Complete controllability

A linear autonomous control system in ⁿ is said to be completely controllable iff there existsT>0 such that eachx ⁿ can be steered to anyy ⁿ in timeT. This paper presents a geometric characterization of this property in the case in which there are constraints on the values which the control maps can assume. A necessary and sufficient condition to get instant controllability (i.e., complete controllability for anyT>0) is also derived. This condition generalizes the well-known Kalman condition to the constrained case.     

Continuous dependence of the controllability set of a nonlinear control system on a parameter

In this note, we consider the mapping which associates a controllability set to each value of the parameter of a system. Sufficient conditions for the existence of breaks of this mapping are obtained. They show that nearly all such mappings have a finite number of breaks.This note was written during the author's stay at the University of Hamburg. He expresses his gratitude to the Institut für Angewandte Mathematik for its warm hospitality.     

On exact controllability and complete stabilizability for linear systems

This work is concerned with the relations between exact controllability and complete stabilizability for linear systems in Hilbert spaces. We give an affirmative answer to the open problem posed by Rabah and Karrakchou [R. Rabah, J. Karrakchou, Exact controllability and complete stabilizability for linear systems in Hilbert spaces, Appl. Math. Lett. 10 (1997) 35–40]. More precisely, if the $C_0$-semigroup $S\left(t\right)$ generated by $A$ is surjective and the pair $(A,B)$ with a bounded operator $B$ is completely stabilizable, then $(A,B)$ is exactly controllable without any additional condition.     

Global null controllability of nonlinear delay equations with controls in a compact set

The question of controlling some nonlinear retarded functional differential equations from an initial function to the zero function is considered. The control setsM() are square integrable functions with values in the unit closed sphereL ₂([t ₀, ),E ^m ) with center the origin. Assuming that the linear approximation of the nonlinear equation is null controllable with some integrable controls on some interval [t ₀,t ₁–2r], wheret ₀ is sufficiently large and wherer>0 is the delay, and assuming that the nonlinear system, withu=0, is uniformly globally asymptotically stable, we show that the nonlinear control process is globally null controllable with controlsuM(). The paper gives conditions which guarantee the stability assumptions, and also indicates conditions which yield the null controllability assumptions of the linear approximation. Our research extends known results on ordinary differential processes.     

The monotone method for controllability of the nonlinear evolution systems

In this paper,we study the controllability of the nonlinear evolution systems.We establish the controllability results by using the monotone operator theory.No compactness assumptions are imposed in the main results.We present an example to illustrate our results.     

Complete controllability of perturbed linear control systems,revisited

LetL ¹(I, ^n,n )×M(I, ^n,m ) be the space of all pairs (A, B), whereA andB are measurable functions from a compact intervalI to ^n,n and ^n,m , respectively, andA is Lebesgue integrable. Also, let this space be endowed with the topology of theL ¹-norm with respect toA and the topology of convergence in measure with respect toB. Then, the set of all pairs (A, B), for which the corresponding linear control system

On the 0-local controllability of a linear control system

The problem of the 0-local controllability of a linear control system when the origin does not belong to the constraint set is considered. Sufficient conditions for the 0-local controllability are presented here, extending a previous theorem by Saperstone and connecting this property with the distance of the constraint set from the origin.     

Self-accessibility of a set with respect to a multivalued field

LetF be a multivalued field on the manifoldM and letN be a submanifold ofM, possibly with boundary. We give a sufficient condition for the self-accessibility property of the pair (N, F), that is:N is contained in the interior of the attainable set fromN at a time smaller or equal toT, for everyT>0 (at the timeT, ifN is compact). To obtain such a condition, a modification of Petrov's implicit function theorem is proved (Ref. 1). Finally, a necessary condition for the self-accessibility property is given.This work was performed under the auspices of the Consiglio Nazionale delle Ricerche, Rome, Italy.     

Approximate and complete controllability of nonlinear systems to a convex target set

The research deals with complete and approximate controllability of the system $\text{(1)}\text{dx}\text{dy}\text{= f(t, x, u)}$, without control restraints to an arbitrary convex target set. First, some characterizations of complete controllability, to the target of (1) and a special case of (1) namely $\text{x}\text{?}\text{= A(t)x + k(t, u)}{}^7$ are given. As a consequence complete controllability is equivalent to null-controllability. Next certain equations are formulated. These are in the same spirit as J. P. Dauer's “A Controllability Technique for Nonlinear Systems” (J. Math. Anal. Appl. oo (1972), 442–451) and are utilized in the main contribution of the paper: Under certain convexity assumption, bounded perturbations of systems which are completely controllable to a fixed target G are completely controllable to G. Without the convexity assumption, but with perturbations satisfying a Lipschitz condition, approximate controllability to G of a perturbed system is equivalent to complete controllability to G of the unperturbed equation.     

