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 2156-8472

eISSN:
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Volume 8, 2018

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Volume 1, 2011

MCRF aims to publish original research as well as expository papers on mathematical control theory and related fields. The goal is to provide a complete and reliable source of mathematical methods and results in this field. The journal will also accept papers from some related fields such as differential equations, functional analysis, probability theory and stochastic analysis, inverse problems, optimization, numerical computation, mathematical finance, information theory, game theory, system theory, etc., provided that they have some intrinsic connections with control theory.

MCRF is edited by a group of international leading experts in mathematical control theory and related fields. A key feature of MCRF is the journal's rapid publication, with a special emphasis on the highest scientific standard. The journal is essential reading for scientists and researchers who wish to keep abreast of the latest developments in the field.

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  • Publishes 4 issues a year in March, June, September and December.
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Asymptotic behavior of a Schrödinger equation under a constrained boundary feedback
Haoyue Cui, Dongyi Liu and Genqi Xu
2018, 8(2) : 383-395 doi: 10.3934/mcrf.2018015 +[Abstract](169) +[HTML](107) +[PDF](434.73KB)
Abstract:

Design of controller subject to a constraint for a Schrödinger equation is considered based on the energy functional of the system. Thus, the resulting closed-loop system is nonlinear and its well-posedness is proven by the nonlinear monotone operator theory and a complex form of the nonlinear Lax-Milgram theorem. The asymptotic stability and exponential stability of the system are discussed with the LaSalle invariance principle and Riesz basis method, respectively. In the end, a numerical simulation illustrates the feasibility of the suggested feedback control law.

Compact perturbations of controlled systems
Michel Duprez and Guillaume Olive
2018, 8(2) : 397-410 doi: 10.3934/mcrf.2018016 +[Abstract](155) +[HTML](75) +[PDF](410.77KB)
Abstract:

In this article we study the controllability properties of general compactly perturbed exactly controlled linear systems with admissible control operators. Firstly, we show that approximate and exact controllability are equivalent properties for such systems. Then, and more importantly, we provide for the perturbed system a complete characterization of the set of reachable states in terms of the Fattorini-Hautus test. The results rely on the Peetre lemma.

Finite element error analysis for measure-valued optimal control problems governed by a 1D wave equation with variable coefficients
Philip Trautmann, Boris Vexler and Alexander Zlotnik
2018, 8(2) : 411-449 doi: 10.3934/mcrf.2018017 +[Abstract](146) +[HTML](72) +[PDF](1072.35KB)
Abstract:

This work is concerned with the optimal control problems governed by a 1D wave equation with variable coefficients and the control spaces \begin{document}$\mathcal M_T$\end{document} of either measure-valued functions \begin{document}$L_{{{w}^{*}}}^{2}\left( I, \mathcal M\left( {\mathit \Omega } \right) \right)$\end{document} or vector measures \begin{document}$\mathcal M({\mathit \Omega }, L^2(I))$\end{document}. The cost functional involves the standard quadratic tracking terms and the regularization term \begin{document}$α\|u\|_{\mathcal M_T}$\end{document} with \begin{document}$α>0$\end{document}. We construct and study three-level in time bilinear finite element discretizations for this class of problems. The main focus lies on the derivation of error estimates for the optimal state variable and the error measured in the cost functional. The analysis is mainly based on some previous results of the authors. The numerical results are included.

A second-order stochastic maximum principle for generalized mean-field singular control problem
Hancheng Guo and Jie Xiong
2018, 8(2) : 451-473 doi: 10.3934/mcrf.2018018 +[Abstract](169) +[HTML](92) +[PDF](436.85KB)
Abstract:

In this paper, we study the generalized mean-field stochastic control problem when the usual stochastic maximum principle (SMP) is not applicable due to the singularity of the Hamiltonian function. In this case, we derive a second order SMP. We introduce the adjoint process by the generalized mean-field backward stochastic differential equation. The keys in the proofs are the expansion of the cost functional in terms of a perturbation parameter, and the use of the range theorem for vector-valued measures.

Stability and output feedback control for singular Markovian jump delayed systems
Jian Chen, Tao Zhang, Ziye Zhang, Chong Lin and Bing Chen
2018, 8(2) : 475-490 doi: 10.3934/mcrf.2018019 +[Abstract](135) +[HTML](62) +[PDF](253.58KB)
Abstract:

This paper is concerned with the admissibility analysis and control synthesis for a class of singular systems with Markovian jumps and time-varying delay. The basic idea is the use of an augmented Lyapunov-Krasovskii functional together with a series of appropriate integral inequalities. Sufficient conditions are established to ensure the systems to be admissible. Moreover, control design via static output feedback (SOF) is derived to achieve the stabilization for singular systems. A new algorithm is built to solve the SOF controllers. Examples are given to show the effectiveness of the proposed method.

