Arbitrary Small Time Research Articles

Optimal approximation of internal controls for a wave-type problem with fractional Laplacian using finite-difference method

We consider a finite-difference semi-discrete scheme for the approximation of internal controls of a one-dimensional evolution problem of hyperbolic type involving the spectral fractional Laplacian. The continuous problem is controllable in arbitrary small time. However, the high frequency numerical spurious oscillations lead to a loss of the uniform (with respect to the mesh size) controllability property of the semi-discrete model in the natural setting. For all initial data in the natural energy space, if we filter the high frequencies of these initial data in an optimal way, we restore the uniform controllability property in arbitrary small time. The proof is mainly based on a (non-classic) moment method.

Robust switching signal estimation for a class of uncertain nonlinear switched systems

ABSTRACTIn this paper, a robust switching signal estimation approach is presented for a class of uncertain nonlinear switched systems using a sliding mode robust observer on the continuous state vector of system. In this approach, the estimation of switching signal is achieved at a bounded and arbitrary small time in spite of un-modelled uncertainties. The developed estimation technique can also be employed on mode-dependent control methods to reach a practical controller for nonlinear switched systems with the inaccessible switching signal. The simulation results illustrate the capability of proposed method.

Minimal controllability time for the heat equation under unilateral state or control constraints

The heat equation with homogeneous Dirichlet boundary conditions is well known to preserve non-negativity. Besides, due to infinite velocity of propagation, the heat equation is null controllable within arbitrary small time, with controls supported in any arbitrarily open subset of the domain (or its boundary) where heat diffuses. The following question then arises naturally: can the heat dynamics be controlled from a positive initial steady state to a positive final one, requiring that the state remains non-negative along the controlled time-dependent trajectory? We show that this state-constrained controllability property can be achieved if the control time is large enough, but that it fails to be true in general if the control time is too short, thus showing the existence of a positive minimal controllability time. In other words, in spite of infinite velocity of propagation, realizing controllability under the unilateral non-negativity state constraint requires a positive minimal time. We establish similar results for unilateral control constraints. We give some explicit bounds on the minimal controllability time, first in 1D by using the sinusoidal spectral expansion of solutions, and then in the multi-dimensional case. We illustrate our results with numerical simulations, and we discuss similar issues for other control problems with various boundary conditions.

Indirect controllability of some linear parabolic systems of m equations with m − 1 controls involving coupling terms of zero or first order

This paper is devoted to the study of the null and approximate controllability for some classes of linear coupled parabolic systems with less controls than equations. More precisely, for a given bounded domain Ω in RN (N∈N⁎), we consider a system of m linear parabolic equations (m⩾2) with coupling terms of first and zero order, and m−1 controls localized in some arbitrary nonempty open subset ω of Ω. In the case of constant coupling coefficients, we provide a necessary and sufficient condition to obtain the null or approximate controllability in arbitrary small time. In the case m=2 and N=1, we also give a generic sufficient condition to obtain the null or approximate controllability in arbitrary small time for general coefficients depending on the space and times variables, provided that the supports of the coupling terms intersect the control domain ω. The results are obtained thanks to the fictitious control method together with an algebraic method and some appropriate Carleman estimates.

A degenerating Cahn–Hilliard system coupled with complete damage processes

In this work, we analytically investigate a degenerating PDE system for phase separation and complete damage processes considered on a nonsmooth time-dependent domain with mixed boundary conditions. The evolution of the system is described by a degenerating Cahn–Hilliard equation for the concentration, a doubly nonlinear differential inclusion for the damage variable and a degenerating quasi-static balance equation for the displacement field. All these equations are highly nonlinearly coupled. Because of the doubly degenerating character of the system, the doubly nonlinear differential inclusion and the nonsmooth domain, the structure of the model is very complex from an analytical point of view.A novel approach is introduced for proving existence of weak solutions for such degenerating coupled system. To this end, we first establish a suitable notion of weak solutions, which consists of weak formulations of the diffusion and the momentum balance equation, a variational inequality for the damage process and a total energy inequality. To show existence of weak solutions, several new ideas come into play. Various results on shrinking sets and its corresponding local Sobolev spaces are used. It turns out that, for instance, on open sets which shrink in time a quite satisfying analysis in Sobolev spaces is possible. The presented analysis can handle highly nonsmooth regions where complete damage takes place. To mention only one difficulty, infinitely many completely damaged regions which are not connected with the Dirichlet boundary may occur in arbitrary small time intervals.

Stochastic Stability in Three-Player Games with Time Delays

We discuss combined effects of stochasticity and time delays in finite-population three-player games with two mixed Nash equilibria and a pure one. We show that if basins of attraction of the stable interior equilibrium and the stable pure one are equal, then an arbitrary small time delay makes the pure one stochastically stable. Moreover, if the basin of attraction of the interior equilibrium is bigger than the one of the pure equilibrium, then there exists a critical time delay where the pure equilibrium becomes stochastically stable.

