Recent zbMATH articles in MSC 93B05https://zbmath.org/atom/cc/93B052023-12-07T16:00:11.105023ZWerkzeugControllability results of Hilfer fractional derivative through integral contractorshttps://zbmath.org/1522.340242023-12-07T16:00:11.105023Z"Jothimani, K."https://zbmath.org/authors/?q=ai:jothimani.kasthurisamy"Valliammal, N."https://zbmath.org/authors/?q=ai:valliammal.n"Alsaeed, S."https://zbmath.org/authors/?q=ai:alsaeed.suliman"Nisar, Kottakkaran S."https://zbmath.org/authors/?q=ai:sooppy-nisar.kottakkaran"Ravichandran, C."https://zbmath.org/authors/?q=ai:ravichandran.chokkalingamSummary: The paper sheds light on Hilfer's controllability facts of neutral fractional system. Originally, the mild solution is derived using semigroup theory and the Laplace transform approach. Controllability of the Hilfer fractional system in non-dense domain using integral contractor which employs the sequence technique with the advantage that the nonlinear function does not meet the Lipschitz condition. To support the computed results, an appropriate examples are discussed.Geometric control of eigenfunctions of Schrödinger operatorshttps://zbmath.org/1522.350482023-12-07T16:00:11.105023Z"Macià, Fabricio"https://zbmath.org/authors/?q=ai:macia.fabricioSummary: We review the role of the Geometric Control Condition in establishing the observability property from an open set for solutions to the wave, Schrödinger, and eigenfunction equations. We show how to construct surfaces of revolution for which the observability property holds under strictly weaker conditions on the observation set than their counterparts for the wave and Schrödinger equations. We also introduce a class of Schrödinger operators on the two-dimensional sphere for which observability for eigenfunctions holds provided the observation region intersects only three fixed geodesics on the sphere, which only depend on the potential.
For the entire collection see [Zbl 1515.35010].Null controllability of strongly degenerate parabolic equationshttps://zbmath.org/1522.351922023-12-07T16:00:11.105023Z"Benoit, Antoine"https://zbmath.org/authors/?q=ai:benoit.antoine"Loyer, Romain"https://zbmath.org/authors/?q=ai:loyer.romain"Rosier, Lionel"https://zbmath.org/authors/?q=ai:rosier.lionelSummary: We consider linear one-dimensional strongly degenerate parabolic equations with measurable coefficients that may be degenerate or singular. Taking 0 as the point where the strong degeneracy occurs, we assume that the coefficient \(a = a(x)\) in the principal part of the parabolic equation is such that the function \(x \rightarrow x/a(x)\) is in \(L^p(0,1)\) for some \(p > 1\). After establishing some spectral estimates for the corresponding elliptic problem, we prove that the parabolic equation is null controllable in the energy space by using one boundary control.Sufficient conditions for the controllability of wave equations with a transmission condition at the interfacehttps://zbmath.org/1522.353242023-12-07T16:00:11.105023Z"Gagnon, Ludovick"https://zbmath.org/authors/?q=ai:gagnon.ludovickSummary: We consider waves travelling in two different mediums each endowed with a different constant speed of propagation. At the interface between the two mediums, the refraction of the rays of the optic geometry obeys the Snell's law. We provide sufficient conditions on the geometry of the mediums and on the speed of propagation for the exact boundary controllability.The exact boundary controllability of nodal profile for entropy solutions to 1-D quasilinear hyperbolic systems of conservation laws with linearly degenerate negative (resp., positive) characteristic fieldshttps://zbmath.org/1522.353292023-12-07T16:00:11.105023Z"Li, Tatsien"https://zbmath.org/authors/?q=ai:li.tatsien"Yu, Lei"https://zbmath.org/authors/?q=ai:yu.leiSummary: We consider the exact boundary controllability of nodal profile for entropy solutions to 1-D quasilinear hyperbolic systems of conservation laws \(\partial_tH(u)+\partial_xF(u)=0\), \(t>0\), \(0<x<L\) under the assumption that all negative (resp., positive) characteristic fields are linearly degenerate. We prove that for small initial-boundary data, if the total variations of the nodal profile and the boundary data given at \(x=0\) (resp., \(x=L\)) are sufficiently small, then there exists a time \(T\) and boundary controls at \(x=L\) (resp., \(x=0\)), such that the corresponding mixed initial-boundary