Dynamical analysis of particular class of time-delay control systems

U disertaciji su razmatrani problemi dinamike analize posebnih klasa sistema sa istim vremenskim kašnjenjem. Prošireni su osnovni rezultati na polju ljapunovske stabilnosti linearnih, vremenski diskretnih sistema sa istim vremenskim kašnjenjem. Data je Ljapunov–Krasovski metoda za vremenski diskretne sisteme sa istim vremenskim kašnjenjem. Prezentovani su potrebni i dovoljni uslovi asimptotske stabilnosti, zavisne od isto vremenskog kašnjenja, linearnih, vremenski kontinualnih i diskretnih sistema sa istim vremenskim kašnjenjem. Dati su dovoljni uslovi asimptotske stabilnosti, nezavisne od isto vremenskog kašnjenja, klase linearnih, perturbovanih sistema sa višestrukim vremenskim kašnjenjem. Prezentovani su dovoljni uslovi D–stabilnosti klase linearnih, vremenski diskretnih sistema sa istim vremenskim kašnjenjem. Dati su dovoljni uslovi eksponencijalne stabilnosti vremenski diskretnih sistema sa istim vremenskim kašnjenjem i perturbacijama. Prezentovani su potrebni i dovoljni uslovi kvadratne stabilnosti linearnih, vremenski diskretnih sistema sa istim vremenskim kašnjenjem u stanju i neodreenostima. Potrebni i dovoljni uslovi asimptotske stabilnosti, zavisni od isto vremenskog kašnjenja, velikih, linearnih, vremenski kontinualnih i diskretnih sistema sa istim vremenskim kašnjenjem, su dati. Prouena je stabilnost velikih, intervalnih, vremenski kontinualnih i diskretnih sistema sa istim vremenskim kašnjenjem. Izvedeni su novi dovoljni kriterijumi, zavisni i nezavisni od isto vremenskog kašnjenja, stabilnosti na konanom vremenskom intervalu i atraktivne praktine stabilnosti linearnih, vremenski kontinualnih i diskretnih sistema sa istim vremenskim kašnjenjem, kao i odgovarajui rezultati koji se tiu problema praktine nestabilnosti. Istražen je problema stabilnosti na konanom vremenskom intervalu za klasu linearnih, vremenski diskretnih sistema sa vremenski promenljivim kašnjenjem. Numeriki primeri su dati da demonstriraju primenu prezentovanih metoda.control systems are considered. Some of the basic results in the area of Lyapunov stability of linear, discrete time–delay systems are extended. A Lyapunov–Krasovskii method for discrete time–delay systems is gived. Necessary and sufficient conditions for delay–dependent asymptotic stability of linear, continuous and discrete time–delay systems is offered. Sufficient conditions, independent of delay, for asymptotic stability of a particular class of linear perturbed time–delay systems with multiple delays are gived. New sufficient conditions for the D–stability of a particular class of linear, discrete time–delay systems are established. Sufficient conditions for the exponential stability of discrete time–delay systems with perturbations are gived. Necessary and sufficient conditions for quadratic stability of uncertain linear discrete systems with state delay are presented. New necessary and sufficient conditions for delay–dependent asymptotic stability of a particular class of large–scale, linear, continuous and discrete time–delay systems are established. The stability of continuous and discrete large–scale time–delay interval systems are considered. A new sufficient delay–dependant and delay–independent criteria for the finite time stability and attractive practical stability of linear continuous and discrete time–delay systems has been derived, as well as corresponding results concerning instability problems. Finite–time stability problem has been investigated for a class of linear discrete time–varying delay systems. Numerical examples are given to demonstrate the application of the proposed methods

Asymptotic forecast uncertainty and the unstable subspace in the presence of additive model error

