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This paper studies the Mittag-Leffler stabilization for unstable infinite dimensional systems actuated by boundary controllers described by time fractional reaction diffusion equations. Via the Riesz basis method, the stable and the unstable part of the considered system are separated. The Kalman rank criterion, which is a classical linear algebra condition, guarantees the stabilizability of the unstable subsystem. Based on these, a controller governed by a finite dimensional system (so called the finite dimensional controller hereafter) is designed to achieve the Mittag-Leffler stability of the closed loop. From infinite to finite, this methodology is a substantial improvement for the existing control laws.

This paper considers boundary control of time fractional order diffusion–wave equation. Both boundary stabilization and disturbance rejection are considered. This paper, for the first time, has confirmed, via hybrid symbolic and numerical simulation studies, that the existing two schemes for boundary stabilization and disturbance rejection for (integer order) wave/beam equations are still valid for time fractional order diffusion–wave equations. The problem definition, the hybrid symbolic and numerical simulation techniques, outlines of the methods for boundary stabilization and disturbance rejection are presented together with extensive simulation results. Different dynamic behaviors are revealed for different fractional orders which may stimulate new future research opportunities.

This paper considers performance output tracking and disturbance rejection for a boundary controlled one-dimensional Euler–Bernoulli beam equation suffered unknown external bounded disturbance. By first proposing a disturbance estimator and then based on this disturbance estimator, we construct a servomechanism to track the reference signal and reject the external disturbance. Four control objectives are achieved: a) the output is exponentially tracking the reference signal; b) the disturbance is rejected; c) all the states of internal-loops are bounded; d) when the disturbance and reference are disconnected, the closed-loop is exponentially stable. Finally, the state of the system is shown to be exponentially tracking the reference state.

This paper is concerned with Caputo–Hadamard fractional Halanay inequality. A useful fractional derivative inequality regarding on x 2 in the Caputo–Hadamard sense is obtained, which makes Halanay inequality more applicable to choose the Lyapunov function and to study the stability of fractional system.

In this paper, we are concerned with the output feedback exponential stabilization problem for a system (plant) described by a one-dimensional Euler–Bernoulli beam equation. The measurements are only the displacement and the angular velocity at the right end. An infinite dimensional estimator is designed to estimate the disturbance. With the estimated disturbance, we propose a state observer that is exponentially convergent to the original system, then design two different kinds of stabilizing controllers: one is based on the velocity feedback, the other is based on the angular velocity feedback. In both cases, by adopting the Riesz basis approach, the exponential stability of the closed-loop systems is built with guaranteeing that all internal systems are uniformly bounded. The numerical experiments are carried out to illustrate the theoretical results.

In this paper, we consider performance output tracking for a boundary controlled one-dimensional wave equation with possibly unknown internal nonlinear uncertainty and external disturbance. We first show that the open-loop system is well-posed and then propose a disturbance estimator. It is shown that the disturbance estimator can estimate successfully the total disturbance that consists of internal uncertainty and external disturbance. An servomechanism based on the estimated total disturbance is then designed. It is shown that the closed-loop system is well-posed. Three control objectives are achieved: (a) the output is tracking the reference signal; (b) all the internal signals are uniformly bounded; (c) the closed-loop system is internally asymptotically stable if both the reference signal and the disturbance vanish or belong to the space H2(0, ∞) and L2(0, ∞), respectively. The unmatched performance output tracking control is first time applied to a system described by the partial differential equation for complete general disturbance rejection and reference tracking purpose. Another key feature of this paper is that we do not use the high-gain to estimate total disturbance for unmatched system. The numerical experiments are carried out to illustrate effectiveness of the proposed control law.

In this paper, we consider boundary output feedback stabilization for a multi-dimensional wave equation with boundary control matched unknown nonlinear internal uncertainty and external disturbance. A new unknown input type extended state observer is proposed to recover both state and total disturbance which consists of internal uncertainty and external disturbance. A key feature of the proposed observer in this paper is that we do not use the high-gain to estimate the disturbance. By the active disturbance rejection control (ADRC) strategy, the total disturbance is compensated (canceled) in the feedback loop, which together with a collocated stabilizing controller without uncertainty, leads to an output feedback stabilizing feedback control. It is shown that the resulting closed-loop system is well-posed and asymptotically stable under weak assumption on internal uncertainty and external disturbance. The numerical experiments are carried out to show the effectiveness of the proposed scheme.

In this paper, we consider boundary output feedback stabilization for a multidimensional wave equation with boundary control matched unknown nonlinear internal uncertainty and external disturbance. A new unknown input type extended state observer is proposed to recover both state and total disturbance which consists of internal uncertainty and external disturbance by using the hidden regularity of the wave equation. By the active disturbance rejection control (ADRC) strategy, the total disturbance is compensated (canceled) in the feedback loop, which together with a collocated stabilizing controller without uncertainty, leads to an output feedback stabilizing feedback control. It is shown that the resulting closed-loop system is well-posed and asymptotically stable under weak assumption on internal uncertainty and external disturbance. The numerical experiments are carried out to show the effectiveness of the proposed scheme.