A MIMO Sampling Rate Dependent Controller

A sampling-rate-dependent (SRD) controller is proposed for the uniform output tracking problem of multiinput multioutput discrete time-varying linear systems. A proportional-integral-derivative (PID) controller is a special case of the proposed controller. The SRD controller aims at achieving uniform output tracking in the sense of attaining arbitrary small steady-state errors as well as arbitrary small settling time. The SRD controller makes use of higher sampling rates for larger variations in the desired trajectories. An analytic approach for finding the parameters of the SRD controller that guarantees reference output trajectory tracking is presented. Four different numerical examples and two different experimental applications are provided in order to illustrate the output tracking capability of the proposed SRD controller. Performance is compared to a PID controller with fuzzy gain scheduling, four multivariable PID controllers, an H <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sup> optimal controller, and an iterative-learning-control algorithm.

Exact Controllability of Scalar Conservation Laws with an Additional Control in the Context of Entropy Solutions

In this paper, we study the exact controllability problem for nonlinear scalar conservation laws on a compact interval with a regular convex flux and in the framework of entropy solutions. With the boundary data and a source term depending only on the time as controls, we provide sufficient conditions for a state to be reachable in arbitrary small time. To do so we introduce a slightly modified wave-front tracking algorithm.

Output integral sliding mode control to stabilize position of a Stewart platform

The problem of the realization of integral sliding mode controllers, based only on output information, is applied to a Stewart platform. This platform has three degrees of freedom and it is used as a remote surveillance devise. We consider the hierarchical sliding mode observer, allowing the reconstruction of the system states from the initial moment if we suppose that there exist ideal sliding modes and equivalent output injections. This allows the implementation of an output integral sliding mode controller ensuring the insensitivity of the state trajectory with respect to the matched uncertainties from the initial moment. The discrete realization output integral sliding mode controller requires the filtration to obtain the equivalent output injections. It is shown that the observation error can be made arbitrarily small after an arbitrary small time without any adjustment of the observer parameters, only by decreasing the sampling step and filter time constant. The results obtained are illustrated by simulations.

Dynamic programming for impulse controls

This paper describes the theory of feedback control in the class of inputs which allow delta-functions and their derivatives. It indicates a modification of dynamic programming techniques appropriate for such problems. Introduced are physically realizable bang-bang-type approximations of the “ideal” impulse-type solutions. These may also serve as “fast” feedback controls which solve the terminal control problem in arbitrary small time. Examples of damping high-order oscillations in finite time are presented.

Tracking Control for an Ellipsoidal Submarine Driven by Kirchhoff's Laws

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> In this paper, we study the control of an ellipsoid immersed in an infinite volume of ideal fluid. The dynamics of the uncontrolled body are given by Kirchhoff's laws. The control system is underactuated: one control is an acceleration along an axis of the ellipsoid and two are angular accelerations around the other two axes. By adopting a backstepping viewpoint, we prove that the position and the attitude of the solid can be forced to approximately follow any given path, using fast-oscillating controls. Moreover, we prove that the controlled mechanical system (which includes the impulses) is completely controllable in an arbitrary small time. </para>

Stochastic Control with Exit Time and Constraints, Application to Small Time Attainability of Sets

We study the existence of a solution of controlled stochastic differential equations remaining in a given set of constraints at any time smaller than the exit time of a given open set. We also investigate the small time attainability of a given closed setK, i.e., the property that, for all arbitrary small time horizonT and for all initial condition in a sufficiently small neighborhood ofK, there exists a solution to the controlled stochastic differential equation which reachesK beforeT.

On the stabilization and stability robustness against small delays of some damped wave equations

In this paper we consider a system which can be modeled by two different one-dimensional damped wave equations in a bounded domain, both parameterized by a nonnegative damping constant. We assume that the system is axed at one end and is controlled by a boundary controller at the other end. We consider two problems, namely the stabilization and the stability robustness of the closed-loop system against arbitrary small time delays in the feedback loop. We propose a class of dynamic boundary controllers and show that these controllers solve the stabilization problem when the damping coefficient is nonnegative and the stability robustness problem when the damping coefficient is strictly positive.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Solutions of Langevin equations with multiplicative colored noise

A Fokker-Planck equation for a nonlinear Langevin equation with multiplicative gaussian noise with arbitrary covariance and small coherence time is derived. An exact steady-state solution of this Fokker-Planck equation is obtained for a lorentzian power spectrum of the driving noise. The crucial influence of the finite coherence time of the noise on the critical properties of the system is exhibited.