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We study the output feedback exponential stabilization of a one-dimensional unstable wave equation, where the boundary input, given by the Neumann trace at one end of the domain, is the sum of the control input and the total disturbance. The latter is composed of a nonlinear uncertain feedback term and an external bounded disturbance. Using the two boundary displacements as output signals, we design a disturbance estimator that does not use high gain. It is shown that the disturbance estimator can estimate the total disturbance in the sense that the estimation error signal is in L2[0;1). Using the estimated total disturbance, we design an observer whose state is exponentially convergent to the state of original system. Finally, we design an observer-based output feedback stabilizing controller. The total disturbance is approximately canceled in the feedback loop by its estimate. The closed-loop system is shown to be exponentially stable while guaranteeing that all the internal signals are uniformly bounded. [ABSTRACT FROM AUTHOR]

In this paper, we consider boundary feedback stabilization for unstable time fractional reaction diffusion equations. New state feedback controls with actuation on one end are designed by the backstepping method for both Dirichlet and Neumann boundary controls. By the Riesz basis approach and the fractional Lyapunov method, we prove t he existence and uniqueness and the Mittag--Leffler stability for the closed-loop systems. For both cases, the observers and the observerbased output feedback are designed to stabilize the systems. [ABSTRACT FROM AUTHOR]

We study Jeffreys-type overdamped second order linear systems with observed outputs in the setting of Hilbert spaces. The state equation comes from an overdamped second order linear partial differential equation which is wave-like but was proposed to describe heat conduction. It results from adopting the Jeffreys law of constitutive relation for heat flux, rather than the usual Fourier law. Sufficient conditions for infinite-time admissibility of the system observation operator and system observability are obtained. In the general case, we obtain the infinite-time admissibility from that of the first order Cauchy system, which is done by employing the Hardy space approach. In the special case when the operator in the state equation is negative definite, we derive the infinite-time admissibility and system observability using a semigroup approach. Illustrative examples are given. [ABSTRACT FROM AUTHOR]

This article is devoted to investigating the existence of solutions to fractional multi-point boundary-value problems at resonance in a Hilbert space. More precisely, the dimension of the kernel of the fractional differential operator with the boundary conditions be any positive integer. We point out that the problem is new even when the system under consideration is reduced to a second-order ordinary differential system with resonant boundary conditions. We show that the considered system admits at least a solution by applying coincidence degree theory first introduced by Mawhin. An example is presented to illustrate our results. [ABSTRACT FROM AUTHOR]

In this paper, by applying the coincidence degree theory which was first introduced by Mawhin, we obtain an existence result for the fractional three-point boundary value problems in ℝn, where the dimension of the kernel of fractional differential operator with the boundary conditions can take any value in {1,2,... n}. This is our novelty. Several examples are presented to illustrate the result. [ABSTRACT FROM AUTHOR]

In this paper, an algorithm is developed to reject time and spatially varying boundary disturbances from a multidimensional Kirchhoff plate via boundary control. The disturbance and control input are assumed to be matched. The active disturbance rejection control approach is adopted for developing the algorithm. A state feedback scheme is designed to estimate the disturbance based on an infinite number of ordinary differential equations obtained from the original multidimensional system using infinitely many time-dependent test functions. The proposed control law cancels the disturbance using its estimated value. All subsystems in the closed loop are shown to be asymptotically stable. Simulation results are presented to validate the theoretical conclusions and to exhibit the reduction in the peaking phenomenon due to the use of time varying gains instead of constant high gains. [ABSTRACT FROM AUTHOR]

We study a class of integro--differential equation systems in Hilbert spaces which fits the Gurtin--Pipkin model of heat conduction with memory. Well-posedness of the system state equation is established using the operator semigroup approach. Sufficient conditions are then obtained under which infinite-time admissibility of the system observation operator follows from finite-time admissibility of the same observation operator for the associated second order Cauchy system without the first order term and convolution terms. This is done by combining the semigroup approach with the Hardy space method. An illustrative example is given, and the obtained theoretical results are verified by numerical simulations. [ABSTRACT FROM AUTHOR]

A linear control system with quadratic cost functional over infinite time horizon is considered without assuming a controllability/stabilizability condition and the global integrability condition for the nonhomogeneous term of the state equation and the weight functions in the linear terms in the running cost rate function. Classical approaches do not apply for such problems. Existence and nonexistence of overtaking optimal controls in various cases are established. Some concrete examples are presented. These results show that the overtaking optimality approach can be used to solve some of the above-mentioned problems and at the same time, the limitation of this approach is also revealed. [ABSTRACT FROM AUTHOR]