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Basic phenomena of H+and H− ion concentration process are considered, and state apace models are derived for a basic continuous flow system. Inherent non-linearity of the system can he avoided by a change of variables, hut due to the logarithmic measurement via pH the corresponding feedback control system is highly nonlinear. A basic feedback system is proved to be globally asymptotically stable. Through an augmentation of the basic model it is shown that a relatively small non-ideality in a real system is sufficient to make it unstable. On the other hand, a steady state analysis show that the concentration equilibrium is highly sensitive to deviations of parameters related to the controlling chemical reagent, if a concentrated acid or base is being neutralized in a single stage.

Nonlinear analysis has shown that when the buckling of a nuclear reactor with negative feedback is increased, the flux, under appropriate conditions, will proceed to a new asymptotically stable state. This contrasts with the linear theory which predicts a runaway. The method of''coordinate stretching'' has been used to obtain the asymptotic solution of a nonlinear nuclear reactor under the combined effect of an initial positive disturbance and a negative feedback based on the Newton's law of cooling. The minimum stability condition is derived by requiring that a bounded new equilibrium state exist. This condition sets an upper limit to the magnitude of the initial disturbance beyond which an equilibrium solution does not exist. Furthermore, the magnitude of the equilibrium flux is determined explicitly in terms of several relevant physical properties of the system: feedback coefficient, energy production rate, and rate of energy transfer to coolant. (auth)

In this paper it is shown that a number of ambiguous points arising in transient stability analysis of power systems can be clarified if partial or output stability concepts are used instead of stato stability. This solves the question of the number of state variables and the validity of using positive semi-definite Lyapunov functions. It is further discussed how a very general Lur'e-type Lyapunov function can be derived for the power system stability problem, which unifies and generalizes numerous Lyapunov functions available in the technical literature.

Volterra integral equations of the form u ′ ( t ) = − ∫ 0 t a ( t − τ ) u ( τ ) d τ , u ( 0 ) = 1 u’(t) = - \smallint _0^ta(t - \tau )u(\tau )d\tau ,u(0) = 1 , are considered, where a ( t ) ∈ C ( 0 , ∞ ) ∩ L 1 ( 0 , 1 ) a(t) \in C(0,\infty ) \cap {L_1}(0,1) . Explicit asymptotic forms are obtained for the solutions, when the kernels a ( t ) a(t) have a specific asymptotic representation.

The Hilbert resolution space setting is used to obtain some new sufficient conditions for the existence and causality of an inverse system. These conditions, are applicable when the system under consideration is characterized by a behavior which is special, in some sense, with respect to the notion of causality. Relevant conceptual connections with recent problems in system sensitivity and stability theory are pointed out.

A model of a distributed-boundary control system is considered. Assume the uncontrolled system possesses an exponential asymptotically stable zero solution. We then construct suboptimal feedback controls for the distributed and boundary control problems via the direct method of Liapunov. Furthermore, existence-uniqueness of the synthesized control systems is proven by applying the theory of nonlinear semigroups and maximal dissipative sets. Applications to diffusion equations are given.

Consider the non-linear difference equations where x, y, j, and g are; q-vectors, fx(n, x) is continuous in x, and the origin is asymptotically self-invariant relative to (1). It is shown that if the origin is uniformly stable in variation (uniformly asymptotically stable in variation), and if the perturbation g satisfies then the origin is eventually uniformly stable (eventually uniform-asymptotically stable) for (2). These stability results are obtained via a non-linear variation of constants formula.

