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Mathematical Control & Related Fields

March 2019 , Volume 9 , Issue 1

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Optimal control of the coefficient for the regional fractional $p$-Laplace equation: Approximation and convergence
Harbir Antil and Mahamadi Warma
2019, 9(1): 1-38 doi: 10.3934/mcrf.2019001 +[Abstract](4108) +[HTML](923) +[PDF](617.03KB)

In this paper we study optimal control problems with the regional fractional \begin{document} $p$\end{document}-Laplace equation, of order \begin{document} $s \in \left( {0,1} \right)$ \end{document} and \begin{document} $p \in \left[ {2,\infty } \right)$ \end{document}, as constraints over a bounded open set with Lipschitz continuous boundary. The control, which fulfills the pointwise box constraints, is given by the coefficient of the regional fractional \begin{document} $p$\end{document}-Laplace operator. We show existence and uniqueness of solutions to the state equations and existence of solutions to the optimal control problems. We prove that the regional fractional \begin{document} $p$\end{document}-Laplacian approaches the standard \begin{document} $p$\end{document}-Laplacian as \begin{document} $s$ \end{document} approaches 1. In this sense, this fractional \begin{document} $p$\end{document}-Laplacian can be considered degenerate like the standard \begin{document} $p$\end{document}-Laplacian. To overcome this degeneracy, we introduce a regularization for the regional fractional \begin{document} $p$\end{document}-Laplacian. We show existence and uniqueness of solutions to the regularized state equation and existence of solutions to the regularized optimal control problem. We also prove several auxiliary results for the regularized problem which are of independent interest. We conclude with the convergence of the regularized solutions.

Decay rates for stabilization of linear continuous-time systems with random switching
Fritz Colonius and Guilherme Mazanti
2019, 9(1): 39-58 doi: 10.3934/mcrf.2019002 +[Abstract](3237) +[HTML](817) +[PDF](467.0KB)

For a class of linear switched systems in continuous time a controllability condition implies that state feedbacks allow to achieve almost sure stabilization with arbitrary exponential decay rates. This is based on the Multiplicative Ergodic Theorem applied to an associated system in discrete time. This result is related to the stabilizability problem for linear persistently excited systems.

Robust optimal investment and reinsurance of an insurer under Jump-diffusion models
Xin Zhang, Hui Meng, Jie Xiong and Yang Shen
2019, 9(1): 59-76 doi: 10.3934/mcrf.2019003 +[Abstract](3868) +[HTML](1192) +[PDF](526.9KB)

This paper studies a robust optimal investment and reinsurance problem under model uncertainty. The insurer's risk process is modeled by a general jump process generated by a marked point process. By transferring a proportion of insurance risk to a reinsurance company and investing the surplus into the financial market with a bond and a share index, the insurance company aims to maximize the minimal expected terminal wealth with a penalty. By using the dynamic programming, we formulate the robust optimal investment and reinsurance problem into a two-person, zero-sum, stochastic differential game between the investor and the market. Closed-form solutions for the case of the quadratic penalty function are derived in our paper.

On Algebraic condition for null controllability of some coupled degenerate systems
Ait Ben Hassi El Mustapha, Fadili Mohamed and Maniar Lahcen
2019, 9(1): 77-95 doi: 10.3934/mcrf.2019004 +[Abstract](3076) +[HTML](751) +[PDF](445.76KB)

In this paper we will generalize the Kalman rank condition for the null controllability to \begin{document}$n$\end{document}-coupled linear degenerate parabolic systems with constant coefficients, diagonalizable diffusion matrix, and \begin{document}$m$\end{document}-controls. For that we prove a global Carleman estimate for the solutions of a scalar \begin{document}$2n$\end{document}-order parabolic equation then we infer from it an observability inequality for the corresponding adjoint system, and thus the null controllability.

