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## Gimbal Suspension Gyro: Stability, Bifurcation, and Chaos

### Journal of Dynamical and Control Systems (2001-07-01) 7: 339-351 , July 01, 2001

In this paper the solutions of three problems, devoted to the investigation of the dynamics of balanced gimbal suspension gyro (GSG) which is installed at the immovable base are presented. In the first problem the stability of stationary motion (SM) of the GSG is analyzed under which the planes of the internal and external rings of GSG are orthogonal. The moment of viscous friction and the moment which is the function of the deviation angle of the internal ring act on the external axis. The analysis of stability of SM is carried out by means of construction of Lyapunov's function with the estimation of the attraction domain. In the second problem the mechanism of the stability loss of the SM under the transfer of the bifurcational parameter through the critical value is presented. In this case the periodic motion originates (Andronov-Hopf bifurcation). The orbital stability condition of the periodic motion is found The third problem investigates the forced vibration of the GSG under the action on the internal ring of the perturbed moment which is the sum of the small moment of viscous friction and moment with small amplitude and fixed frequency. Here we consider the case where the projection of the angular moment on the axis of the external ring is equal to zero. In case of absence of the perturbations the SMs under which the external and internal rings are orthogonal or lie in the common plane are stable and unstable, respectively. For the unperturbed system the equation of the separatrix which passes through the hyperbolic points is found. For the determination of the condition of intersection of the separatrices in the perturbed system, the distance between them is calculated (Melnikov's distance). In the space of parameters the domain in which this distance can change the sign is distinguished and it is the feature of the chaotic motion arising.

## Dual-Sampling-Rate Moving-Horizon Control of a Class of Linear Systems with Input Saturation and Plant Uncertainty

### Journal of Optimization Theory and Applications (2003-03-01) 116: 485-516 , March 01, 2003

Moving-horizon control is a type of sampled-data feedback control in which the control over each sampling interval is determined by the solution of an open-loop optimal control problem. We develop a dual-sampling-rate moving-horizon control scheme for a class of linear, continuous-time plants with strict input saturation constraints in the presence of plant uncertainty and input disturbances. Our control scheme has two components: a slow-sampling moving-horizon controller for a nominal plant and a fast-sampling state-feedback controller whose function is to force the actual plant to emulate the nominal plant. The design of the moving-horizon controller takes into account the nonnegligible computation time required to compute the optimal control trajectory.

We prove the local stability of the resulting feedback system and illustrate its performance with simulations. In these simulations, our dual-sampling-rate controller exhibits performance that is considerably superior to its single-sampling-rate moving-horizon controller counterpart.

## On Stability in Two-Stage Stochastic Nonlinear Programming

### Asymptotic Statistics (1994-01-01): 329-340 , January 01, 1994

Two-stage stochastic programming problems are very often assigned to practical optimization problems with random elements. Especially, these models are employed if the basic solution should be determined without knowing the random parameter realization and if the obtained effect can be corrected by a new optimization problem (called the recourse problem) depending on the random elements realization. It is well-known that then the total problem depends on the random elements only through the corresponding probability measure. Consequently, the probability measure can be treated as a parameter in such problems and it is surely reasonable to study the stability with respect to it. The aim of this paper is to study the stability of two-stage nonlinear programming problem with respect to the distribution function. Of course, the linear case is also included in our consideration.

## The stability of monomial functions on a restricted domain

### aequationes mathematicae (2006-09-01) 72: 100-109 , September 01, 2006

### Summary.

Let (
$$G,\circ$$
) be a power-associative and square-symmetric groupoid, *Y* a Banach space. We prove that a mapping *f* : *G*→ *Y* is a monomial function of degree *m* if and only if for every δ > 0 there exists a weakly bounded set *V*_{δ}⊂ *G* such that
$$ ||\Delta ^{m}_{y} f(x) - m!f(y)|| \leq \delta ,\quad (x,y) \notin V_{\delta } \times V_{\delta } . $$

## Implicit Multifunction Theorems

### Set-Valued Analysis (1999-09-01) 7: 209-238 , September 01, 1999

We prove a general implicit function theorem for multifunctions with a metric estimate on the implicit multifunction and a characterization of its coderivative. Traditional open covering theorems, stability results, and sufficient conditions for a multifunction to be metrically regular or pseudo-Lipschitzian can be deduced from this implicit function theorem. We prove this implicit multifunction theorem by reducing it to an implicit function/solvability theorem for functions. This approach can also be used to prove the Robinson–Ursescu open mapping theorem. As a tool for this alternative proof of the Robinson–Ursescu theorem, we also establish a refined version of the multidirectional mean value inequality which is of independent interest.

## Second-order explicit characteristic difference schemes for quasilinear hyperbolic systems

### Computing (1985-03-01) 35: 85-91 , March 01, 1985

We present two-step, second-order explicit characteristic difference schemes for the numerical solution of initialvalue problems for quasilinear hyperbolic system and show that the method is stable for systems with constant coefficients.

## Stability for the Minkowski measure of convex domains of constant width

### Journal of Geometry (2013-12-01) 104: 505-513 , December 01, 2013

In 1988, H. Groemer gave a stability theorem for the area of convex domains of constant width. In this paper, we obtain a stability theorem for the well-known Minkowski measure of asymmetry for convex domains of constant width.

## Stability and superconvergence analysis of the FDTD scheme for the 2D Maxwell equations in a lossy medium

### Science China Mathematics (2011-12-01) 54: 2693-2712 , December 01, 2011

This paper is concerned with the stability and superconvergence analysis of the famous finite-difference time-domain (FDTD) scheme for the 2D Maxwell equations in a lossy medium with a perfectly electric conducting (PEC) boundary condition, employing the energy method. To this end, we first establish some new energy identities for the 2D Maxwell equations in a lossy medium with a PEC boundary condition. Then by making use of these energy identities, it is proved that the FDTD scheme and its time difference scheme are stable in the discrete *L*^{2} and *H*^{1} norms when the CFL condition is satisfied. It is shown further that the solution to both the FDTD scheme and its time difference scheme is second-order convergent in both space and time in the discrete *L*^{2} and *H*^{1} norms under a slightly stricter condition than the CFL condition. This means that the solution to the FDTD scheme is superconvergent. Numerical results are also provided to confirm the theoretical analysis.

## A New ADI Scheme for Solving Three-Dimensional Parabolic Differential Equations

### Journal of Scientific Computing (1997-12-01) 12: 361-369 , December 01, 1997

A new alternating-direction implicit (ADI) scheme for solving three-dimensional parabolic differential equations has been developed based on the idea of regularized difference scheme. It is unconditionally stable and second-order accurate. Further, it overcomes the drawback of the Douglas scheme and is to be very well to simulate fast transient phenomena and to efficiently capture steady state solutions of parabolic differential equations. Numerical example is illustrated.

## Self-bounded controlled invariants versus stabilizability

### Journal of Optimization Theory and Applications (1986-02-01) 48: 245-263 , February 01, 1986

Self-bounded controlled and self-hidden conditioned invariant subspaces, recently introduced by the authors for a more direct and neat handling of some fundamental concepts of the geometric approach, such as controllability subspaces, are proved in this paper to be very useful tools also in dealing with synthesis problems with stability requirements.

Definitions concerning stability of invariants and stabilizability of controlled invariants, simple and self-bounded, are first presented and discussed. In particular, it is shown that a more straightforward definition for controlled invariant stabilizability allows a simpler development of the theory, Then, some fundamental results relating self-boundedness to stabilizability are derived. For the sake of completeness, all statements are dualized to conditioned invariants, simple and self-hidden.