It is well understood that dynamic instability is among the primary drivers of forecast uncertainty in chaotic, physical systems. Data assimilation techniques have been designed to exploit this phenomenon, reducing the effective dimension of the data assimilation problem to the directions of rapidly growing errors. Recent mathematical work has, moreover, provided formal proofs of the central hypothesis of the assimilation in the unstable subspace methodology of Anna Trevisan and her collaborators: for filters and smoothers in perfect, linear, Gaussian models, the distribution of forecast errors asymptotically conforms to the unstable-neutral subspace. Specifically, the column span of the forecast and posterior error covariances asymptotically align with the span of backward Lyapunov vectors with nonnegative exponents. Earlier mathematical studies have focused on perfect models, and this current work now explores the relationship between dynamical instability, the precision of observations, and the evolution of forecast error in linear models with additive model error. We prove bounds for the asymptotic uncertainty, explicitly relating the rate of dynamical expansion, model precision, and observational accuracy. Formalizing this relationship, we provide a novel, necessary criterion for the boundedness of forecast errors. Furthermore, we numerically explore the relationship between observational design, dynamical instability, and filter boundedness. Additionally, we include a detailed introduction to the multiplicative ergodic theorem and to the theory and construction of Lyapunov vectors. While forecast error in the stable subspace may not generically vanish, we show that even without filtering, uncertainty remains uniformly bounded due its dynamical dissipation. However, the continuous reinjection of uncertainty from model errors may be excited by transient instabilities in the stable modes of high variance, rendering forecast uncertainty impractically large. In the context of ensemble data assimilation, this requires rectifying the rank of the ensemble-based gain to account for the growth of uncertainty beyond the unstable and neutral subspace, additionally correcting stable modes with frequent occurrences of positive local Lyapunov exponents that excite model errors

Turing conditions for pattern forming systems on evolving manifolds

The study of pattern-forming instabilities in reaction-diffusion systems on growing or otherwise time-dependent domains arises in a variety of settings, including applications in developmental biology, spatial ecology, and experimental chemistry. Analyzing such instabilities is complicated, as there is a strong dependence of any spatially homogeneous base states on time, and the resulting structure of the linearized perturbations used to determine the onset of instability is inherently non-autonomous. We obtain general conditions for the onset and structure of diffusion driven instabilities in reaction-diffusion systems on domains which evolve in time, in terms of the time-evolution of the Laplace-Beltrami spectrum for the domain and functions which specify the domain evolution. Our results give sufficient conditions for diffusive instabilities phrased in terms of differential inequalities which are both versatile and straightforward to implement, despite the generality of the studied problem. These conditions generalize a large number of results known in the literature, such as the algebraic inequalities commonly used as a sufficient criterion for the Turing instability on static domains, and approximate asymptotic results valid for specific types of growth, or specific domains. We demonstrate our general Turing conditions on a variety of domains with different evolution laws, and in particular show how insight can be gained even when the domain changes rapidly in time, or when the homogeneous state is oscillatory, such as in the case of Turing-Hopf instabilities. Extensions to higher-order spatial systems are also included as a way of demonstrating the generality of the approach

Tippe Top Inversion as a Dissipation-Induced Instability

By treating tippe top inversion as a dissipation-induced instability, we explain tippe top inversion through a system we call the modified Maxwell--Bloch equations. We revisit previous work done on this problem and follow Or's mathematical model [SIAM J. Appl. Math., 54 (1994), pp. 597--609]. A linear analysis of the equations of motion reveals that the only equilibrium points correspond to the inverted and noninverted states of the tippe top and that the modified Maxwell--Bloch equations describe the linear/spectral stability of these equilibria. We supply explicit criteria for the spectral stability of these states. A nonlinear global analysis based on energetics yields explicit criteria for the existence of a heteroclinic connection between the noninverted and inverted states of the tippe top. This criteria for the existence of a heteroclinic connection turns out to agree with the criteria for spectral stability of the inverted and noninverted states. Throughout the work we support the analysis with numerical evidence and include simulations to illustrate the nonlinear dynamics of the tippe top