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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.

  • AIMS is a member of COPE. All AIMS journals adhere to the publication ethics and malpractice policies outlined by COPE.
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Preface A tribute to professor Eduardo Casas on his 60TH birthday
2018, 8(1) : i-ii doi: 10.3934/mcrf.201801i +[Abstract](34) +[HTML](9) +[PDF](93.68KB)
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](4) +[HTML](2) +[PDF](640.17KB)

We discuss an optimal control problem governed by a quasilinear parabolic PDE including mixed boundary conditions and Neumann boundary control, as well as distributed control. Second order necessary and sufficient optimality conditions are derived. The latter leads to a quadratic growth condition without two-norm discrepancy. Furthermore, maximal parabolic regularity of the state equation in Bessel-potential spaces \begin{document} $H_D^{-\zeta,p}$ \end{document} with uniform bound on the norm of the solution operator is proved and used to derive stability results with respect to perturbations of the nonlinear differential operator.

Optimal voltage control of non-stationary eddy current problems
Fredi Tröltzsch  and  Alberto Valli 
2018, 8(1) : 35-56 doi: 10.3934/mcrf.2018002 +[Abstract](3) +[HTML](3) +[PDF](474.31KB)

A mathematical model is set up that can be useful for controlled voltage excitation in time-dependent electromagnetism.The well-posedness of the model is proved and an associated optimal control problem is investigated. Here, the controlfunction is a transient voltage and the aim of the control is the best approximation of desired electric and magnetic fields insuitable \begin{document} $L^2$ \end{document}-norms.Special emphasis is laid on an adjoint calculus for first-order necessary optimality conditions.Moreover, a peculiar attention is devoted to propose a formulation for which the computational complexity of the finite element solution method is substantially reduced.

Second-order necessary conditions for optimal control of semilinear elliptic equations with leading term containing controls
Hongwei Lou  and  Jiongmin Yong 
2018, 8(1) : 57-88 doi: 10.3934/mcrf.2018003 +[Abstract](3) +[HTML](3) +[PDF](498.21KB)

An optimal control problem for a semilinear elliptic equation of divergenceform is considered. Both the leading term and the semilinear term of the state equationcontain the control. The well-known Pontryagin type maximum principle for the optimal controls is the first-order necessary condition. When such a first-order necessary condition is singular in some sense, certain type of the second-order necessary condition will come in naturally. The aim of this paper is to explore such kind of conditions for our optimal control problem.

Drag minimization for the obstacle in compressible flow using shape derivatives and finite volumes
Anna Kaźmierczak  , Jan Sokolowski  and  Antoni Zochowski 
2018, 8(1) : 89-115 doi: 10.3934/mcrf.2018004 +[Abstract](2) +[HTML](3) +[PDF](541.74KB)

In the paper the shape optimization problem for the static, compressible Navier-Stokes equations is analyzed. The drag minimizing of an obstacle immersed in the gas stream is considered. The continuous gradient of the drag is obtained by application of the sensitivity formulas derived in the works of one of the co-authors. The numerical approximation scheme uses mixed Finite Volume - Finite Element formulation. The novelty of our numerical method is a particular choice of the regularizing term for the non-homogeneous Stokes boundary value problem, which may be tuned to obtain the best accuracy. The convergence of the discrete solutions for the model under considerations is proved. The non-linearity of the model is treated by means of the fixed point procedure. The numerical example of an optimal shape is given.

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](2) +[HTML](3) +[PDF](625.38KB)

This paper deals with the optimal control of a mathematical model for the evolution of a low-grade glioma (LGG). We will consider a model of the Fischer-Kolmogorov kind for two compartments of tumor cells, using ideas from Galochkina, Bratus and Pérez-García [10] and Pérez-García [17]. The controls are of the form $(t_1, \dots, t_n; d_1, \dots, d_n)$, where $t_i$ is the $i$-th administration time and $d_i$ is the $i$-th applied radiotherapy dose. In the optimal control problem, we try to find controls that maximize, in an admissible class, the first time at which the tumor mass reaches a critical value $M_{*}$. We present an existence result and, also, some numerical experiments (in the previous paper [7], we have considered and solved a very similar control problem where tumoral cells of only one kind appear).

