Conditions For Null Controllability Research Articles

Sufficient Criteria for Stabilization Properties in Banach Spaces

We study abstract sufficient criteria for cost-uniform open-loop stabilizability of linear control systems in a Banach space with a bounded control operator, which build up and generalize a sufficient condition for null-controllability in Banach spaces given by an uncertainty principle and a dissipation estimate. For stabilizability these estimates are only needed for a single spectral parameter and, in particular, their constants do not depend on the growth rate w.r.t. this parameter. Our result unifies and generalizes earlier results obtained in the context of Hilbert spaces. As an application we consider fractional powers of elliptic differential operators with constant coefficients in Lp(Rd)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_p(\\mathbb {R}^d)$$\\end{document} for p∈[1,∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p\\in [1,\\infty )$$\\end{document} and thick control sets.

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 Algebraic condition for null controllability of some coupled degenerate systems

In this paper we will generalize the Kalman rank condition for the null controllability to $n$-coupled linear degenerate parabolic systems with constant coefficients, diagonalizable diffusion matrix, and $m$-controls. For that we prove a global Carleman estimate for the solutions of a scalar $2n$-order parabolic equation then we infer from it an observability inequality for the corresponding adjoint system, and thus the null controllability.

Sharp geometric condition for null-controllability of the heat equation on $$\mathbb {R}^d$$ R d and consistent estimates on the control cost

In this note we study the control problem for the heat equation on $\mathbb{R}^d$, $d\geq 1$, with control set $\omega\subset\mathbb{R}^d$. We provide a necessary and sufficient condition (called $(\gamma, a)$-\emph{thickness}) on $\omega$ such that the heat equation is null-controllable in any positive time. We give an estimate of the control cost with explicit dependency on the characteristic geometric parameters of the control set. Finally, we derive a control cost estimate for the heat equation on cubes with periodic, Dirichlet, or Neumann boundary conditions, where the control sets are again assumed to be thick. We show that the control cost estimate is consistent with the $\mathbb{R}^d$ case.

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.

Necessary condition for null controllability in many-server heavy traffic

Throughput sub-optimality (TSO), introduced in Atar and Shaikhet [Ann. Appl. Probab. 19 (2009) 521-555] for static fluid models of parallel queueing networks, corresponds to the existence of a resource allocation, under which the total service rate becomes greater than the total arrival rate. As shown in Atar, Mandelbaum and Shaikhet [Ann. Appl. Probab. 16 (2006) 1764-1804] and Atar and Shaikhet (2009), in the many server Halfin-Whitt regime, TSO implies null controllability (NC), the existence of a routing policy under which, for every finite $T$, the measure of the set of times prior to $T$, at which at least one customer is in the buffer, converges to zero in probability at the scaling limit. The present paper investigates the question whether the converse relation is also true and TSO is both sufficient and necessary for the NC behavior. In what follows we do get the affirmation for systems with either two customer classes (and possibly more service pools) or vice-versa and state a condition on the underlying static fluid model that allows the extension of the result to general structures.

Resolvent conditions for the control of parabolic equations

Since the seminal work of Russell and Weiss in 1994, resolvent conditions for various notions of admissibility, observability and controllability, and for various notions of linear evolution equations have been investigated intensively, sometimes under the name of infinite-dimensional Hautus test. This paper sets out resolvent conditions for null-controllability in arbitrary time: necessary for general semigroups, sufficient for analytic normal semigroups. For a positive self-adjoint operator A, it gives a sufficient condition for the null-controllability of the semigroup generated by −A which is only logarithmically stronger than the usual condition for the unitary group generated by iA. This condition is sharp when the observation operator is bounded. The proof combines the so-called “control transmutation method” and a new version of the “direct Lebeau–Robbiano strategy”. The improvement of this strategy also yields interior null-controllability of new logarithmic anomalous diffusions.

On the null-controllability of diffusion equations

This work studies the null-controllability of a class of abstract parabolic equations. The main contribution in the general case consists in giving a short proof of an abstract version of a sufficient condition for null-controllability which has been proposed by Lebeau and Robbiano. We do not assume that the control operator is admissible. Moreover, we give estimates of the control cost. In the special case of the heat equation in rectangular domains, we provide an alternative way to check the Lebeau-Robbiano spectral condition. We then show that the sophisticated Carleman and interpolation inequalities used in previous literature may be replaced by a simple result of Turan. In this case, we provide explicit values for the constants involved in the above mentioned spectral condition. As far as we are aware, this is the first proof of the null-controllability of the heat equation with arbitrary control domain in a n-dimensional open set which avoids Carleman estimates.

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.

Geometric Criterion for Controllability under Arbitrary Constraints on the Control

In this paper, necessary and sufficient conditions for null-controllability of a linear system under geometric constraints on the control are given, without the assumption that the origin is an equilibrium point of the system. The criterion for controllability uses the concept of a return condition on an interval which is introduced in the paper. This condition generalizes the existence of an equilibrium point.

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.