value problem admits a unique entropy solution as the limit of front tracking approximate solutions, which takes the given nodal profile at the boundary \(x=0\) (resp., \(x=L\)) from a time larger than \(T\) to infinity. Moreover, if the nodal profile decays to zero when \(t\to+\infty\), then the entropy solution, together with the boundary control, possesses the same decay rate as the nodal profile.An explicit time for the uniform null controllability of a linear Korteweg-de Vries equationhttps://zbmath.org/1522.354372023-12-07T16:00:11.105023Z"Carreño, Nicolás"https://zbmath.org/authors/?q=ai:carreno.nicolas"Loyola, Cristóbal"https://zbmath.org/authors/?q=ai:loyola.cristobalSummary: In this paper, we consider a linear Korteweg-de Vries equation posed in a bounded interval and study the time dependency with respect to the interval length and the transport coefficient, for which the uniform null controllability holds as the dispersion coefficient goes to zero. We consider two cases of boundary controls. First, only one control on the left end of the interval, and then, two controls acting on the right. The strategy is based on the combination of an exponential dissipation inequality and suitable Carleman estimates for each case.Biorthogonal functions for complex exponentials and an application to the controllability of the Kawahara equation via a moment approachhttps://zbmath.org/1522.354462023-12-07T16:00:11.105023Z"Pazoto, Ademir F."https://zbmath.org/authors/?q=ai:pazoto.ademir-fernando"Vieira, Miguel D. Soto"https://zbmath.org/authors/?q=ai:vieira.miguel-d-sotoSummary: The paper deals with the controllability properties of the Kawahara equation posed on a periodic domain. We show that the equation is exactly controllable by means of a control depending only on time and acting on the system through a given shape function in space. Firstly, the exact controllability property is established for the linearized system through a Fourier expansion of solutions and the analysis of a biorthogonal sequence to a family of complex exponential functions. Finally, the local controllability of the full system is derived by combining the analysis of the linearized system, a fixed point argument and some Bourgain smoothing properties of the Kawahara equation on a periodic domain.Unique continuation and time decay for a higher-order water wave modelhttps://zbmath.org/1522.354472023-12-07T16:00:11.105023Z"Pazoto, Ademir F."https://zbmath.org/authors/?q=ai:pazoto.ademir-fernando"Vieira, Miguel D. Soto"https://zbmath.org/authors/?q=ai:vieira.miguel-d-sotoSummary: This work is devoted to prove the exponential decay for the energy of solutions of a higher order Korteweg-de Vries (KdV)-Benjamin-Bona-Mahony (BBM) equation on a periodic domain with a localized damping mechanism. Following the method in [\textit{L. Rosier} and \textit{B.-Y. Zhang}, J. Differ. Equations 254, No. 1, 141--178 (2013; Zbl 1256.35122)], which combines energy estimates, multipliers and compactness arguments, the problem is reduced to prove the Unique Continuation Property (UCP) for weak solutions of the model. Then, this is done by deriving Carleman estimates for a system of coupled elliptic-hyperbolic equations.Controllability conditions for Fredholm integrodifferential equations with degenerate kernel in Banach spaceshttps://zbmath.org/1522.450102023-12-07T16:00:11.105023Z"Zhuravlev, V. P."https://zbmath.org/authors/?q=ai:zhuravlev.viktor-filippovich"Honhalo, N. V."https://zbmath.org/authors/?q=ai:honhalo.n-v"Slyusarenko, I. P."https://zbmath.org/authors/?q=ai:slyusarenko.i-pSummary: By the theory of generalized inversion of operators and integral operators, we establish a criterion for the solvability and find the general form of solutions of an integrodifferential equation with degenerate kernel and control in a Banach space. The general form of control for which these solutions exist is also determined.On the exact controllability of a semilinear evolution equation with an unbounded operatorhttps://zbmath.org/1522.471122023-12-07T16:00:11.105023Z"Chernov, A. V."https://zbmath.org/authors/?q=ai:chernov.aleksei-vyacheslavovich|chernov.andrei-vladimirovich|chernov.andrew-v|chernov.alexey.1Summary: For the Cauchy problem associated with a controlled semilinear evolution equation with an unbounded maximal monotone operator in a Hilbert space, sufficient conditions are obtained for exact controllability to a given final state. Here a generalization of the Browder-Minty theorem and results on the total global solvability of this equation obtained by the author earlier [Differ. Equ. 49, No. 4, 517--527 (2013; Zbl 1279.47079); translation from Differ. Uravn. 