On the switching behavior of sparse optimal controls for the one-dimensional heat equation
Fredi Tröltzsch  and  Daniel Wachsmuth 
2018, 8(1) : 135-153 doi: 10.3934/mcrf.2018006 +[Abstract](2) +[HTML](3) +[PDF](462.7KB)

An optimal boundary control problem for the one-dimensional heat equation is considered. The objective functional includes a standard quadratic terminal observation, a Tikhonov regularization term with regularization parameter $ν$, and the $L^1$-norm of the control that accounts for sparsity. The switching structure of the optimal control is discussed for $ν ≥ 0$. Under natural assumptions, it is shown that the set of switching points of the optimal control is countable with the final time as only possible accumulation point. The convergence of switching points is investigated for $ν \searrow 0$.

Frequency-sparse optimal quantum control
Gero Friesecke  , Felix Henneke  and  Karl Kunisch 
2018, 8(1) : 155-176 doi: 10.3934/mcrf.2018007 +[Abstract](3) +[HTML](4) +[PDF](748.6KB)

A new class of cost functionals for optimal control of quantum systems which produces controls which are sparse in frequency and smooth in time is proposed. This is achieved by penalizing a suitable time-frequency representation of the control field, rather than the control field itself, and by employing norms which are of $L^1$ or measure form with respect to frequency but smooth with respect to time.

We prove existence of optimal controls for the resulting nonsmooth optimization problem, derive necessary optimality conditions, and rigorously establish the frequency-sparsity of the optimizers. More precisely, we show that the time-frequency representation of the control field, which a priori admits a continuum of frequencies, is supported on only finitely many frequencies. These results cover important systems of physical interest, including (infinite-dimensional) Schrödinger dynamics on multiple potential energy surfaces as arising in laser control of chemical reactions. Numerical simulations confirm that the optimal controls, unlike those obtained with the usual $L^2$ costs, concentrate on just a few frequencies, even in the infinite-dimensional case of laser-controlled chemical reactions.

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](2) +[HTML](5) +[PDF](3968.17KB)

Air pollution is one of the most important environmental problems nowadays. In large metropolitan areas, the main source of pollution is vehicular traffic. Consequently, the search for traffic measures that help to improve pollution levels has become a hot topic today. In this article, combining a 1D model to simulate the traffic flow over a road network with a 2D model for pollutant dispersion, we present a tool to search for traffic operations that are optimal in terms of pollution. The utility of this tool is illustrated by formulating the problem of the expansion of a road network as a problem of optimal control of partial differential equations. We propose a complete algorithm to solve the problem, and present some numerical results obtained in a realistic situation posed in the Guadalajara Metropolitan Area (GMA), Mexico.

Error analysis for global minima of semilinear optimal control problems
Ahmad Ahmad Ali  , Klaus Deckelnick  and  Michael Hinze 
2018, 8(1) : 195-215 doi: 10.3934/mcrf.2018009 +[Abstract](2) +[HTML](3) +[PDF](1052.67KB)

In [2] we consider an optimal control problem subject to a semilinear elliptic PDE together with its variational discretization, where we provide a condition which allows to decide whether a solution of the necessary first order conditions is a global minimum. This condition can be explicitly evaluated at the discrete level. Furthermore, we prove that if the above condition holds uniformly with respect to the discretization parameter the sequence of discrete solutions converges to a global solution of the corresponding limit problem. With the present work we complement our investigations of [2] in that we prove an error estimate for those discrete global solutions. Numerical experiments confirm our analytical findings.

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](5) +[HTML](2) +[PDF](577.16KB)

The paper deals with finite element approximations of elliptic Dirichlet boundary control problems posed on two-dimensional polygonal domains. Error estimates are derived for the approximation of the control and the state variables. Special features of unconstrained and control constrained problems as well as general quasi-uniform meshes and superconvergent meshes are carefully elaborated. Compared to existing results, the convergence rates for the control variable are not only improved but also fully explain the observed orders of convergence in the literature. Moreover, for the first time, results in nonconvex domains are provided.

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](8) +[HTML](4) +[PDF](584.0KB)

This paper is concerned with an optimal control problem governed by a non-smooth semilinear elliptic equation. We show that the control-to-state mapping is directionally differentiable and precisely characterize its Bouligand sub-differential. By means of a suitable regularization, first-order optimality conditions including an adjoint equation are derived and afterwards interpreted in light of the previously obtained characterization. In addition, the directional derivative of the control-to-state mapping is used to establish strong stationarity conditions. While the latter conditions are shown to be stronger, we demonstrate by numerical examples that the former conditions are amenable to numerical solution using a semi-smooth Newton method.

