Search

We study a system composed of a free quantum particle trapped in a box whose walls can change their position. We prove the global approximate controllability of the system: any initial pure state can be driven arbitrarily close to any target pure state in the Hilbert space of the free particle with a predetermined final position of the box. To this purpose we consider weak solutions of the Schrödinger equation and use a stability theorem for the time-dependent Schrödinger equation. [ABSTRACT FROM AUTHOR]

For a coupled system of wave equations with locally distributed controls, we prove that the exactly synchronizable state is independent of applied controls and then can be uniquely determined by the initial data if and only if the system is not approximately controllable or, equivalently, if and only if Kalman's rank condition (3.2) holds. [ABSTRACT FROM AUTHOR]

We consider the 3D Navier--Stokes system driven by an additive finite-dimensional control force. The purpose of this paper is to show how the approximate controllability of this system can be derived from the approximate controllability of the Euler system linearized around some suitable trajectory. The proof presented here is shorter than the previous ones obtained by Lie algebraic methods and gives some new information about the structure of the control. The dimension of the control space provided by this approach is larger, but it is still uniform with respect to the viscosity. [ABSTRACT FROM AUTHOR]

Let Ω ⊂ RN a bounded domain with a Lipschitz continuous boundary. We study the controllability of the space-time fractional diffusion equation {Dtαu + (−∆)su = 0 in (0, T ) × Ω, u = gχ(0, T ) ×O in (0, T ) × (RN \ Ω) u(0, ·) = u0 in Ω, where u = u(t, x) is the state to be controlled and g = g(t, x) is the control function which is localized in a subset O of RN \ Ω. Here, 0 < α ≤ 1, 0 < s < 1 and T > 0 be real numbers. After giving an explicit representation of solutions, we show that the system is always approximately controllable for every T > 0, u0 ∈ L²(Ω) and g ∈ D((0, T ) × O) where O ⊆ RN \ Ω is any open set. The results obtained are sharp in the sense that such a system is never null controllable if 0 < α < 1. The proof of our result is based on a new unique continuation principle for the eigenvalues problem associated with the fractional Laplace operator subject to the zero exterior boundary condition that we have established. [ABSTRACT FROM AUTHOR]

The improved Boussinesq equation is studied in this paper. Control properties for this equation posed on a bounded interval are first considered. When the control acts through the Dirichlet boundary condition the linearized system is proved to be approximately but not spectrally controllable. In a second part, the equation is posed on the one-dimensional torus and distributed moving controls are considered. Under some condition on the velocity at which the control moves, exact controllability results for both linear and nonlinear improved Boussinesq equations are obtained applying the moment method and a fixed point argument. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the control systems governed by fractional evolution differential equations involving Riemann--Liouville fractional derivatives in Banach spaces. Our main purpose in this article is to establish suitable assumptions to guarantee the existence and uniqueness results of mild solutions. Under these conditions, the approximate controllability of the associated fractional evolution systems involving Riemann--Liouville fractional derivatives is formulated and proved. [ABSTRACT FROM AUTHOR]

This paper is addressed to a study of the null/approximate controllability for a class of coupled systems governed by two linear forward stochastic parabolic equations with fractional order spatial differential operators. Our method is based on the Lebeau--Robbiano strategy. The key is to establish a suitable observability estimate for some coupled fractional order backward stochastic parabolic systems with terminal states in finite dimensional spaces. Compared to deterministic coupled parabolic systems, the coupling appearing in diffusion terms in the stochastic case introduces quite interesting new phenomena. We present a somewhat surprising counterexample to show that the controllability of coupled stochastic parabolic systems is not robust with respect to the coupling coefficient in diffusion terms. This indicates that the usual Carleman-type estimate approach does not seem to work for our controllability problem. Moreover, our controllability results for parabolic systems with fractional order spatial differential operators through one control are new, even when the considered system degenerates to a deterministic one. [ABSTRACT FROM AUTHOR]

We consider the approximate control of solitons in generalized Korteweg--de Vries equations. By introducing a suitable internal bilinear control on the equation, we prove that any soliton is approximate null controllable, and, moreover, any soliton can be accelerated to any particular positive velocity, after a suitable large amount of time. Precise estimates on the error terms and the rate of decay in the approximate null controllability result are also given. Our method introduces a new insight into the control of nonlinear objects from the point of view of interaction and collision problems for nonlinear dispersive equations, recently developed by Martel and Merle [Ann. Math. (2), 174 (2011), pp. 757-857], [Invent. Math., 183 (2011), pp. 563-648]. It can be applied in principle to several other models with soliton solutions. [ABSTRACT FROM AUTHOR]

In this paper we consider the one-dimensional compressible Navier-Stokes system linearized about a constant steady state (Q0, 0) with Q0 > 0. We study the controllability and stabilizability of this linearized system. We establish that the linearized system is null controllable for regular initial data by an interior control acting everywhere in the velocity equation. We prove that this result is sharp by showing that the null controllability cannot be achieved by a localized interior control or by a boundary control acting only in the velocity equation. On the other hand, we show that the system is approximately controllable. We also show that the system is not stabilizable with a decay rate e-ωt for ω > ω0, where ω0 is an accumulation point of the real eigenvalues of the linearized operator. [ABSTRACT FROM AUTHOR]

We study an optimal control problem for viscosity solutions of a Hamilton-Jacobi equation describing the propagation of a one-dimensional graph with the control being the speed function. The existence of an optimal control is proved together with an approximate controllability result in the H-1-norm. We prove convergence of a discrete optimal control problem based on a monotone finite difference scheme and describe some numerical results. [ABSTRACT FROM AUTHOR]

This paper addresses a study of the controllability of a class of Newtonian filtration equations, with nonnegative constraints on the control and state variables. When the control enters the system through the whole domain where the equation evolves, we characterize the set of nonnegative targets which are approximately controllable at any time T > 0. The proof combines the Fenchel–Rockafellar duality theory and a fixed point argument. When the control is restricted to be active in a proper open subset of the whole domain, we prove a negative controllability result by means of a localization technique which reflects the underlying obstruction phenomenon in the system. [ABSTRACT FROM AUTHOR]

In this paper, we study the approximate controllability and noncontrollability for a class of nonlinear degenerate integrodifferential control systems. Based on the property of finite propagation for the disturbances, we prove noncontrollability. We also establish sufficient conditions for approximate controllability of the system in L2 (Ω). Moreover, we prove the global existence and uniqueness of generalized solutions. [ABSTRACT FROM AUTHOR]

The new necessary and sufficient conditions, which are formulated in terms of convergence of a certain sequence of operators involving the resolvent of the negative of the controllability operator, are found for deterministic linear stationary control systems to be completely and approximately controllable, respectively. These conditions are applied to study the S-controllability (a property of attaining an arbitrarily small neighborhood of each point in the state space with a probability arbitrarily close to one) and C-controllability (the S-controllability fortified with some uniformity) of stochastic systems. It is shown that the S-controllability (the C-controllability) of a partially observable linear stationary control system with an additive Gaussian white noise disturbance on all the intervals [0,T] for T > 0 is equivalent to the approximate (complete) controllability of its deterministic part on all the intervals [0,T] for T > 0. [ABSTRACT FROM AUTHOR]