Stabilization of multidimensional wave equation with locally boundary fractional dissipation law under geometric conditions
Mohammad Akil and Ali Wehbe
2019, 9(1): 97-116 doi: 10.3934/mcrf.2019005 +[Abstract](3227) +[HTML](663) +[PDF](704.28KB)

In this paper, we consider a multidimensional wave equation with boundary fractional damping acting on a part of the boundary of the domain. First, combining a general criteria of Arendt and Batty with Holmgren's theorem we show the strong stability of our system in the absence of the compactness of the resolvent and without any additional geometric conditions. Next, we show that our system is not uniformly stable in general, since it is the case of the interval. Hence, we look for a polynomial decay rate for smooth initial data for our system by applying a frequency domain approach combining with a multiplier method. Indeed, by assuming that the boundary control region satisfy some geometric conditions and by using the exponential decay of the wave equation with a standard damping, we establish a polynomial energy decay rate for smooth solutions, which depends on the order of the fractional derivative.

Insensitizing controls for a semilinear parabolic equation: A numerical approach
Franck Boyer, Víctor Hernández-Santamaría and Luz De Teresa
2019, 9(1): 117-158 doi: 10.3934/mcrf.2019007 +[Abstract](3030) +[HTML](612) +[PDF](956.46KB)

In this paper, we study the insensitizing control problem in the discrete setting of finite-differences. We prove the existence of a control that insensitizes the norm of the observed solution of a 1-D semi discrete parabolic equation. We derive a (relaxed) observability estimate that yields a controllability result for the cascade system arising in the insensitizing control formulation. Moreover, we deal with the problem of computing numerical approximations of insensitizing controls for the heat equation by using the Hilbert Uniqueness Method (HUM). We present various numerical illustrations.

Randomized algorithms for stabilizing switching signals
Niranjan Balachandran, Atreyee Kundu and Debasish Chatterjee
2019, 9(1): 159-174 doi: 10.3934/mcrf.2019009 +[Abstract](2716) +[HTML](623) +[PDF](458.36KB)

Qualitative behaviour of switched systems has attracted considerable research attention in the recent past. In this article we study 'how likely' is it for a family of systems to admit stabilizing switching signals. A weighted digraph is associated to a switched system in a natural fashion, and the switching signal is expressed as an infinite walk on this digraph. We provide a linear time probabilistic algorithm to find cycles on this digraph that have a desirable property (we call it "contractivity"), and under mild statistical hypotheses on the connectivity and weights of the digraph, demonstrate that there exist uncountably many stabilizing switching signals derived from such cycles. Our algorithm does not require the vertex and edge weights to be stored in memory prior to its application, has a learning/exploratory character, and naturally fits very large scale systems.

Extension of the strong law of large numbers for capacities
Zengjing Chen, Weihuan Huang and Panyu Wu
2019, 9(1): 175-190 doi: 10.3934/mcrf.2019010 +[Abstract](3185) +[HTML](776) +[PDF](403.95KB)

In this paper, with a new notion of exponential independence for random variables under an upper expectation, we establish a kind of strong laws of large numbers for capacities. Our limit theorems show that the cluster points of empirical averages not only lie in the interval between the upper expectation and the lower expectation with lower probability one, but such an interval also is the unique smallest interval of all intervals in which the limit points of empirical averages lie with lower probability one. Furthermore, we also show that the cluster points of empirical averages could reach the upper expectation and lower expectation with upper probability one.

Boundary controllability for a one-dimensional heat equation with a singular inverse-square potential
Umberto Biccari
2019, 9(1): 191-219 doi: 10.3934/mcrf.2019011 +[Abstract](2995) +[HTML](553) +[PDF](523.99KB)

We analyze controllability properties for the one-dimensional heat equation with singular inverse-square potential

For any \begin{document}$\mu<1/4$\end{document}, we prove that the equation is null controllable through a boundary control \begin{document}$f\in H^1(0, T)$\end{document} acting at the singularity point x = 0. This result is obtained employing the moment method by Fattorini and Russell.

Erratum on: Controllability of the heat and wave equations and their finite difference approximations by the shape of the domain
Jonathan Touboul
2019, 9(1): 221-222 doi: 10.3934/mcrf.2019006 +[Abstract](2832) +[HTML](589) +[PDF](200.41KB)

2019  Impact Factor: 0.857




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