On the optimal controllability time for linear hyperbolic systems with time-dependent coefficients

Asymptotically exact discontinuous Galerkin error estimates for linear symmetric hyperbolic systems

We present an a posteriori error analysis for the discontinuous Galerkin discretization error of first-order linear symmetrizable hyperbolic systems of partial differential equations with smooth solutions. We perform a local error analysis by writing the local error as a series and showing that its leading term can be expressed as a linear combination of Legendre polynomials of degree p p and p + 1 p+1 . We apply these asymptotic results to show that projections of the error are pointwise O ( h p + 2 ) \mathcal {O}(h^{p+2}) -superconvergent. We solve relatively small local problems to compute efficient and asymptotically exact estimates of the finite element error. We present computational results for several linear hyperbolic systems in acoustics and electromagnetism.

Discontinuous Galerkin error estimation for linear symmetric hyperbolic systems

We consider linear hyperbolic systems with a stable rank 1 relaxation term and establish that the characteristic polynomial for the individual Fourier components of the solution can be written as a convex combination of the characteristic polynomials for the formal stiff and non-stiff limits. This allows us to provide a direct and elementary proof of the equivalence between linear stability and the subcharacteristic condition. In a similar vein, a maximum principle follows: The velocity of each individual Fourier component is bounded by the minimum and maximum eigenvalues of the non-stiff limit system.

We are concerned about the controllability of a general linear hyperbolic system of the form $\partial_t w (t, x) = \Sigma(x) \partial_x w (t, x) + \gamma C(x) w(t, x) $ ($\gamma \in \mathbb{R}$) in one space dimension using boundary controls on one side. More precisely, we establish the optimal time for the null and exact controllability of the hyperbolic system for generic $\gamma$. We also present examples which yield that the generic requirement is necessary. In the case of constant $\Sigma$ and of two positive directions, we prove that the null-controllability is attained for any time greater than the optimal time for all $\gamma \in \mathbb{R}$ and for all $C$ which is analytic if the slowest negative direction can be alerted by both positive directions. We also show that the null-controllability is attained at the optimal time by a feedback law when $C \equiv 0$. Our approach is based on the backstepping method paying a special attention on the construction of the kernel and the selection of controls.

Hyperbolic systems in one-dimensional space are frequently used in the modeling of many physical systems. In our recent works we introduced time-independent feedbacks leading to finite stabilization in optimal time of homogeneous linear and quasilinear hyperbolic systems. In this work we present Lyapunov’s functions for these feedbacks and use estimates for Lyapunov’s functions to rediscover the finite stabilization results.

Sampled-data distributed control for homo-directional linear hyperbolic system with spatially sampled state measurements

ABSTRACTThis paper presents a backstepping solution for the output feedback control of general linear heterodirectional hyperbolic PDE-ODE systems with spatially varying coefficients. Thereby, the ODE is coupled to the PDE in-domain and at the uncontrolled boundary, whereas the ODE is coupled with the latter boundary. For the state feedback design, a two-step backstepping approach is developed, which yields the conventional kernel equations and additional decoupling equations of simple form. In order to implement the state feedback controller, the design of observers for the PDE-ODE systems in question is considered, whereby anti-collocated measurements are assumed. Exponential stability with a prescribed convergence rate is verified for the closed-system pointwise in space. The resulting compensator design is illustrated for a 4 × 4 heterodirectional hyperbolic system coupled with a third-order ODE modelling a dynamic boundary condition.

According to the spatial dimension, equation type, and time sequence, the latest progress in controllability of stochastic linear systems and some unsolved problems are introduced. Firstly, the exact controllability of stochastic linear systems in finite dimensional spaces is discussed. Secondly, the exact, exact null, approximate, approximate null, and partial approximate controllability of stochastic linear systems in infinite dimensional spaces are considered. Thirdly, the exact, exact null and impulse controllability of stochastic singular linear systems in finite dimensional spaces are investigated. Fourthly, the exact and approximate controllability of stochastic singular linear systems in infinite dimensional spaces are studied. At last, the controllability and observability for a type of time-varying stochastic singular linear systems are studied by using stochastic GE-evolution operator in the sense of mild solution in Banach spaces, some necessary and sufficient conditions are obtained, the dual principle is proved to be true, an example is given to illustrate the validity of the theoretical results obtained in this part, and a problem to be solved is introduced. The main purpose of this paper is to facilitate readers to fully understand the latest research results concerning the controllability of stochastic linear systems and the problems that need to be further studied, and attract more scholars to engage in this research.

