List of Figures. Index.

For a number of applications it is important to know the location of the boundary-layer transition from laminar to turbulent. At present it is generally recognized that the onset of turbulence is directly connected with the loss of stability of the initial laminar flow. In the overwhelming majority of cases experimental data on the influence of various factors upon the transition location agree well with the calculated data concerning the influence of these factors on the boundary-layer stability, i.e. the theory of stability may be used successfully to predict various experimental dependencies. The boundary-layer stability and the transition are considerably affected by heat transfer from the surface of the streamlined body. But, in this case, experimental data on the transition do not always correspond to the results of the stability theory. In particular, experimental works concerning the effect of cooling of the model surface on the supersonic boundary layer transition yield contradictory results (see e.g. Gaponov & Maslov 1980; Morkovin 1969). Some of the contradictions were removed by Demetriades (1978) and Lysenko & Maslov (1981), but on the whole the problem cannot be considered solved, primarily owing to the fact that many theoretical results have not yet been experimentally confirmed. In the present paper the experimental study of development of small natural disturbances in the boundary layer of a cooled flat plate for Mach numbers M = 2, 3 and 4 is described. It confirms the main conclusions of the linear theory of hydrodynamic stability concerning the fact that surface cooling: (i) stabilizes the first-mode disturbances; (ii) destabilizes the second-mode disturbances; (iii) may lead to the region of unstable frequencies of the first mode being divided into two; (iv) does not affect the interaction of acoustic waves and the supersonic boundary layer.

Summer school on the theory of stability and stabilization of motion

The main results on the three-dimensional theory of stability of compressible and incompressible hyperelastic simply connected bodies under uniform compression are analyzed. The problems are classified according to the type of loading (dead or follower loads, acting on the whole or a part of the surface) and the type of boundary conditions (the same conditions on the whole surface or different conditions on different parts of the surface). Approaches based on the three-dimensional linearized theory of stability (theory of finite subcritical deformations, first and second theories of small subcritical deformations, incremental-deformation theory, theory of small average rotations) and an approximate approach (linear equations; the loading parameter is approximately included in the boundary conditions) to solving these problems are discussed. Related problems (rock pressure manifestations, folding in the Earth’s crust, wave-like formations on the surface of structural members, stability of laminated composite materials) are briefly commented. Regarding isotropic compressible and incompressible hyperelastic materials, the three-dimensional theory of elastic stability, theory of finite subcritical deformations, and first and second theories of small subcritical deformations are considered. The sufficient conditions for the applicability of the static (Euler’s) method, the sufficient conditions for the stability of equilibrium state, and the general solutions to anti-plane, plane, and spatial problems for homogeneous subcritical states are formulated. The exact solutions, obtained with the above-mentioned general approaches, for compressible and incompressible isotropic bodies (strip, rectangular and circular plates, circular cylinder, sphere, and body of arbitrary geometry) under dead or follower loading are presented. These solutions were obtained for isotropic hyperelastic materials with an arbitrary elastic potential under uniform (hydrostatic, biaxial, or triaxial) pressure. The reviewed results were originally reported in the author’s monograph (A. N. Guz, Stability of Elastic Bodies under Uniform Compression [in Russian], Naukova Dumka, Kyiv (1979)) and articles listed in the References

Special Section Authors

Structure functions are apparently unfamiliar to most engineers and the unifying role they play in oscillator instability theory has largely gone unrecognized. This paper introduces and places into perspective the role which Kolmogorov structure functions have in theory. It is demonstrated that the rms fractional frequency deviation (phase accumulation) introduced by Cutler and Searle is related to the first phase structure function; the two-sample Allan variance is related to the second phase structure function. In addition, it is shown how the two-sample Allan variance is related to the rms fractional frequency deviation under suitable conditions. The L-sample Allan variance is also identified in terms of the first phase structure function; it is shown to be an asymptotically unbiased estimator of the rms fractional frequency deviation squared if the latter existL The utility of higher order structure functions of frequency and phase in the theory of instability is also demonstrated; in particular, how the frequency drift and "flicker"-type noise convergence problems can be overcome.

On some topological invariants of algebraic functions associated to the Young stratification of polynomials

Under Gromov–Hausdorff convergence, and equivariant Gromov–Hausdorff convergence, we prove stability results of Wasserstein spaces over certain classes of singular and non-singular spaces. For example, we obtain an analogue of Perelman’s stability theorem on Wasserstein spaces.

Improved stability theorems of random nonlinear time delay systems and their application

Robust control of scissor-like elements based systems

Stability and stabilization analysis of fractional‐order linear time‐invariant (FO‐LTI) systems with different derivative orders is studied in this paper. First, by using an appropriate linear matrix function, a single‐order equivalent system for the given different‐order system is introduced by which a new stability condition is obtained that is easier to check in practice than the conditions known up to now. Then the stabilization problem of fractional‐order linear systems with different fractional orders via a dynamic output feedback controller with a predetermined order is investigated, utilizing the proposed stability criterion. The proposed stability and stabilization theorems are applicable to FO‐LTI systems with different fractional orders in one or both of 0 < α < 1 and 1 ≤ α < 2 intervals. Finally, some numerical examples are presented to confirm the obtained analytical results.

This study reviews the main results in the literature on the integration of competitive equilibrium theory and optimal growth theory, in particular those concerning the convergence of an equilibrium path to a stationary state (stability theorems). JEL Classification Numbers: C6, D9.

For the fixed-time nonlinear system control problem, a new fixed-time stability (FxTS) theorem and an integral sliding mode surface are proposed to balance the control speed and energy consumption. We discuss the existing fixed time inequalities and set up less conservative inequalities to study the FxTS theorem. The new inequality differs from other existing inequalities in that the parameter settings are more flexible. Under different parameter settings, the exact upper bound on settling time in four cases is discussed. Based on the stability theorem, a new integral sliding mode surface and sliding mode controller are proposed. The new control algorithm is successfully applied to the fixed-time control of chaotic four-dimensional Lorenz systems and permanent magnet synchronous motor systems. By comparing the numerical simulation results of this paper's method and traditional fixed-time sliding mode control (SMC), the flexibility and superiority of the theory proposed in this paper are demonstrated. Under the same parameter settings, compared to the traditional FxTS SMC, it reduces the convergence time by 18%, and the estimated upper bound of the fixed time reduction in waiting time is 41%. In addition, changing the variable parameters can improve the convergence velocity.