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Exponential stabilization to time-dependent trajectories for the incompressible Navier–Stokes equations is achieved with explicit feedback controls. The fluid is contained in a given two-dimensional spatial domain. An appropriate finite number of actuators is constructed and the control force is, at each time instant, a linear combination of such actuators. Each actuator has its vorticity supported in a small subdomain. The velocity field is subject to Lions boundary conditions. Simulations are presented showing the stabilizing performance of the proposed feedback. The results also apply to a class of observer design problems. [ABSTRACT FROM AUTHOR]

In this paper, we consider the exact boundary controllability of nodal profile for nonautonomous quasilinear hyperbolic systems on a finite time interval and obtain that there is the exact boundary controllability of nodal profile for any given initial time t 0 ∈ R so long as the controllability time T > 0 is suitably large, but T depends on t 0 in general. As applications, we discuss the exact boundary controllability of nodal profile for the Saint-Venant system with a nonuniform subcritical steady state and the system of traffic flow depending on time. [ABSTRACT FROM AUTHOR]

In this paper, we study the impulse controllability of a multi-dimensional heat equation with dynamic boundary conditions in a bounded smooth domain. Using a recent approach based on finite-time stabilization, we show that the system is impulse null controllable at any positive time via impulse controls supported in a nonempty open subset of the physical domain. Furthermore, we infer an explicit estimate for the exponential decay of the solution. The proof of the main result combines a logarithmic convexity estimate and some spectral properties associated to dynamic boundary conditions. In our setting, the nature of the equations, which couple intern-boundary phenomena, makes it necessary to go into quite sophisticated estimates incorporating several boundary terms. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the performance output tracking for a Schrödinger PDE-ODE cascaded system with external disturbances enter in all possible channels. The main challenge of the problem is the fact that the disturbances are non-collocated to the controller. By proper trajectory planning approach, this difficulty can be overcome by converting non-collocated configurations into the collocated ones. Then a state observer is designed in terms of the tracking errors. Finally, the feedback control is proposed by applying the backstepping technique. The stability of the closed-loop system and the exponential convergence of the regulation error are proved. Some numerical simulations illustrate that the proposed approach is very effective. [ABSTRACT FROM AUTHOR]

This paper considers the feedback stabilization of infinite-dimensional semi-linear systems with discrete multi-delays. Sufficient conditions for an appropriate sequence of feedback control are given to ensure exponential and weak stabilization of the considered system. Moreover, applications with numerical simulation to wave equations and heat equation are presented. [ABSTRACT FROM AUTHOR]

In this paper, we consider the approximation of abstract thermoelastic models. It is by now well known that approximated systems are not in general uniformly exponentially or polynomially stable with respect to the discretization parameter, even if the continuous system has this property. Our goal in this paper is to study the uniform exponential/polynomial stability of a sequence of a system of weakly coupled thermoelastic models. We prove that when 0 ≤ β < 1 2 , the total energy of solutions is not uniformly exponentially stable, but it decays uniformly polynomially to zero. Finally, the results are applied to space semi-discretizations of thermoelastic beam equation in a bounded interval with homogeneous Dirichlet boundary conditions. We consider finite element, spectral element and finite difference semi-discretizations. Finally, we illustrate the mathematical results with several numerical experiments. [ABSTRACT FROM AUTHOR]

We prove the controllability and the existence of the time optimal control for semilinear parabolic equations with the homogenous Neumann boundary condition via multiplicative controls. It is worth pointing out that there is no restriction on the growth of the nonlinearity f(s) with respect to the variable s in the equation, which is a remarkable difference compared to the semilinear parabolic system with additive locally distributed controls. [ABSTRACT FROM AUTHOR]

A class of coupled parabolic PDEs with time delay is considered. We treat the problem of boundary exponential stabilization where the goal is to present a state feedback law with actuation on only one end of the domain which provide the exponential stability of the closed-loop system. We consider both the Dirichlet and Neumann boundary conditions. The corresponding proposed control law is given in explicit form. Numerical simulations are carried out to show the effectiveness of the obtained results. [ABSTRACT FROM AUTHOR]

This paper deals with the application of Stackelberg-Nash strategies to the control to quasi-linear parabolic equations in dimensions 1D, 2D, or 3D. We consider two followers, intended to solve a Nash multi-objective equilibrium; and one leader satisfying the controllability to the trajectories. [ABSTRACT FROM AUTHOR]