On an Tz-manifold X, let n be a dynamical system that has an asymptotically stable periodic orbit I: (In our usage, i‘ is asymptotically stable iff inside any neighborhood FV of r, there is an open invariant set V containing r such that, for any other neighborhood Z of r, there is a time T < GO with the property that ?T( b’ x [T, CO)) C Z.) Our principal result is that if U is any neighborhood of r in X, and m& is any dynamical system on X resulting from a. “sufficiently small” perturbation of n, then rLd has a periodic orbit contained in zi. This is a partial extension to dynamical systems of a classical result concerning autonomous differential equations of the form k = f(~, .u) whose first variation satisfies certain conditions [2, p. 3521. In the classical case, the periodic solution of the perturbed system is also asymptotically orbitally stable. We include an example of a dynamical system on E3 with a single asymptotically stable (in the sense of the definirion above) orbit. Small perturbations of this system are constructed whose resulting periodic orbit is not even stable, much less asymptotically stable. A dynamical system on a topological space X is a continuous mapping ‘ir: X x R -+ X (R = real number line) such that z(“Y, 0) = x k’,z E X and “(S, t1 + tz) = T+T(x, t& t3) vx E x and t1 , tz E R. In this paper we will write ~(x, t) as ZH, and use ArrB to denote n(a y\ B), ip C X, B CR. We will use without reference certain elementary properties of dynamical systems. The reader will find in either [I, 4, or 61 a good introduction to dynamical systems. The set of all dynamical systems on X with the compactopen topology, we term D(X).

Asymptotic stability of the zero solution of \[\dot x(t) = - a(t)x(t) + P(t,x_t )\] is studied with a direct method of Razumikhin. If $| {P(t,\varphi )} | \leqq \alpha (t)\theta \| \varphi \|$ for some $\theta 0$, then zero is exponentially stable; if the condition on $a(t)$ is weakened to $a(t) \geqq 0$ and $\int ^\infty a(t)dt = \infty $, then zero is asymptotically stable.

For a large class of functional differential equations of the neutral type, sufficient conditions for set stability and uniform set stability are developed. The results arc obtained via Lyapunov function arguments and are extension of the recent work of Heinen and Dimino (1973).

Sufficient conditions for the stability and asymptotic stability of the null solution of a neutral functional differential equation are established. These stability results are obtained by employing the idea of a Lyapunov function which has a strongly negative definite derivative on an annulus, and they represent an extension of the recent work of Haddock (1971).

We derive, by use of the Lyapunov theory for difference equations, a criterion for the asymptotic stability of a system governed by a nonautonomous difference equation of second order. The results are pertinent to the design of a simple digital frequency-shift-keying (FSK) oscillator. A feature of our analysis that is of major importance is the technique used in our application of the Lyapunov theory. It is shown that the technique yields the best stability criterion that can be obtained by the use of any positive definite quadratic form as a Lyapunov function. Thus we show how one might overcome one of the major obstacles to the practical application of the Lyapunov theory, i.e., the problem of choosing an appropriate (in some meaningful sense) Lyapunov function.

This paper considers discrete-time systems of the Lur'e-Postnikov class where the linear part is not asymptotically stable and the nonlinear characteristic satisfies only partially the usual sector condition. Estimates of the resulting finite regions of absolute ultimate boundedness are calculated by means of a quadratic Liapunov function.

In this paper a controller design incorporating a non-linear state reconstruotor is considered for a class of unstable non-linear plants. A basic assumption is that if all state variables x of the process to be controlled were available for measurement, then there is some control law Φ(x) that would result in an asymptotically stable closed-loop system. When only a limited number of measurements are available, it is shown that a non-linear state reconstructor can be used to provide estimates x of the process state. Conditions are given for the stability of the controlled system using Φ(x). Simulation results of the performance of a non-linear regulator are included.

From the Vlasov equation a reduced kinetic equation, defined as the drift-kinetic equation, is derived by the small gyroradius expansion theory. The reduced equation is of the form df dt = 0 where f is the distribution of guiding centers, and the time derivative is taken along the particle orbits in the drift approximation. The derived kinetic equation is taken for the basis of the study of equilibria and electrostatic stability in axisymmetric, toroidal geometry. From the integral mode equation derived in the stability analysis, the theory of the current driven and the ballooning drift mode is developed and applied to tokamak plasmas.

