Recent zbMATH articles in MSC 93B05https://zbmath.org/atom/cc/93B052022-11-17T18:59:28.764376ZWerkzeugStability, boundedness and controllability of solutions of measure functional differential equationshttps://zbmath.org/1496.340042022-11-17T18:59:28.764376Z"Andrade da Silva, F."https://zbmath.org/authors/?q=ai:andrade-da-silva.f"Federson, M."https://zbmath.org/authors/?q=ai:federson.marcia"Toon, E."https://zbmath.org/authors/?q=ai:toon.eduardIn the present paper, converse Lyapunov results on uniform boundedness for the very general class of generalized differential equations are established. Relations between stability and boundedness of solutions are also obtained. Using Lyapunov techniques, asymptotic controllability is characterized as well. As the theory of measure functional differential equations is a particular case of this wide setting, corresponding theorems are derived for measure functional differential equations.
Reviewer: Bianca-Renata Satco (Suceava)A new approach on the approximate controllability of fractional differential evolution equations of order \(1<r<2\) in Hilbert spaceshttps://zbmath.org/1496.340212022-11-17T18:59:28.764376Z"Raja, M. Mohan"https://zbmath.org/authors/?q=ai:raja.m-mohan"Vijayakumar, V."https://zbmath.org/authors/?q=ai:vijayakumar.velusamy"Udhayakumar, R."https://zbmath.org/authors/?q=ai:udhayakumar.r"Zhou, Yong"https://zbmath.org/authors/?q=ai:zhou.yongSummary: This manuscript is mainly focusing on approximate controllability for fractional differential evolution equations of order \(1<r<2\) in Hilbert spaces. We consider a class of control systems governed by the fractional differential evolution equations. By using the results on fractional calculus, cosine and sine functions of operators, and Schauder's fixed point theorem, a new set of sufficient conditions are formulated which guarantees the approximate controllability of fractional differential evolution systems. The results are established under the assumption that the associated linear system is approximately controllable. Then, we develop our conclusions to the ideas of nonlocal conditions. Lastly, we present theoretical and practical applications to support the validity of the study.Controllability of fractional evolution systems of Sobolev type via resolvent operatorshttps://zbmath.org/1496.340972022-11-17T18:59:28.764376Z"Yang, He"https://zbmath.org/authors/?q=ai:yang.he"Zhao, Yanjie"https://zbmath.org/authors/?q=ai:zhao.yanjieSummary: In this paper, we consider the nonlocal controllability of \(\alpha\in (1,2)\)-order fractional evolution systems of Sobolev type in abstract spaces. By utilizing fixed point theorems and the theory of resolvent operators we establish some sufficient conditions for the nonlocal controllability of Sobolev-type fractional evolution systems.A discussion on the approximate controllability of Hilfer fractional neutral stochastic integro-differential systemshttps://zbmath.org/1496.341112022-11-17T18:59:28.764376Z"Dineshkumar, C."https://zbmath.org/authors/?q=ai:dineshkumar.c"Udhayakumar, R."https://zbmath.org/authors/?q=ai:udhayakumar.r"Vijayakumar, V."https://zbmath.org/authors/?q=ai:vijayakumar.velusamy"Nisar, Kottakkaran Sooppy"https://zbmath.org/authors/?q=ai:sooppy-nisar.kottakkaranSummary: This manuscript is mainly focusing on the approximate controllability of Hilfer fractional neutral stochastic integro-differential equations. The principal results of this article are proved based on the theoretical concepts related to the fractional calculus and Schauder's fixed-point theorem. Initially, we discuss the approximate controllability of the fractional evolution system. Then, we extend our results to the concept of nonlocal conditions. Finally, we provide theoretical and practical applications to assist in the effectiveness of the discussion.A study on controllability of impulsive fractional evolution equations via resolvent operatorshttps://zbmath.org/1496.341162022-11-17T18:59:28.764376Z"Gou, Haide"https://zbmath.org/authors/?q=ai:gou.haide"Li, Yongxiang"https://zbmath.org/authors/?q=ai:li.yongxiangSummary: In this article, we study the controllability for impulsive fractional integro-differential evolution equation in a Banach space. The discussions are based on the Mönch fixed point theorem as well as the theory of fractional calculus and the \((\alpha ,\beta)\)-resolvent operator, we concern with the term \(u^\prime(\cdot)\) and finding a control \(v\) such that the mild solution satisfies \(u(b)=u_b\) and \(u^\prime b)=u^\prime_b\). Finally, we present an application to support the validity study.A new approach on approximate controllability of fractional evolution inclusions of order \(1<r<2\) with infinite delayhttps://zbmath.org/1496.341202022-11-17T18:59:28.764376Z"Raja, M. Mohan"https://zbmath.org/authors/?q=ai:raja.m-mohan"Vijayakumar, V."https://zbmath.org/authors/?q=ai:vijayakumar.velusamy"Udhayakumar, R."https://zbmath.org/authors/?q=ai:udhayakumar.rSummary: This manuscript is mainly focusing on the approximate controllability of fractional differential evolution inclusions of order \(1<r<2\) with infinite delay. We study our primary outcomes by using the theoretical concepts about fractional calculus, cosine, and sine function of operators and Dhage's fixed point theorem. Initially, we prove the approximate controllability for the fractional evolution system. The results are established under the assumption that the associated linear system is approximately controllable. Then, we develop our conclusions to the ideas of nonlocal conditions. Finally, we present theoretical and practical applications to support the validity of the study.On the extremum control problem with pointwise observation for a parabolic equationhttps://zbmath.org/1496.352242022-11-17T18:59:28.764376Z"Astashova, I. V."https://zbmath.org/authors/?q=ai:astashova.irina-v"Lashin, D. A."https://zbmath.org/authors/?q=ai:lashin.d-a"Filinovskiy, A. V."https://zbmath.org/authors/?q=ai:filinovskii.alexei-vladislavovichSummary: In this paper we consider a control problem with pointwise observation for a one-dimensional parabolic equation which arises in a mathematical model of climate control in industrial greenhouses. We study a general equation with variable diffusion coefficient, convection coefficient, and depletion potential. For the extremum problem of minimizing an integral weighted quadratic cost functional, we establish the existence and uniqueness of a minimizing function. We also study exact controllability and dense controllability of the problem. Necessary conditions for an extremum are obtained, and qualitative properties of the minimizing function are studied.Equivalent one-dimensional first-order linear hyperbolic systems and range of the minimal null control time with respect to the internal coupling matrixhttps://zbmath.org/1496.352452022-11-17T18:59:28.764376Z"Hu, Long"https://zbmath.org/authors/?q=ai:hu.long"Olive, Guillaume"https://zbmath.org/authors/?q=ai:olive.guillaumeSummary: In this paper, we are interested in the minimal null control time of one-dimensional first-order linear hyperbolic systems by one-sided boundary controls. Our main result is an explicit characterization of the smallest and largest values that this minimal null control time can take with respect to the internal coupling matrix. In particular, we obtain a complete description of the situations where the minimal null control time is invariant with respect to all the possible choices of internal coupling matrices. The proof relies on the notion of equivalent systems, in particular the backstepping method, a canonical \textit{LU}-decomposition for boundary coupling matrices and a compactness-uniqueness method adapted to the null controllability property.Local null controllability of a class of non-Newtonian incompressible viscous fluidshttps://zbmath.org/1496.353132022-11-17T18:59:28.764376Z"de Carvalho, Pitágoras Pinheiro"https://zbmath.org/authors/?q=ai:de-carvalho.pitagoras-pinheiro"Límaco, Juan"https://zbmath.org/authors/?q=ai:limaco.juan"Menezes, Denilson"https://zbmath.org/authors/?q=ai:menezes.denilson"Thamsten, Yuri"https://zbmath.org/authors/?q=ai:thamsten.yuriSummary: We investigate the null controllability property of systems that mathematically describe the dynamics of some non-Newtonian incompressible viscous flows. The principal model we study was proposed by O. A. Ladyzhenskaya, although the techniques we develop here apply to other fluids having a shear-dependent viscosity. Taking advantage of the Pontryagin Minimum Principle, we utilize a bootstrapping argument to prove that sufficiently smooth controls to the forced linearized Stokes problem exist, as long as the initial data in turn has enough regularity. From there, we extend the result to the nonlinear problem. As