This paper deals with the controllability of the second-grade fluids, a class of non-Newtonian of differential type, on a two-dimensional torus. Using the method of Agrachev and Sarychev (J. Math Fluid Mech., 7(1):108–52 (2005)), Agrachev and Sarychev (Commun Math Phys., 265(3):673–97 (2006)), and of Shirikyan (Commun Math Phys., 266(1):123–51 (2006)), we prove that the system of second-grade fluids is approximately controllable by a finite-dimensional control force. [ABSTRACT FROM AUTHOR]

In this paper, we study the approximate controllability of a class of semilinear systems governed by coupled degenerate parabolic equations. The equations may be weakly degenerate and strongly degenerate on a portion of the lateral boundary. We prove that the control systems are approximately controllable by using the Kakutani fixed point theorem and the controls can be taken to be of quasi bang-bang form. [ABSTRACT FROM AUTHOR]

In this paper, we consider the output feedback exponential stabilization problem of ODE-Schrödinger cascade systems with the external disturbance. We propose a new extended state observer (ESO) that estimates both state and disturbance by the three output signals, then design a stabilizing control law by utilizing the backstepping technique. The resulting closed-loop system is shown to be exponentially stable guaranteeing that all internal systems involved are uniformly bounded. Finally, some numerical experiments are carried out to verify the effectiveness of the proposed control law. [ABSTRACT FROM AUTHOR]

This paper investigates global strong stabilization of time-delayed bilinear systems on a real Hilbert state space. Sufficient conditions for strong global stabilization are formulated in terms of observability like assumptions. An explicit decay rate of the stabilized state is provided. Illustrative examples are presented. [ABSTRACT FROM AUTHOR]

The 3D Navier–Stokes system, under Lions boundary conditions, is proven to be approximately controllable provided a suitable saturating set does exist. An explicit saturating set for 3D rectangles is given. [ABSTRACT FROM AUTHOR]

In this paper, we consider a one-dimensional Schrödinger equation under a joint linear feedback control at an arbitrary internal point ξ (0 < ξ < 1) . It is shown that the pointwise control system is asymptotically stable if ξ is either an irrational number or a rational number satisfying ξ ≠ 2 l / (2 m − 1) for any positive integers l , m (1 ≤ l ≤ m − 1) ; further, the system is exponentially stable if ξ is a rational number satisfying ξ ≠ 2 l / (2 m − 1) for any positive integers l , m (1 ≤ l ≤ m − 1) . Moreover, we consider the Schrödinger equation under a joint feedback control where the observation is suffered from a given time delay. A Luenberger observer is designed at the time interval when the observation signal is available, while a predictor is designed at the time interval when the observation signal is not available. A natural control law is constructed based on the estimated state. The closed-loop system is shown to be exponentially stable for the smooth initial value. Finally, numerical simulations demonstrate effectiveness of the dynamic output feedback controller. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the controllability for the one-dimensional plate equation in intervals with a moving boundary. This equation models the vertical displacement of a point x at time t in a bar with uniform cross section. We assume the ends of the bar with small and uniform variations. More precisely, we have introduced functions α(t) and β(t) modeling the motion of these ends. We present the following results: the existence and uniqueness of Nash equilibrium, the approximate controllability with respect to the leader control, and the optimality system for the leader control. [ABSTRACT FROM AUTHOR]

We consider the output regulation of an unstable reaction-diffusion PDE in the presence of regulator delay and unmatched disturbances, which are generated by an exosystem. The systematic design procedure of a backstepping state feedback regulator is first presented by mapping the reaction-diffusion PDE cascaded with a transport equation into an error system, which is shown to be exponentially stable with a prescribed rate in a suitable Hilbert space. The regulator design relies on solving regulator equations, and the solvability condition of the regulator equations is then characterized by a transfer function and eigenvalues of the exosystem. Finally, the numerical simulations are provided to illustrate the effect of the regulator. [ABSTRACT FROM AUTHOR]

In this paper, we study the null controllability and approximate controllability for a class of weakly degenerate parabolic equations with memory by means of boundary controls. Unlike the known result for the degenerate parabolic equation, the degenerate parabolic equation with memory in general is not null controllable. This is based on the observability inequality for the adjoint system, which does not hold in the corresponding space. On the other hand, we prove the approximate controllability property of it in a suitable state space with a boundary control, which acts on the degenerate boundary or the nondegenerate boundary. [ABSTRACT FROM AUTHOR]