Attempts to realize optimal fixed-lag smoothers for continuous-time processes have encountered two major difficulties: the presence of delay lines in the smoother, and the instability of most smoothers. This paper considers suboptimal smoothers which still outperform filters and have performance very close to the optimum. Design procedures are exhibited for smoothers that are both finite-dimensional and asymptotically stable, with the procedures depending on the delay line being approximated by a finite-dimensional dynamical system.

A standard procedure for optimizing the transient response of a system of nonlinear differential equations near an asymptotically stable equilibrium point involves choosing the system parameters so that the right-most eigenvalue of the linearized system is furthest to the left in the complex plane. This procedure is shown to also apply for nonlinear systems in infinite dimensional Banach spaces.

We consider linear elasticity equations of the type encountered in the theory 'of small deformations superposed on large, when the large deformation leaves the material in a homogeneous configuration. Said differently, we consider cases where these are linear equations with constant coefficients. Such equations admit a variety of complex exponential solutions. As is discussed by Truesdell & Noll [1, §§ 71, 73], for example, one subset, the plane waves, has been studied in some detail. Another type is encountered in the various studies of surface waves, whose amplitude varies with position, but not time. Judging from conversations I have had with various workers, it seems not to be a matter of common knowledge that there also exist waves whose amplitude increases or decreases exponentially with time. Failure to appreciate this seems to have led to some confusion concerning elastic stability theory. I t thus seems worthwhile to at tempt a more general exploration of complex exponential solutions, hereafter abbreviated as CES. We here establish some of their properties. We restrict our attention to equations of strongly elliptic type. Various interpretations and implications of this condition are discussed by Truesdell & Noll [1, §§ 44, 45, 48, 52, 68, 71, 74, 90]. For reasons discussed by Ericksen [2, 3], elasticities violating this condition are not likely to be observed.

By using the self-consistent stability theory it is shown that during the growth phase of a magnetospheric substorm the stability of the tail configuration decreases. Taking two-dimensionality into account explicitly, we show that the free energy built up during the growth phase can be released when the plasma sheet has become sufficiently thin and/or the normal magnetic field component has been sufficiently reduced. Then the ion-tearing mode will abruptly set in and lead to changes in the magnetic field topology. It is speculated that there is a tendency for a macroscopic neutral line to form fairly close to the earth. Lack of equilibrium will lead to energy dissipation and enhanced precipitation.

A procedure to find the existence and domain of asymptotic stability for a class of systems described by a set of non-linear partial differential equations is presented in the context of semigroup theory. No a priori bound is imposed on the non-linearity which is assumed to be a quad ratic (bilinear) or multilinear form. The linearized system can be non-self-adjoint and need not be asymptotically stable equation by equation. The solution of the stability problem is carried out. in two steps : (a) the development of an L2-like norm that bounds above the sup norm, a requirement to obtain a useful estimate of the non-linear term, and (a)the computation of an estimate of the L2-likc norm of the semigroup generated by the linearized system in terms of its usual L2 norm.

This expository lecture surveys recent progress of the stability theory in Celestial Mechanics with emphasis on the analytical problems. In particular, the old question of convergence of perturbation series are discussed and positive results obtained, in the light of the work by Kolmogorov Arnold and Moser. For the three body problem, classes of quasi-periodic solutions and doubly asymptotic (or homoclinic) orbits are discussed.

The main contribution of this paper is the introduction of a new canonical form for an adaptive observer for a linear multivariable system. The adaptive observer simultaneously estimates the state and the parameters of the unknown plant and is shown to be globally asymptotically stable. The extension of the results to discrete systems and the application of the adaptive observer in the control of helicopter dynamics is also considered in detail.

Liapunov functions are constructed and used to prove stability theorems for critical autonomous systems in which the linear part of the right-hand side has a zero eigenvalue.