SUMMARYIn this paper, we investigate controllability and minimal energy optimal control for Goursat–Darboux problem for the second‐order linear hyperbolic systems with two independent variables. The equation describes a relation between functions u: D→ℝm and z: D→ℝn. We get an integral representation of the Goursat–Darboux problem by means of Riemann's matrix. The first half of the paper considers conditions under which there exists a control u for which the solution z of dynamics satisfies z(x1, y1) = p for any given p. The studied problem is reduced to the moments problem. The optimal control was found in a closed analytic form. Further, degeneracy of the matrix constructed by means of Riemann's matrix is shown to be a necessary and sufficient condition of controllability. Copyright © 2011 John Wiley & Sons, Ltd.

Summary This work focuses on the exact fractional controllability of linear hyperbolic systems over a finite time interval. The control is applied to the internal moving zone of a predefined actuator location in the spatial domain of the system. We use the Hilbert uniqueness method (HUM) to characterize the minimum‐energy control that enables us to achieve the desired fractional state at time . We therefore consider a numerical approach that allows us to determine the explicit formula for optimal control. We also verify the results' applicability using computer simulations for the one‐dimensional wave equation with a moving zone actuator.

This article presents the central finite-dimensional H ∞ controller for linear time-varying systems with unknown parameters, that is suboptimal for a given threshold γ with respect to a modified Bolza–Meyer quadratic criterion including the attenuation control term with the opposite sign. In contrast to the previously obtained results, this article reduces the original H ∞ controller problem to the corresponding H 2 controller problem, using the technique proposed in Doyle et al. [Doyle, J.C., Glover, K., Khargonekar, P.P., and Francis, B.A. (1989), ‘State-space Solutions to Standard H 2 and H Infinity Control Problems’, IEEE Transactions Automatic Control, 34, 831–847]. This article yields the central suboptimal H ∞ controller for linear systems with unknown parameters in a closed finite-dimensional form, based on the corresponding H 2 controller obtained in Basin and Calderon-Alvarez [Basin, M.V., and Calderon-Alvarez, D. (2008), ‘Optimal LQG Controller for Linear Systems with Unknown Parameters’, Journal of The Franklin Institute, 345, 293–302]. Numerical simulations are conducted to verify performance of the designed central suboptimal controller for uncertain linear systems with unknown parameters against the conventional central suboptimal H ∞ controller for linear systems with exactly known parameter values.

In this paper an adaptive control problem for the boundary or point control of a stochastic linear evolution system is formulated and the solution is described. The infinitesimal generator of the evolution system generates a C/sub 0/-semigroup that can model many linear hyperbolic systems and the noise in the system is a cylindrical white noise. The solution of the algebraic Riccati equation for the ergodic control problem with a quadratic cost functional is a continuous function of parameters. A family of least squares estimates of the unknown parameters is exhibited that is strongly consistent. A certainty equivalence adaptive control is constructed that is self-optimizing, that is, the family of average costs using this control converges (almost surely) to the optimal ergodic cost. >

For 1-D linear hyperbolic systems with constant coefficients we introduce the asymptotic controllability and the asymptotic zero controllability in L2 space under the lack of boundary controls and show the duality that they are equivalent, respectively, to the strong observability and the weak observability for the dual system. An example of 4×4 system with only one control is shown to be asymptotically controllable but not exactly controllable.

We consider the fractional controllability of linear hyperbolic systems with an interior pointwise control acting on a moving point ( b ( t ) ) t ≥ 0 in cases where the output function is a Riemann-Liouville fractional derivative of order β ∈ ( 0 , 1 ) . Hence, we employ a low-energy control strategy leveraging the Hilbert Uniqueness Method (HUM) to attain the desired fractional state at time T. In particular, if β = 0 , we obtain the exact controllability of the system state, while gradient controllability of the system state is obtained with β = 1. Furthermore, we approximate a control of minimal norm through a mixed formulation solved by using a Fourier series for the adjoint state linked to the HUM control to validate the practicality of the findings through numerical simulations based on the one-dimensional wave equation featuring a movable pointwise actuator.