49, N0. 4, 535--544 (2013)] are used. As an example, a semilinear wave equation is considered.Asymptotics of a solution to an optimal control problem with integral convex performance index, cheap control, and initial data perturbationshttps://zbmath.org/1522.490172023-12-07T16:00:11.105023Z"Danilin, A. R."https://zbmath.org/authors/?q=ai:danilin.aleksei-rufimovich"Shaburov, A. A."https://zbmath.org/authors/?q=ai:shaburov.aleksandr-aleksandrovichSummary: We consider an optimal control problem in the class of piecewise continuous controls with smooth geometric constraints for a linear system with constant coefficients and an integral convex performance index containing two small parameters (the first of them multiplying the integral term, and the second in the initial data). Such problems are called cheap control problems. It is shown that the limit problem is a problem with terminal performance index. It is established that if the limit problem is actually one-dimensional whereas the initial problem is not, then the asymptotics of the solution can be more complicated. In particular, the asymptotics of the solution may have no expansion in the Poincare sense in any asymptotic sequence of rational functions of the small parameter or its logarithms.Structural controllability in timed continuous Petri netshttps://zbmath.org/1522.930282023-12-07T16:00:11.105023Z"Arzola, César"https://zbmath.org/authors/?q=ai:arzola.cesar"Vázquez, Carlos Renato"https://zbmath.org/authors/?q=ai:vazquez.carlos-renato"Ramírez-Treviño, Antonio"https://zbmath.org/authors/?q=ai:ramirez-trevino.antonio"Silva, Manuel"https://zbmath.org/authors/?q=ai:silva.manuel.1In this paper the authors studied the concepts of net rank-controllability (NRC) and a global structural property. First, a relation between NRC and BIC was introduced: if the system is live as untimed, then, NRC is a sufficient condition for BIC over multiple regions. Moreover, it has been pointed out that in non-live systems NRC is not sufficient nor necessary for BIC. After that, a structural characterization of NRC, in terms of the influence of the controllable transitions and the uncontrollable flow invariants (UFIs), was introduced. It was demonstrated that if total influence holds and there are no UFIs, NRC is guaranteed. Moreover, polynomial-time algorithms for the verification of NRC, and therefore, for the verification of BIC, are provided.
Reviewer: Udhayakumar Ramalingam (Vellore)Controllability for the wave equation on graph with cycle and delta-prime vertex conditionshttps://zbmath.org/1522.930292023-12-07T16:00:11.105023Z"Avdonin, Sergei"https://zbmath.org/authors/?q=ai:avdonin.sergei-anatolevich"Edward, Julian"https://zbmath.org/authors/?q=ai:edward.julian"Leugering, Günter"https://zbmath.org/authors/?q=ai:leugering.gunterSummary: Exact controllability for the wave equation on a metric graph consisting of a cycle and two attached edges is proven. One boundary and one internal control are used. At the internal vertices, delta-prime conditions are satisfied. As a second example, we examine a tripod controlled at the root and the junction, while the leaves are fixed. These examples are key to understanding controllability properties in general metric graphs.Controllability results for second-order integro-differential equations with state-dependent delayhttps://zbmath.org/1522.930302023-12-07T16:00:11.105023Z"Bensalem, Abdelhamid"https://zbmath.org/authors/?q=ai:bensalem.abdelhamid"Salim, Abdelkrim"https://zbmath.org/authors/?q=ai:salim.abdelkrim"Benchohra, Mouffak"https://zbmath.org/authors/?q=ai:benchohra.mouffak"Nieto, Juan J."https://zbmath.org/authors/?q=ai:nieto.juan-joseSummary: The purpose of this study is to use resolvent operators to investigate the existence and the controllability of a mild solution to a second-order semilinear integro-differential problem. To construct our criterion, we use a fixed point theorem in conjunction with measures of noncompactness. A practical example is used to illustrate the obtained results.Partial asymptotic null controllability for semilinear evolution equations in Hilbert