The feasibility of applying various hydrodyanmic instability theories, previously found applicable in the development of tropical cyclone vortices, to the formative stage of a severe thunderstorm when a meso-low-pressure cell is the dominant circulation feature of the storm's immediate environment is investigated. A consistent grid of meteorological data representative of a meso-low-pressure cell is generated using a model previously described by Lin and Martin. Stability theory of atmospheric vortices is then involved to reflect certain characteristic qualities of an assumed superimposed perturbation upon the generated mesoscale data. The resultant growth-rate equation indicates the determining factors for severe thunderstorm development to be the dimensions of the perturbation and the stability characteristics of the meso-low-pressure cell upon which that perturbation is superimposed. The static stability term representative of the meso-low-pressure cell, in particular, appears to dominate to t...

The present state of the stability theory of singular interfaces is reviewed. Stability, as determined by the anisotropy of the kinetic coefficient, and the absolute value of the kinetic coefficient, is discussed for solution and melt growth. The slopes of the hillocks which determine the growth rate of the faces increase with supersaturation and, therefore, a singular surface becomes less stable when the supersaturation increases. Nucleation at apexes is also of great importance for the occurrence of instability. The stability of an infinite planar interface against perturbations is considered. The criterion outlined for the growth from an impure melt contains the logarithmic derivative of the kinetic coefficient in addition to the usual parameters describing constitutional supercooling. A singular face turns out to be stable against short-range perturbations even in highly undercooled melts. The stability of polyhedra against perturbation at the apexes is also considered. Analysis of shapes of two-dimensional crystals growing on substrates is considered in connection with the dipole moments of adatoms and aggregates. The repulsion of an adatom from a two-dimensional crystal having a dipole moment of the same sign is weaker at the apexes than at the center of the facet, skeletal shapes being the result.

This paper derives the necessary and sufficient conditions for a multivariable plant P(s) with asymptotically stable hidden modes to be stabilizable by means of singleloop feedback employing an asymptotically stable controller and feedback sensor. These conditions are completely general and therefore encompass unstable, nonminimum-phase plants as well. For single-input-output plants with zero gain at infinite frequency the conditions reduce to the sole requirement that no plant zeros on the nonnegative real axis of the complex s-plane lie to the left of an odd number of real plant poles, multiple poles counted according to their multiplicities. It has also been possible to derive simple necessary and sufficient conditions for a closed-loop transfer function T(s) to be realizable by asymptotically stable compensation around a prescribed single-input-output plant P(s). It is expected that this latter result can be suitably generalized to the multivariable case. In a real sense, this paper constitutes a continuation of some earlier unpublished work [8] by the first author.

A detailed discussion is given of the quadratic characteristic value problem, x = λPx + ( 1 λ )Qx , where P and Q are compact selfadjoint operators in Hilbert space. Theorems are proven showing that unconditional bases of various types may be constructed from the root vectors of A(λ) = λP + ( 1 λ )Q . These theorems are of interest in the stability theory of linear evolution equations, particularly for application to certain problems in the linearized theory of hydrodynamic stability.

The concept of almost sure sample stability and sample stability in probability are formulated for elastic systems. Using a Koiter type approach these concepts are used in the analysis of imperfection sensitive structures. The applied load and the initial geometric imperfections are introduced into the analysis as random quantities. A compressed beam of finite length on a nonlinear elastic foundation is used in an example calculation.

The notions of constant, discrete-time, and linear dynamical systems over a commutative ring and their corresponding input/output maps are defined and studied. Classical stability theory is generalized to systems over fields complete with respect to a rank-one valuation. The resulting ''p-adic stability theory'' is used to solve the realization problem for matrix sequences over a broad class of integral domains, generalizing results first announced in Rouchaleau, Wyman, and Kalman [Proc. Nat. Acad. Sci. U.S.A. 69 (1972), 3404-3406].