Inverse source problem with a final overdetermination for a fractional diffusion equation
Kenichi Sakamoto and Masahiro Yamamoto
2011, 1(4) : 509-518 doi: 10.3934/mcrf.2011.1.509 +[Abstract](324) +[PDF](316.6KB) Cited By(23)
Strict Lyapunov functions for semilinear parabolic partial differential equations
Frédéric Mazenc and Christophe Prieur
2011, 1(2) : 231-250 doi: 10.3934/mcrf.2011.1.231 +[Abstract](343) +[PDF](929.8KB) Cited By(18)
Sparse stabilization and optimal control of the Cucker-Smale model
Marco Caponigro, Massimo Fornasier, Benedetto Piccoli and Emmanuel Trélat
2013, 3(4) : 447-466 doi: 10.3934/mcrf.2013.3.447 +[Abstract](357) +[PDF](467.3KB) Cited By(18)
A linear-quadratic optimal control problem for mean-field stochastic differential equations in infinite horizon
Jianhui Huang, Xun Li and Jiongmin Yong
2015, 5(1) : 97-139 doi: 10.3934/mcrf.2015.5.97 +[Abstract](352) +[PDF](614.2KB) Cited By(13)
A deterministic linear quadratic time-inconsistent optimal control problem
Jiongmin Yong
2011, 1(1) : 83-118 doi: 10.3934/mcrf.2011.1.83 +[Abstract](393) +[PDF](442.3KB) Cited By(12)
Time-inconsistent optimal control problems and the equilibrium HJB equation
Jiongmin Yong
2012, 2(3) : 271-329 doi: 10.3934/mcrf.2012.2.271 +[Abstract](355) +[PDF](637.9KB) Cited By(11)
Control of a Korteweg-de Vries equation: A tutorial
Eduardo Cerpa
2014, 4(1) : 45-99 doi: 10.3934/mcrf.2014.4.45 +[Abstract](279) +[PDF](701.7KB) Cited By(11)
Global well-posedness and asymptotic behavior of a class of initial-boundary-value problem of the Korteweg-De Vries equation on a finite domain
Ivonne Rivas, Muhammad Usman and Bing-Yu Zhang
2011, 1(1) : 61-81 doi: 10.3934/mcrf.2011.1.61 +[Abstract](384) +[PDF](455.9KB) Cited By(11)
Time-delayed boundary feedback stabilization of the isothermal Euler equations with friction
Martin Gugat and Markus Dick
2011, 1(4) : 469-491 doi: 10.3934/mcrf.2011.1.469 +[Abstract](291) +[PDF](368.5KB) Cited By(10)
Stability estimates for a Robin coefficient in the two-dimensional Stokes system
Muriel Boulakia, Anne-Claire Egloffe and Céline Grandmont
2013, 3(1) : 21-49 doi: 10.3934/mcrf.2013.3.21 +[Abstract](303) +[PDF](695.4KB) Cited By(8)
Preface: A tribute to professor Eduardo Casas on his 60th birthday
Luis Alberto Fernández, Mariano Mateos, Cecilia Pola, Fredi Tröltzsch and Enrique Zuazua
2018, 8(1) : i-ii doi: 10.3934/mcrf.201801i +[Abstract](336) +[HTML](179) +[PDF](93.68KB) PDF Downloads(85)
Second order optimality conditions for optimal control of quasilinear parabolic equations
Lucas Bonifacius and Ira Neitzel
2018, 8(1) : 1-34 doi: 10.3934/mcrf.2018001 +[Abstract](327) +[HTML](190) +[PDF](640.17KB) PDF Downloads(70)
Optimal control of a non-smooth semilinear elliptic equation
Constantin Christof, Christian Meyer, Stephan Walther and Christian Clason
2018, 8(1) : 247-276 doi: 10.3934/mcrf.2018011 +[Abstract](343) +[HTML](183) +[PDF](584.0KB) PDF Downloads(64)
Optimal control of a two-equation model of radiotherapy
Enrique Fernández-Cara, Juan Límaco and Laurent Prouvée
2018, 8(1) : 117-133 doi: 10.3934/mcrf.2018005 +[Abstract](287) +[HTML](150) +[PDF](625.38KB) PDF Downloads(56)
Optimal control of urban air pollution related to traffic flow in road networks
Lino J. Alvarez-Vázquez, Néstor García-Chan, Aurea Martínez and Miguel E. Vázquez-Méndez
2018, 8(1) : 177-193 doi: 10.3934/mcrf.2018008 +[Abstract](379) +[HTML](195) +[PDF](3968.17KB) PDF Downloads(51)
Asymptotic behavior of a Schrödinger equation under a constrained boundary feedback
Haoyue Cui, Dongyi Liu and Genqi Xu
2018, 8(2) : 383-395 doi: 10.3934/mcrf.2018015 +[Abstract](169) +[HTML](107) +[PDF](434.73KB) PDF Downloads(51)
Error estimates for Dirichlet control problems in polygonal domains: Quasi-uniform meshes
Thomas Apel, Mariano Mateos, Johannes Pfefferer and Arnd Rösch
2018, 8(1) : 217-245 doi: 10.3934/mcrf.2018010 +[Abstract](245) +[HTML](173) +[PDF](577.16KB) PDF Downloads(45)
Tikhonov regularization of optimal control problems governed by semi-linear partial differential equations
Frank Pörner and Daniel Wachsmuth
2018, 8(1) : 315-335 doi: 10.3934/mcrf.2018013 +[Abstract](256) +[HTML](119) +[PDF](476.25KB) PDF Downloads(43)
Compact perturbations of controlled systems
Michel Duprez and Guillaume Olive
2018, 8(2) : 397-410 doi: 10.3934/mcrf.2018016 +[Abstract](155) +[HTML](75) +[PDF](410.77KB) PDF Downloads(36)
Frequency-sparse optimal quantum control
Gero Friesecke, Felix Henneke and Karl Kunisch
2018, 8(1) : 155-176 doi: 10.3934/mcrf.2018007 +[Abstract](299) +[HTML](138) +[PDF](748.6KB) PDF Downloads(35)

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