a byproduct, we devise a quasi-Newton algorithm to compute the states and a control, which we prove to converge in an appropriate sense. We finish the work with some numerical experiments.The fractional Schrödinger equation on compact manifolds: global controllability resultshttps://zbmath.org/1496.353542022-11-17T18:59:28.764376Z"Capistrano-Filho, Roberto de A."https://zbmath.org/authors/?q=ai:capistrano-filho.roberto-de-a"Pampu, Ademir B."https://zbmath.org/authors/?q=ai:pampu.ademir-bSummary: The goal of this work is to prove global controllability and stabilization properties for the fractional Schrödinger equation on \(d\)-dimensional compact Riemannian manifolds without boundary \((M, g)\). To prove our main results we use techniques of pseudo-differential calculus on manifolds. More precisely, by using microlocal analysis, we are able to prove propagation of regularity which together with the so-called \textit{Geometric Control Condition} and \textit{Unique Continuation Property} help us to prove global control results for the system under consideration. As a main novelty this manuscript presents the relation between the geometric control condition and the controllability for the fractional Schrödinger equation.Control and numerical approximation of fractional diffusion equationshttps://zbmath.org/1496.651552022-11-17T18:59:28.764376Z"Biccari, Umberto"https://zbmath.org/authors/?q=ai:biccari.umberto"Warma, Mahamadi"https://zbmath.org/authors/?q=ai:warma.mahamadi"Zuazua, Enrique"https://zbmath.org/authors/?q=ai:zuazua.enriqueSummary: The aim of this chapter is to give a broad panorama of the control properties of fractional diffusive models from a numerical analysis and simulation perspective. We do this by surveying several research results we obtained in the last years, focusing in particular on the numerical computation of controls, though not forgetting to recall other relevant contributions which can be currently found in the literature of this prolific field. Our reference model will be a non-local diffusive dynamics driven by the fractional Laplacian on a bounded domain \(\Omega\). The starting point of our analysis will be a Finite Element approximation for the associated elliptic model in one and two space-dimensions, for which we also present error estimates and convergence rates in the \(L^2\) and energy norm. Secondly, we will address two specific control scenarios: firstly, we consider the standard interior control problem, in which the control is acting from a small subset \(\omega \subset \Omega\). Secondly, we move our attention to the exterior control problem, in which the control region \(\mathcal{O} \subset \Omega^c\) is located outside \(\Omega\). This exterior control notion extends boundary control to the fractional framework, in which the non-local nature of the models does not allow for controls supported on \(\partial \Omega\). We will conclude by discussing the interesting problem of simultaneous control, in which we consider families of parameter-dependent fractional heat equations and we aim at designing a unique control function capable of steering all the different realizations of the model to the same target configuration. In this framework, we will see how the employment of stochastic optimization techniques may help in alleviating the computational burden for the approximation of simultaneous controls. Our discussion is complemented by several open problems related with fractional models which are currently unsolved and may be of interest for future investigation.
For the entire collection see [Zbl 1492.49003].Controlling swarms toward flocks and millshttps://zbmath.org/1496.930192022-11-17T18:59:28.764376Z"Carrillo, José A."https://zbmath.org/authors/?q=ai:carrillo.jose-antonio"Kalise, Dante"https://zbmath.org/authors/?q=ai:kalise.dante"Rossi, Francesco"https://zbmath.org/authors/?q=ai:rossi.francesco"Trélat, Emmanuel"https://zbmath.org/authors/?q=ai:trelat.emmanuelRandom matrices and controllability of dynamical systemshttps://zbmath.org/1496.930232022-11-17T18:59:28.764376Z"Leventides, John"https://zbmath.org/authors/?q=ai:leventides.john"Poulios, Nick"https://zbmath.org/authors/?q=ai:poulios.nick"Poulios, Costas"https://zbmath.org/authors/?q=ai:poulios.costasSummary: We introduce the concept of \(\epsilon\)-uncontrollability for random linear systems, i.e. linear systems in which the usual matrices