The mean-square stability of elastic bodies in supersonic flow is studied. The external load consists of pressure fluctuations in the turbulent boundary layer, and of the pressure perturbation depending on the elastic body deformations according to the “piston theory”. For solving the stochastic boundary problem, both the given and sought random functions are resolved in the modes of vibration of an elastic system in vacuo . Galerkin's method yields an infinite set of linear algebraic equations for the response spectra, solved by reduction. It is shown that the reduced set is stable in the mean-square if the corresponding deterministic set is asymptotically stable, and moreover if a certain inequality is satisfied regarding the response characteristics. It is also shown that the mean-square stability falls within the pre-flutter range of the system parameters. The actual computations for the mean-square stability of a clamped-clamped beam in turbulent supersonic flow in two- and three-mode approximations will be presented at a later date.

Several techniques for investigating the multiplicity and stability of open isothermal enzymatic reactors are discussed and some of the pitfalls in previous thinking pointed out. The example which is used to illustrate these methods exhibits several interesting features. Among these is the existence of a stable oscillatory state which surrounds a unique steady state which is asymptotically stable to certain finite disturbances.

We introduce the concepts of the degree and the order of synchronism on the basis of a mathematical model of the emergence of synchronization in the form of an asymptotically stable integral torus in the phase plane. We investigate the existence conditions for synchronisms in a dynamic system described by differential equations with rapidly rotating phases. As an application we examine synchronisms in a system of quasi-Hamiltonian objects. In recent years the phenomena of synchronization and resonance in dynamic systems have been subjected to intensive study, in particular, in connection with the question of the synchronization of satellites [1, 2] and of mechanical vibrators [3]. On the mathematical side the appearance of synchronization is closely connected with the theory of differential equations with rapidly rotating phases. Here in the first place we must mention the works specified in [4–9].

In [l], B.S. Razumikhin gave conditions for the stability and asymptotic stability of the zero state of systems of ordinary differential equations involving a fixed finite time delay. His conditions make use of Liapunov functions defined on the finite-dimensional state space rather than on the space of functions continuous on the interval of delay; for methods involving functions of the latter type, cf., for example, Yoshizawa 121. In a recent paper [3], the author has used Liapunov functions of Razumikhin type to give conditions sufficient for the stability of the zero state of a system of ordinary differential equations involving an interval of delay which becomes unbounded as t + + co; an example of such a system is an integrodifferential equation of Volterra type

The stability of a quasi-linear system is investigated by assuming stability properties of associated linear systems. By this method, the task of finding suitable Lyapunov functions may be avoided in certain cases.

Practically, predictor-corrector methods for the solution of differential equations are widely used. In these numerical methods, the question of stability is most important. For a single differential equation Crane and Lambert (1962) studies the stability of a fourth-order generalized corrector formula; Chase (1962) and subsequently Emanuel (1963) demonstrated for particular methods that the overall integration procedure does not generally possess the same stability properties as the corrector except when the step of integration tends to zero. In the present paper it is shown that it is possible to develop a necessary and sufficient condition for the stability of general predictor-corrector methods for the solution of systems of differential equations.

Some recent results in stability theory are used to derive criteria for the BIBO stability of two broad classes of lumped-distributed networks. The first class of networks consists of “feedback” networks whose forward gain is one of the ABCD parameters of an arbitrary nonuniform RC transmission line, and whose feedback gain is constant. The second class consists of networks which are arbitrary interconnections of active elements and open and short-circuited commensurate RC lines.

Simple, easy-to-apply estimates are obtained for the norms, both Euclidean and generalized, of the solutions of linear time-varying vector differential equations, and for the settling times in systems which are asymptotically stable. Both eigenvalue and Gerschgorin estimates are obtained.

The paper deals with responses of a non-linear multiple lumped dynamical system bounded in a large neighbourhood of the single, asymptotically stable equilibrium point. The sufficient, analytic criteria for the non-oscillation of such responses have been introduced in previous work (cf. Skowronaki and Shannon 1972). On the basis of the latter, the author discusses a geometric interpretation of necessary and sufficient criteria for non-oscillatory responses applied to the synthetic design of the relevant systems.