This paper develops systematically the output feedback exponential stabilization for a one-dimensional unstable/anti-stable wave equation where the control boundary suffers from both internal nonlinear uncertainty and external disturbance. Using only two displacement signals, we propose a disturbance estimator that not only can estimate successfully the disturbance in the sense that the error is in L2(0, ∞) but also is free high-gain. With the estimated disturbance, we design a state observer that is exponentially convergent to the state of original system. An observer-based output feedback stabilizing control law is proposed. The disturbance is then canceled in the feedback loop by its approximated value. The closed-loop system is shown to be exponentially stable and it can be guaranteed that all internal signals are uniformly bounded.

We solve the output regulator problem for a system described by a Neumann boundary controlled anti-stable one-dimensional Schrodinger equation by the backstepping approach. The output to be controlled is not required to be measured and its observation operator is assumed to be admissible for a certain Co-semigroup that is related to the original system, while the measurements used for control are located at the boundary First we show that the open-loop system is well-posed. Then we find a state feedback control law to solve the output regulator problem by the backstepping method. We design a finite-dimensional reference observer and an infinite-dimensional disturbance observer. Based on this observer, we propose an output feedback controller to achieve output regulation.

This work aims to design the disturbance rejection controllers for three classes of fractional heat equations. Based on Filippov’s theory, the existence conclusion for the partial differential inclusion solution (PDIS) is established for fractional heat equations with discontinuous boundary conditions. Boundary control strategies are designed directly without the use of any robust control method to respectively achieve the power-law type stabilization and the asymptotical stabilization for fractional heat equations without and with time delay, respectively. A numerical example is included to illustrate the obtained results.

We consider boundary output feedback stabilization for a multi-dimensional Kirchhoff plate with boundary observation suffered from a general external disturbance. We adopt for the first time the active disturbance rejection control approach to stabilization of multi-dimensional partial differential equations under corrupted output feedback. In terms of this approach, the disturbance is estimated by a relatively independent estimator, based on (possibly) an infinite number of ordinary differential equations reduced from the original PDEs by infinitely many time-dependent test functions. This gives a state observer, an additional result via this approach. The disturbance is compensated in the feedback-loop. As a result, the control law can be designed almost as the same as that for the system without disturbance. We show that with a time varying gain properly designed, the observer driven by the disturbance estimator is convergent; and that all subsystems in the closed-loop are asymptotically stable. We also provide numerical simulations which demonstrate the convergence results and underline the effect of the time varying high gain estimator.

This paper investigates the controllability of fractional damped dynamical systems with control delay. Necessary and sufficient conditions for the controllability of fractional damped dynamical systems with control delay are established. The obtained conditions are easily checkable. An example is also given to illustrate the theory.

This paper is concerned with the approximate controllability of the semilinear fractional evolution equations with nonlocal and impulsive conditions. Our main results are obtained by utilizing the technique of approximate solution and the theory of fixed point. In addition, the impulsive functions in this paper are supposed to be continuous and the nonlocal item is divided into two cases: Lipschitz continuous and only continuous, which generalizes the previous contributions. Finally two examples are worked out to illustrate our obtained results.

We consider parameter estimation and stabilization for a one-dimensional Schrodinger equation with an unknown constant disturbance suffered from the boundary observation at one end and the non-collocated control at other end. An adaptive observer is designed in terms of measured position with unknown constant by the Lyapunov functional approach. By a backstepping transformation for infinite-dimensional systems, it is shown that the resulting closed-loop system is asymptotically stable. Meanwhile, the estimated parameter is shown to be convergent to the unknown parameter as time goes to infinity. The numerical experiments are carried out to illustrate the proposed approach.

We consider boundary stabilization for a multi-dimensional wave equation with boundary control matched disturbance that depends on both time and spatial variables. The active disturbance rejection control (ADRC) approach is adopted in investigation. An disturbance estimator is designed to estimate, in real time, the disturbance, and the disturbance is canceled in the feedback loop with its approximation. All subsystems in the closed-loop are shown to be asymptotically stable. The numerical experiments are carried out to illustrate the convergence and effect of peaking value reduction.

The main aim of this paper is to study the global existence of solutions of initial value problems for nonlinear fractional differential equations(FDEs) on the half-axis, which is fundamental in the basic theory of FDEs and important in stability analysis of this kind of equations. In this paper, we are concerned with the nonlinear FDE D 0 + α x ( t ) = f ( t , x ) , t ∈ ( 0 , + ∞ ) , 0 α ≤ 1 , where D 0 + α is the standard Riemann–Liouville fractional derivative, subject to the initial value condition lim t → 0 + t 1 − α x ( t ) = u 0 . By constructing a special Banach space and employing fixed-point theorems, some sufficient conditions are obtained to guarantee the global existence of solutions on the interval [ 0 , + ∞ ) . Moreover, in the case α = 1 , existence results of solutions of initial value problems for ordinary differential equations on the half-axis are also obtained. An interesting example is also included.

In this comment, we point out that there are four errors leading to question of the validity of Theorem 3 and Proposition 4 of a recent paper Ran, Wang, and Dong (2016a).