Water artificial circulation for eutrophication control
Aurea Martínez  , Francisco J. Fernández  , Lino J. Alvarez-Vázquez  and  Christian Clason 
2018, 8(1) : 277-313 doi: 10.3934/mcrf.2018012 +[Abstract](3) +[HTML](10) +[PDF](4633.98KB)

This work analyzes, from a mathematical point of view, the artificial mixing of water -by means of several pairs collector/injector that set up a circulation pattern in the waterbody -in order to prevent the undesired effects of eutrophication. The environmental problem is formulated as a constrained optimal control problem of partial differential equations, where the state system is related to the velocity of water and to the concentrations of the different species involved in the eutrophication processes, and the cost function to be minimized represents the volume of recirculated water. In the main part of the work, the wellposedness of the problem and the existence of an optimal control is demonstrated. Finally, a complete numerical algorithm for its computation is presented, and some numerical results for a realistic problem are also given.

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](2) +[HTML](3) +[PDF](476.25KB)

In this article, we consider the Tikhonov regularization of an optimal control problem of semilinear partial differential equations with box constraints on the control. We derive a-priori regularization error estimates for the control under suitable conditions. These conditions comprise second-order sufficient optimality conditions as well as regularity conditions on the control, which consists of a source condition and a condition on the active sets. In addition, we show that these conditions are necessary for convergence rates under certain conditions. We also consider sparse optimal control problems and derive regularization error estimates for them. Numerical experiments underline the theoretical findings.

Operator-valued backward stochastic Lyapunov equations in infinite dimensions, and its application
Qi Lü  and  Xu Zhang 
2018, 8(1) : 337-381 doi: 10.3934/mcrf.2018014 +[Abstract](1) +[HTML](3) +[PDF](650.51KB)

We establish the well-posedness of operator-valued backward stochastic Lyapunov equations in infinite dimensions, in the sense of \begin{document}$ V $\end{document}-transposition solution and of relaxed transposition solution. As an application, we obtain a Pontryagin-type maximum principle for the optimal control of controlled stochastic evolution equations.

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](73) +[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](63) +[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](71) +[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](89) +[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](74) +[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](71) +[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](69) +[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](64) +[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](54) +[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](59) +[PDF](695.4KB) Cited By(8)
Preface A tribute to professor Eduardo Casas on his 60TH birthday
2018, 8(1) : i-ii doi: 10.3934/mcrf.201801i +[Abstract](34) +[HTML](9) +[PDF](93.68KB) PDF Downloads(13)
Water artificial circulation for eutrophication control
Aurea Martínez  , Francisco J. Fernández  , Lino J. Alvarez-Vázquez  and  Christian Clason 
2018, 8(1) : 277-313 doi: 10.3934/mcrf.2018012 +[Abstract](3) +[HTML](10) +[PDF](4633.98KB) PDF Downloads(7)
Necessary conditions for a weak minimum in a general optimal control problem with integral equations on a variable time interval
Andrei V. Dmitruk  and  Nikolai P. Osmolovski 
2017, 7(4) : 507-535 doi: 10.3934/mcrf.2017019 +[Abstract](99) +[HTML](2) +[PDF](452.1KB) PDF Downloads(7)
Controllability of fractional dynamical systems: A functional analytic approach
Venkatesan Govindaraj  and  Raju K. George 
2017, 7(4) : 537-562 doi: 10.3934/mcrf.2017020 +[Abstract](70) +[HTML](2) +[PDF](452.1KB) PDF Downloads(4)
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](89) +[PDF](614.2KB) PDF Downloads(4)
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](8) +[HTML](4) +[PDF](584.0KB) PDF Downloads(4)
Optimal control of a vector-host epidemics model
Qingkai Kong  , Zhipeng Qiu  , Zi Sang  and  Yun Zou 
2011, 1(4) : 493-508 doi: 10.3934/mcrf.2011.1.493 +[Abstract](51) +[PDF](556.4KB) PDF Downloads(4)
Stability and controllability of a wave equation with dynamical boundary control
Bopeng Rao  , Laila Toufayli  and  Ali Wehbe 
2015, 5(2) : 305-320 doi: 10.3934/mcrf.2015.5.305 +[Abstract](58) +[PDF](389.5KB) PDF Downloads(3)
Exact controllability of scalar conservation laws with strict convex flux
Adimurthi   , Shyam Sundar Ghoshal  and  G. D. Veerappa Gowda 
2014, 4(4) : 401-449 doi: 10.3934/mcrf.2014.4.401 +[Abstract](98) +[PDF](657.7KB) PDF Downloads(3)
Exact controllability for the Lamé system
Belhassen Dehman  and  Jean-Pierre Raymond 
2015, 5(4) : 743-760 doi: 10.3934/mcrf.2015.5.743 +[Abstract](191) +[PDF](428.1KB) PDF Downloads(3)

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