This paper deals with the null controllability of the system governed by coupled degenerate equations with the first order terms. The null controllability of the system with two controls is proved firstly, then the system with one control is shown to be null controllable by constructing the control function. To investigate the null controllability, we establish the Carleman estimate and the observability inequality for the adjoin system by using the Carleman estimate for the single degenerate equation. [ABSTRACT FROM AUTHOR]

We design here explicit finite-dimensional boundary stabilizing feedbacks for the parabolic Poiseuille profile of an infinite 3-D channel flow. The controller acts on the normal component of the velocity, and it is easy to manipulate from the computational point of view. The stability is achieved without any a priori condition on the viscosity coefficient, that is on the Reynolds number; however, the lower the value of the viscosity is, the higher the dimension of the feedback is. [ABSTRACT FROM AUTHOR]

In this paper, we consider the output-feedback exponential stabilization of Timoshenko beam with the boundary control and input distributed delay. Suppose that the outputs of controllers are of the forms $\alpha _{1}u_{1}(t)+\beta _{1}u_{1}(t-\tau )+{\int }_{-\tau }^{0}g_{1}(\eta )u_{1} (t+\eta )d\eta $ and $\alpha _{2}u_{2}(t)+\beta _{2}u_{2}(t-\tau ) +{\int }_{-\tau }^{0}g_{2}(\eta )u_{2}(t+\eta )d\eta $ respectively, where u( t) and u( t) are the inputs of controllers. Using the tricks of the Luenberger observer and partial state predictor, we translate the system with delay into a system without delay. And then, we design the feedback controls to stabilize the system without delay. Finally, we prove that under the choice of such controls, the original system also is stabilized exponentially. [ABSTRACT FROM AUTHOR]

In this work, the stabilization problem of the nonlinear vibrating Timoshenko systems of Kirchhoff-type with boundary control conditions is considered. By virtue of the multiplier method, the explicit energy decay rates for solutions of the system are established, depending on boundary control feedback. In the view of control, the result of this work implies that, by choosing suited feedback boundary controls, the Kirchhoff-type Timoshenko system can be achieved by various decay rates, not only exponential and polynomial. [ABSTRACT FROM AUTHOR]

We will consider the exact controllability of the distributed system, governed by string equation with memory. It will be proved that this mechanical system can be driven to an equilibrium point in a finite time, the absolute value of the distributed control function being bounded. In this case, the memory kernel is a linear combination of exponentials. [ABSTRACT FROM AUTHOR]

We consider a problem of stabilization of a one-dimensional anti-stable linearized Schrödinger equation subject to boundary control. The controller is designed through the estimated state and is designed in the case that only displacement is available. The method of 'backstepping' is adopted in the investigation. We then combine the control and observer designs into an output-feedback compensator and prove exponential stability of the closed-loop system. [ABSTRACT FROM AUTHOR]

The dynamical behavior of a kind of elastic hybrid system is considered. This system consists of a nonhomogeneous Timoshenko beam linked at its boundary to a rigid body. The non-collocated input terms are required in the boundary feedback controls. It is proved that this closed loop system is well-posed. By a complete eigenfrequency analysis, it is shown that this infinite-dimensional hybrid system satisfies spectrum-determined growth condition. The stability of this system is discussed and some simulations are given to show that this kind of hybrid system can be exponentially stable under certain condition. [ABSTRACT FROM AUTHOR]

In this paper we consider a star-shaped network of Timoshenko beams, with feedback acting on the common vertex of the star. Suppose that the exterior vertices of this network are clamped, and at the common vertex the displacement and rotational angles are continuous. For this network, feedback controllers are designed at the common vertex so as to move the beams back to their equilibrium position. We show that the operator determined by the closed loop system has a compact resolvent and generates a C0 semigroup in an appropriate Hilbert space. Under certain conditions, we prove that the closed loop system is asymptotically stable. We also show that there is a sequence of the generalized eigenvectors of the system operator, which forms a Riesz basis with parentheses. Hence the spectrum determined growth condition holds. [ABSTRACT FROM AUTHOR]

We consider the Timoshenko system without any internal or boundary damping. The system is rather complemented by a heat equation of an unusual type. This type of thermal diffusion has been introduced by Green and Naghdi. It produces a dissipation of energy only when supplemented by a certain type of damping. In this work we consider the case of a thermo-viscoelastic damping and prove that it is strong enough to drive the system to the equilibrium state in an exponential manner. [ABSTRACT FROM AUTHOR]