On the 3BD‐closed set B3({4,5,6})

Let B₃(K) = {v:? an S(3,K,v)}. For K = {4} or {4,6}, B₃(K) has been determined by Hanani, and for K = {4, 5} by a previous paper of the author. In this paper, we investigate the case of K = {4,5,6}. It is easy to see that if v ∈ B₃ ({4, 5, 6}), then v ≡ 0, 1, 2 (mod 4). It is known that B₃{4, 6}) = {v > 0: v ≡ 0 (mod 2)} ? B₃({4,5,6}) by Hanani and that B₃({4, 5}) = {v > 0: v ≡ 1, 2, 4, 5, 8, 10 (mod 12) and v ≠ 13} ? B₃({4, 5, 6}). We shall focus on the case of v ≡ 9 (mod 12). It is proved that B₃({4,5,6}) = {v > 0: v ≡ 0, 1, 2 (mod 4) and v ≠ 9, 13}. © 2003 Wiley Periodicals, Inc.     

Approximate controllability for trajectories of a delay Volterra control system

For the abstract delay Volterra control system,

Flow control in gas networks: Exact controllability to a given demand

We consider a network of pipelines where the flow is controlled by a number of compressors. The consumer demand is described by desired boundary traces of the system state. We present conditions that guarantee the existence of compressor controls such that after a certain finite time the state at the consumer nodes is equal to the prescribed data. We consider this problem in the framework of continuously differentiable states. We give an explicit construction of the control functions for the control of compressor stations in gas distribution networks. Copyright © 2010 John Wiley & Sons, Ltd.     

Erratum: flow control in gas networks: exact controllability to a given demand

We correct a technical error in the paper of Gugat, Herty, Schleper, Math. Methods Appl. Sci. 34 (2011), where a framework for controllability of quasi–linear hyperbolic systems has been studied. The application to the case of gas networks is specified in more detail in the current work. Copyright © 2014 John Wiley & Sons, Ltd.     

A small generating set for the 3‐BD closed set of block sizes ≥ 6

Let v, k be positive integers and k ≥ 3, then K_k = : {v: v ≥ k} is a 3‐BD closed set. Two finite generating sets of 3‐BD closed sets K₄ and K₅ are obtained by H. Hanani [5] and Qiurong Wu [12] respectively. In this article we show that if v ≥ 6, then v ∈ B₃(K,1), where K = {6,7,…,41,45,46,47,51,52,53,83,84}\{22,26}; that is, we show that K is a generating set for K₆. Finally we show that v ∈ B₃(6,20) for all v ∈ K\{35,39,40,45}. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 128–136, 2008     

Approximate controllability of a delayed semilinear control system with growing nonlinear term

This paper deals with the approximate controllability of a semilinear control system with delay, where the nonlinear term satisfies the linear growth condition. Sufficient conditions for the approximate controllability of a semilinear control system have been established by assuming that the corresponding linear control system with delay is approximately controllable. To prove our results, the Schauder fixed point theorem is applied and instead of a Co-semigroup associated with the mild solution of the system we use the so-called fundamental solution.     

On a Decomposition of Complete Graphs

It is shown, among other results, that for any prime power q, the complete graph on 1+q+q ²+q ³ vertices can be decomposed into a union of 1+q Siamese Strongly Regular Graphs S R G(1+q+q ²+q ³,q+q ²,q–1,q+1) sharing 1+q ² cliques of size 1+q. Acknowledgments.The authors are indebted to a referee for a very extensive report and for many suggestions which improved the presentation of the paper tremendously.AMS Subject Numbers: 05B05, 05B20, 05E30This work was completed while the first author was on sabbatical leave visiting Institute for studies in theoretical Physics and Mathematics, (IPM), in Tehran, Iran. Support and hospitality is appreciated. Supported by an NSERC operating grant.     

The Kalman condition for the boundary controllability of coupled parabolic systems. Bounds on biorthogonal families to complex matrix exponentials

This paper is devoted to prove the controllability to trajectories of a system of n one-dimensional parabolic equations when the control is exerted on a part of the boundary by means of m controls. We give a general Kalman condition (necessary and sufficient) and also present a construction and sharp estimates of a biorthogonal family in L²(0,T;C) to {tje−Λkt}.     

Approximate controllability of a fractional stochastic partial integro-differential systems via noncompact operators

Objective: in this article, we discuss the approximate controllability problems of a new class of fractional impulsive stochastic partial integro-differential systems in separable Hilbert spaces. Methods: by applying the fractional calculus, the measure of noncompactness, properties of fractional resolvent operators and fixed point theorems. Results: we prove our main results without the hypotheses of compactness on the operator generated by the linear part of systems. Instead we suppose that the nonlinear term only satisfies a weakly compactness condition. Conclusion: the approximate controllability for the control systems with noncompact operators is established. Finally, an example is given for the illustration of the obtained theoretical results.