have been replaced by random matrices. We also estimate the \(\varepsilon\)-uncontrollability in the case where the matrices come from the Gaussian orthogonal ensemble. Our proof utilizes tools from systems theory, probability theory and convex geometry.Strong targeted controllability of multi-agent systems with time-varying topologies over finite fieldshttps://zbmath.org/1496.930242022-11-17T18:59:28.764376Z"Lu, Zehuan"https://zbmath.org/authors/?q=ai:lu.zehuan"Zhang, Zhiqiang"https://zbmath.org/authors/?q=ai:zhang.zhiqiang"Ji, Zhijian"https://zbmath.org/authors/?q=ai:ji.zhijianSummary: In this paper, we study the strong targeted controllability of multi-agent systems with time-varying topologies over finite fields from a graph theoretic perspective. We consider leader-follower networks, where only some of the agents are governed by external control inputs, namely the leaders. We develop a graph-theoretic characterization for strong targeted controllability of multi-agent systems with time-varying topologies over finite fields. Specifically, we show that a time-varying multi-agent system is strongly targeted controllable over any finite field if the graphs of the system satisfy certain properties. Our main results are applicable to time-varying topologies with arbitrary fields and coupling weights. We also present a necessary and sufficient condition for the strong structural controllability of the time-invariant multi-agent systems.Suppression of vertical plasma displacements by control system of plasma unstable vertical position in D-shaped tokamakhttps://zbmath.org/1496.930252022-11-17T18:59:28.764376Z"Mitrishkin, Yu. V."https://zbmath.org/authors/?q=ai:mitrishkin.yuri-v"Korenev, P. S."https://zbmath.org/authors/?q=ai:korenev.p-s"Konkov, A. E."https://zbmath.org/authors/?q=ai:konkov.a-e"Kartsev, N. M."https://zbmath.org/authors/?q=ai:kartsev.n-mSummary: Ensuring the stability of the vertical position of plasma is a paramount task of magnetic control for modern D-shaped tokamaks. The introduction of an additional horizontal field coil located near the vacuum vessel provides more than an order of magnitude larger vertical controllability domain than the use of a pair of P6\&P12 poloidal field coils in the IGNITOR tokamak project. Two robust systems for controlling the vertical position of plasma are synthesized by the \(H_\infty \)-optimization theory method and by tuning PID controllers in a cascade system by the method of linear matrix inequalities. The results of mathematical modeling of control systems show that the modernization of the poloidal system of the IGNITOR tokamak project leads to an increase in the stability margin of the plasma vertical position control system, the quality of control when rejecting disturbances, and the reliability of the tokamak.On the solution of generalized Lyapunov equations for a class of continuous bilinear time-varying systemshttps://zbmath.org/1496.930572022-11-17T18:59:28.764376Z"Yadykin, I. B."https://zbmath.org/authors/?q=ai:yadykin.igor-b"Galyaev, I. A."https://zbmath.org/authors/?q=ai:galyaev.i-a"Vershinin, Yu. A."https://zbmath.org/authors/?q=ai:vershinin.yu-aSummary: We have developed a method and algorithms for solving the generalized Lyapunov equation for a wide class of continuous time-varying bilinear systems based on the Gramian method and an iterative solution construction method proposed earlier for such equations. The approach consists in diagonalizing the original system, obtaining a separable spectral decomposition of the Gramian of the time-invariant linear part in terms of the combination spectrum of the dynamics matrix of the linear part, applying the spectral decomposition of the kernel matrix of the solution obtained at the previous step at each iteration step, and then aggregating the matrix entries. A spectral decomposition of the Gramians of controllability and observability of a time-varying bilinear system is obtained as the sum of sub-Gramian matrices corresponding to pair combinations of the eigenvalues of the dynamics matrix of the linear part. A new method and algorithm for entry-by-entry calculation of matrices for solving the generalized Lyapunov equation for bilinear systems has been developed. The fundamental novelty of the approach lies in the transfer of calculations from the solution matrix to the calculation of the sequence of its entries at each iteration step.