# American Institute of Mathematical Sciences

ISSN:
2156-8472

eISSN:
2156-8499

All Issues

## Mathematical Control & Related Fields

March 2018 , Volume 8 , Issue 1

Special Issue in Honor of Eduardo Casas for his 60th birthday

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2018, 8(1): i-ii doi: 10.3934/mcrf.201801i +[Abstract](3641) +[HTML](317) +[PDF](93.68KB)
Abstract:
2018, 8(1): 1-34 doi: 10.3934/mcrf.2018001 +[Abstract](4066) +[HTML](343) +[PDF](640.17KB)
Abstract:

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.

2018, 8(1): 35-56 doi: 10.3934/mcrf.2018002 +[Abstract](2716) +[HTML](208) +[PDF](474.31KB)
Abstract:

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.

2018, 8(1): 57-88 doi: 10.3934/mcrf.2018003 +[Abstract](3021) +[HTML](203) +[PDF](498.21KB)
Abstract:

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.

2018, 8(1): 89-115 doi: 10.3934/mcrf.2018004 +[Abstract](2770) +[HTML](260) +[PDF](541.74KB)
Abstract:

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.

2018, 8(1): 117-133 doi: 10.3934/mcrf.2018005 +[Abstract](3133) +[HTML](292) +[PDF](625.38KB)
Abstract:

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).

2018, 8(1): 135-153 doi: 10.3934/mcrf.2018006 +[Abstract](2444) +[HTML](180) +[PDF](462.7KB)
Abstract:

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 $ν \searrow 0$.

2018, 8(1): 155-176 doi: 10.3934/mcrf.2018007 +[Abstract](3191) +[HTML](227) +[PDF](748.6KB)
Abstract:

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.

2018, 8(1): 177-193 doi: 10.3934/mcrf.2018008 +[Abstract](3720) +[HTML](336) +[PDF](3968.17KB)
Abstract:

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.

2018, 8(1): 195-215 doi: 10.3934/mcrf.2018009 +[Abstract](2954) +[HTML](201) +[PDF](1052.67KB)
Abstract:

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.

2018, 8(1): 217-245 doi: 10.3934/mcrf.2018010 +[Abstract](2903) +[HTML](324) +[PDF](577.16KB)
Abstract:

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.

2018, 8(1): 247-276 doi: 10.3934/mcrf.2018011 +[Abstract](3832) +[HTML](340) +[PDF](584.0KB)
Abstract:

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.

2018, 8(1): 277-313 doi: 10.3934/mcrf.2018012 +[Abstract](2730) +[HTML](369) +[PDF](4633.98KB)
Abstract:

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.

2018, 8(1): 315-335 doi: 10.3934/mcrf.2018013 +[Abstract](3401) +[HTML](237) +[PDF](476.25KB)
Abstract:

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.

Qi Lü and
2018, 8(1): 337-381 doi: 10.3934/mcrf.2018014 +[Abstract](2671) +[HTML](230) +[PDF](650.51KB)
Abstract:

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.

2018  Impact Factor: 1.292