Sufficient Conditions For Null Controllability Research Articles

Null Controllability of Hilfer Fractional Stochastic Differential Inclusions

This paper gives the null controllability for nonlocal stochastic differential inclusion with the Hilfer fractional derivative and Clarke subdifferential. Sufficient conditions for null controllability of nonlocal Hilfer fractional stochastic differential inclusion are established by using the fixed-point approach with the proof that the corresponding linear system is controllable. Finally, the theoretical results are verified with an example.

Null controllability of Hilfer fractional stochastic integrodifferential equations with noninstantaneous impulsive and Poisson jump

Abstract In this paper, we investigate the sufficient conditions for null controllability of noninstantaneous impulsive Hilfer fractional stochastic integrodifferential system with the Rosenblatt process and Poisson jump. The required results are obtained based on fractional calculus, stochastic analysis, and Sadovskii’s fixed point theorem. Finally, an example is given to illustrate the obtained results.

Geometric conditions for the null-controllability of hypoelliptic quadratic parabolic equations with moving control supports

We study the null-controllability of some hypoelliptic quadratic parabolic equations posed on the whole Euclidean space with moving control supports, and provide necessary or sufficient geometric conditions on the moving control supports to ensure null-controllability. The first class of equations is the one associated to non-autonomous Ornstein–Uhlenbeck operators satisfying a generalized Kalman rank condition. In particular, when the moving control supports comply with the flow associated to the transport part of the Ornstein–Uhlenbeck operators, a necessary and sufficient condition for null-controllability on the moving control supports is established. The second class of equations is the class of accretive non-selfadjoint quadratic operators with zero singular spaces for which some sufficient geometric conditions on the moving control supports are also given to ensure null-controllability.

Null controllability results for stochastic delay systems with delayed perturbation of matrices

Abstract In this paper, we adopt a notation of delayed perturbation of matrices to study controllability of stochastic delay systems in finite dimensional spaces. By using the delayed perturbation of matrix, we write an explicit formula of solutions to linear stochastic delay systems. We present the necessary and sufficient conditions for null controllability of linear stochastic delay systems by using perturbed controllability matrix and rank correlation method. Thereafter, sufficient conditions are derived for null controllability of semilinear stochastic delay systems under the proved results of corresponding linear stochastic system is null controllable, stochastic linear controllability operator, appropriate hypothesis on semilinear term and stochastic delay analysis techniques. The obtained theoretical results is verified through numerical simulation.

On null Controllability of fractional stochastic differential system with fractional Brownian motion

ABSTRACT In this paper, we are dealing with the null controllability of fractional stochastic differential system with fractional Brownian motion. The null controllability for Sobolev-type Hilfer fractional stochastic differential system with fractional Brownian motion of Clarke's subdifferential type is studied. The sufficient conditions for null controllability of fractional stochastic differential system with fractional Brownian motion and control on the boundary are established. Finally, an example is given to illustrate the obtained results.

On the lack of controllability of fractional in time ODE and PDE

This paper is devoted to the analysis of the problem of controllability of fractional (in time) ordinary and partial differential equations (ODE/PDE). The fractional time derivative introduces some memory effects on the system that need to be taken into account when defining the notion of control. In fact, in contrast with the classical ODE and PDE control theory, when driving these systems to rest, one is required not only to control the value of the state at the final time, but also the memory accumulated by the long-tail effects that the fractional derivative introduces. As a consequence, the notion of null controllability to equilibrium needs to take into account both the state and the memory term. The existing literature so far is only concerned with the problem of partial controllability in which the state is controlled, but the behavior of the memory term is ignored. In the present paper, we consider the full controllability problem and show that, due to the memory effects, even at the ODE level, controllability cannot be achieved in finite time. This negative result holds even for finite-dimensional systems in which the control is of full dimension. Consequently, the same negative results hold also for fractional PDE, regardless of their parabolic or hyperbolic nature. This negative result exhibits a completely opposite behavior with respect to the existing literature on classical ODE and PDE control where sharp sufficient conditions for null controllability are well known.

On the boundary controllability of non-scalar parabolic systems

This Note is concerned with the boundary controllability of non-scalar linear parabolic systems. More precisely, two coupled one-dimensional linear parabolic equations are considered. We show that, with boundary controls, the situation is much more complex than for similar distributed control systems. In our main result, we provide necessary and sufficient conditions for null controllability. To cite this article: E. Fernández-Cara et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).