spaceshttps://zbmath.org/1522.930312023-12-07T16:00:11.105023Z"Boujallal, Lahoucine"https://zbmath.org/authors/?q=ai:boujallal.lahoucine(no abstract)Finite-approximate controllability of impulsive \(\psi\)-Caputo fractional evolution equations with nonlocal conditionshttps://zbmath.org/1522.930322023-12-07T16:00:11.105023Z"Ding, Yonghong"https://zbmath.org/authors/?q=ai:ding.yonghong"Li, Yongxiang"https://zbmath.org/authors/?q=ai:li.yongxiang(no abstract)Boundary approximate controllability under positivity constraints of infinite-dimensional control systemshttps://zbmath.org/1522.930332023-12-07T16:00:11.105023Z"Gantouh, El Gantouh"https://zbmath.org/authors/?q=ai:gantouh.el-gantouhSummary: This paper focuses on boundary approximate controllability under positivity constraints of a wide range of infinite-dimensional control systems. We develop frequency-domain controllability criteria. Firstly, we derive a controllability result under positivity constraints on the control for such systems. Then, and more importantly, we provide a necessary and sufficient condition for controllability under positivity constraints on the control and the state. The obtained results are applied to the controllability of transportation and heat conduction networks. In particular, provided that the underlying graph is strongly connected, the controllability under positivity constraints on the control/state of transport network systems is fully characterized by a Kalman-type rank condition. For a system of heat equations with Robin boundary conditions on a path-like network, we establish approximate controllability under positivity state constraint with a single positive input through the starting node. However, we prove the lack of controllability under unilateral control constraint.Controllability and observability of linear time-varying fractional systemshttps://zbmath.org/1522.930342023-12-07T16:00:11.105023Z"Jolić, Maja"https://zbmath.org/authors/?q=ai:jolic.maja"Konjik, Sanja"https://zbmath.org/authors/?q=ai:konjik.sanja(no abstract)Null controllability of \(\psi\)-Hilfer implicit fractional integro-differential equations with \(\psi\)-Hilfer fractional nonlocal conditionshttps://zbmath.org/1522.930352023-12-07T16:00:11.105023Z"Kerboua, Mourad"https://zbmath.org/authors/?q=ai:kerboua.mourad"Bouacida, Ichrak"https://zbmath.org/authors/?q=ai:bouacida.ichrak"Segni, Sami"https://zbmath.org/authors/?q=ai:segni.samiSummary: In this paper, we study exact null controllability of a new class of non local \(\psi\)-Hilfer implicit fractional integro-differential equations in Hilbert space. The results are obtained by using semigroup theory, \(\psi\)-Hilfer fractional calculus and Banach's fixed point theorem. Finally, we provide an example to illustrate the applicability of our results.On some problems of controllability and observability for differential-algebraic systems with aftereffecthttps://zbmath.org/1522.930362023-12-07T16:00:11.105023Z"Khartovskiĭ, V. E."https://zbmath.org/authors/?q=ai:khartovskii.vadim-evgenevichSummary: For linear autonomous completely regular differential-algebraic systems with aftereffect, criteria for the solvability of problems of complete null-controllability, null-controllability, and finite observability are obtained.Constructive controllability for incompressible vector fieldshttps://zbmath.org/1522.930372023-12-07T16:00:11.105023Z"Kryzhevich, Sergey"https://zbmath.org/authors/?q=ai:kryzhevich.sergei-gennadevich"Stepanov, Eugene"https://zbmath.org/authors/?q=ai:stepanov.eugeneAuthors' abstract: We give a constructive proof of a global controllability result for an autonomous system of ODEs guided by bounded locally Lipschitz and divergence free (i.e. incompressible) vector field, when the phase space is the whole Euclidean space and the vector field satisfies so-called vanishing mean drift condition. For the case when the ODE is defined over some smooth compact connected Riemannian manifold, we significantly strengthen the assertion of the known controllability theorem in absence of nonholonomic constraints by proving that one can find a control steering the state vector from one given point to another by using the observations of only the state vector, i.e., in other words, by changing slightly the vector field, and such a change can be made small not only in uniform, but also in Lipschitz (i.e.\( C^1\)) topology.
For the entire collection see [Zbl 1515.93013].
Reviewer: Krishnan Balachandran (Coimbatore)Boundary exact controllability for the longitudinal vibrations of a bar in memoriam to Professor Luiz Adauto Medeiroshttps://zbmath.org/1522.930382023-12-07T16:00:11.105023Z"Milla Miranda, M."https://zbmath.org/authors/?q=ai:milla-miranda.manuel"Louredo, Aldo T."https://zbmath.org/authors/?q=ai:louredo.aldo-trajanoSummary: This paper is concerned with the exact controllability of a system that describes the small vibrations of a bar which is clamped at one and in the other end is glued a mass. To obtain the exact controllability of this system, we will use the HUM method due to \textit{J. L. Lions} [Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 2: Perturbations. (Exact controllability, perturbations and stabilization of distributed systems. Vol. 2: Perturbations). Paris etc.: Masson (1988; Zbl 0653.93003)].Approximate controllability results for the Sobolev type fractional delay impulsive integrodifferential inclusions of order \(r \in (1,2)\) via sectorial operatorhttps://zbmath.org/1522.930392023-12-07T16:00:11.105023Z"Raja, M. Mohan"https://zbmath.org/authors/?q=ai:raja.m-mohan|raja.marimuthu-mohan"Vijayakumar, V."https://zbmath.org/authors/?q=ai:vijayakumar.velusamy|vijayakumar.vaidehi(no abstract)Control problems for energy harvester model and interpolation in Hardy spacehttps://zbmath.org/1522.930402023-12-07T16:00:11.105023Z"Shubov, Marianna A."https://zbmath.org/authors/?q=ai:shubov.marianna-aSummary: Three control problems for the system of two coupled differential equations governing the dynamics of an energy harvesting model are studied. The system consists of the equation of an Euler-Bernoulli beam model and the equation representing the Kirchhoff's electric circuit law. Both equations contain coupling terms representing the inverse and direct piezoelectric effects. The system is reformulated as a single evolution equation in the state space of 3-component functions. The control is introduced as a separable forcing term \(\mathbf{g}(x)f(t)\) on the right-hand side of the operator equation. The first control problem deals with an explicit construction of \(f(t)\) that steers an initial state to zero on a time interval \([0, T]\). The second control problem deals with the construction of \(f(t)\) such that the voltage output is equal to some given function \(\mathbf{v}(t)\) (with \(\mathbf{g}(x)\) being given as well). The third control problem deals with an explicit construction of both the force profile, \( \mathbf{g}(x)\), and the control, \(f(t)\), which generate the desired voltage output \(\mathbf{v}(t)\). Interpolation theory in the Hardy space of analytic functions is used in the solution of the second and third problems.Conditions for pointwise controllability and pointwise observability of linear time-invariant singularly perturbed systems with delayhttps://zbmath.org/1522.930412023-12-07T16:00:11.105023Z"Tsekhan, O. B."https://zbmath.org/authors/?q=ai:tsekhan.o-bSummary: For a linear time-invariant singularly perturbed system with finite lumped delay in state variables and with an observable linear output without delay, multipoint finite-dimensional boundary value problems are considered -- the problems of pointwise controllability and pointwise observability. The duality of the considered singularly perturbed control and observation systems is established. By the method of defining equations, necessary, sufficient conditions for pointwise controllability and pointwise observability, which are valid for all sufficiently small values of the singularity parameter, are obtained independent of the parameter. The conditions are expressed in terms of the matrix parameters of the original system, have a rank type, and are formulated in terms of solutions of recurrent algebraic matrix defining equations, which are constructed according to the original systems by simple rules.Exact controllability of forward and backward stochastic difference systemhttps://zbmath.org/1522.930422023-12-07T16:00:11.105023Z"Wang, Wenjing"https://zbmath.org/authors/?q=ai:wang.wenjing"Xu, Juanjuan"https://zbmath.org/authors/?q=ai:xu.juanjuan"Zhang, Huanshui"https://zbmath.org/authors/?q=ai:zhang.huanshuiSummary: In this paper, we study the exact controllability of forward and backward stochastic difference system (FBSDS) with multiplicative noise. Different from the standard system governed by forward stochastic difference equation (FSDE), the inclusion of both initial and terminal conditions in FBSDS complicates the solvability which also leads to the challenge of exact controllability. To overcome the difficulty, we transform the FBSDS into an equivalent backward stochastic difference equation. Accordingly, we derive both the necessary and sufficient Gramian matrix criterion and the Rank criterion for the exact controllability of FBSDS, which is the main contribution of the paper.Two multiobjective problems for stochastic degenerate parabolic equationshttps://zbmath.org/1522.930432023-12-07T16:00:11.105023Z"Yu, Yongyi"https://zbmath.org/authors/?q=ai:yu.yongyi"Zhang, Ji-Feng"https://zbmath.org/authors/?q=ai:zhang.jifengSummary: This paper is devoted to studying two multiobjective problems for stochastic degenerate parabolic equations. The first one is a hierarchical control problem, in which the controls are classified into a pair of leaders and a pair of followers. For each pair of leaders, a Nash equilibrium is searched for a noncooperative game problem. The aim of the pair of leaders is to achieve null controllability of the system. The other multiobjective problem is an inverse initial problem under a Nash equilibrium strategy. In contrast to the classical inverse initial problem, an optimization problem for stochastic degenerate parabolic equations is first investigated. Then, the conditional stability of determining initial information is derived through terminal observation.