Tangent sets, viability for differential inclusions and applications

Let X be a real Banach space, let A : D ( A ) ⊆ X ↝ X be a given operator, K a nonempty and possible non-open subset in D ( A ) ¯ , F : K ↝ X a given multi-function. In this lecture, we consider the differential inclusion, u ′ ( t ) ∈ A u ( t ) + F ( u ( t ) ) , with (a) A = 0 , (b) A linear and (c) A nonlinear and, in each one of these cases, we give a short survey of the most important and very recent necessary and sufficient conditions for viability expressed in terms of tangent sets and A - quasi-tangent sets to K at a given point ξ ∈ K , concepts recently introduced by the authors. From a rather long list of applications, we confined ourselves only to: solutions in moving sets, a comparison result for a reaction–diffusion system, a comparison result for a nonlinear diffusion inclusion and a sufficient condition for null controllability.

Necessary and Sufficient Conditions for Viability for Nonlinear Evolution Inclusions

We consider the viability problem for nonlinear evolutions inclusions of the form u′(t) ∈ Au(t) + F(u(t)), where A is an m-dissipative (possible nonlinear and multi-valued) operator acting in a Banach space X, K⊆X is a nonempty, locally closed set and \(F:K{\user1{ \rightsquigarrow }}X\) is with nonempty, convex, closed and bounded values. We define the concept of A-quasi-tangent set to K at a given point ξ ∈ K and we prove a necessary condition for C0-viability expressed in terms of this new tangency concept. We next show that, under various natural extra-assumptions, the necessary condition is also sufficient. We extend the results to the quasi-autonomous case, we deal with the existence of noncontinuable or even global C0-solutions and, as applications, we deduce a comparison result and a sufficient condition for null controllability.

Queueing systems with many servers: Null controllability in heavy traffic

A queueing model has J≥2 heterogeneous service stations, each consisting of many independent servers with identical capabilities. Customers of I≥2 classes can be served at these stations at different rates, that depend on both the class and the station. A system administrator dynamically controls scheduling and routing. We study this model in the central limit theorem (or heavy traffic) regime proposed by Halfin and Whitt. We derive a diffusion model on ℝI with a singular control term that describes the scaling limit of the queueing model. The singular term may be used to constrain the diffusion to lie in certain subsets of ℝI at all times t>0. We say that the diffusion is null-controllable if it can be constrained to $\mathbb {X}_{-}$, the minimal closed subset of ℝI containing all states of the prelimit queueing model for which all queues are empty. We give sufficient conditions for null controllability of the diffusion. Under these conditions we also show that an analogous, asymptotic result holds for the queueing model, by constructing control policies under which, for any given 0<ɛ<T<∞, all queues in the system are kept empty on the time interval [ɛ,T], with probability approaching one. This introduces a new, unusual heavy traffic “behavior”: On one hand, the system is critically loaded, in the sense that an increase in any of the external arrival rates at the “fluid level” results with an overloaded system. On the other hand, as far as queue lengths are concerned, the system behaves as if it is underloaded.

Controllability for a Class of Parallelly Connected Polynomial Systems

Null controllability for a class of parallelly connected discrete-time polynomial systems is considered. We prove for this class of systems that a necessary and sufficient condition for null controllability of the parallel connection is that all its subsystems are null controllable. Consequently, the controllability test splits into a number of easy-to-check tests for the subsystems. The test for complete controllability is also presented and it is subtly different from the null controllability test. A similar statement is then given for complete controllability of a class of parallelly connected continuous-time polynomial systems. The result is somewhat unexpected when compared with the classical linear systems result. We identify the phenomenon which shows the difference between the linear and nonlinear cases.

Null controllability of semilinear integrodifferential systems in Banach space

Sufficient conditions for null controllability of semilinear integrodifferential systems with unbounded linear operators in Banach space are established. The results are obtained using semigroup of linear operators, fractional powers of operators, and the Schauder fixed point theorem. An application to partial integrodifferential equations is given.

Finite-time null controllability for a class of linear evolution equations on a Banach space with control constraints

We present some necessary and sufficient conditions for null controllability for a class of general linear evolution equations on a Banach space with constraints on the control space. We also present a result on the existence of time-optimal controls and some partial results on the maximum principle. Some interesting insights that can be obtained from these results are discussed, and the paper is concluded with an application to a boundary control problem.