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Mathematical Control & Related Fields

June 2019 , Volume 9 , Issue 2

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Construction of the minimum time function for linear systems via higher-order set-valued methods
Robert Baier and Thuy T. T. Le
2019, 9(2): 223-255 doi: 10.3934/mcrf.2019012 +[Abstract](1359) +[HTML](541) +[PDF](2855.92KB)
Abstract:

The paper is devoted to introducing an approach to compute the approximate minimum time function of control problems which is based on reachable set approximation and uses arithmetic operations for convex compact sets. In particular, in this paper the theoretical justification of the proposed approach is restricted to a class of linear control systems. The error estimate of the fully discrete reachable set is provided by employing the Hausdorff distance to the continuous-time reachable set. The detailed procedure solving the corresponding discrete set-valued problem is described. Under standard assumptions, by means of convex analysis and knowledge of the regularity of the true minimum time function, we estimate the error of its approximation. Higher-order discretization of the reachable set of the linear control problem can balance missing regularity (e.g., if only Hölder continuity holds) of the minimum time function for smoother problems. To illustrate the error estimates and to demonstrate differences to other numerical approaches we provide a collection of numerical examples which either allow higher order of convergence with respect to time discretization or where the continuity of the minimum time function cannot be sufficiently granted, i.e., we study cases in which the minimum time function is Hölder continuous or even discontinuous.

A partially observed non-zero sum differential game of forward-backward stochastic differential equations and its application in finance
Jie Xiong, Shuaiqi Zhang and Yi Zhuang
2019, 9(2): 257-276 doi: 10.3934/mcrf.2019013 +[Abstract](1463) +[HTML](542) +[PDF](430.53KB)
Abstract:

In this article, we study a class of partially observed non-zero sum stochastic differential game based on forward and backward stochastic differential equations (FBSDEs). It is required that each player has his own observation equation, and the corresponding Nash equilibrium control is required to be adapted to the filtration generated by the observation process. To find the Nash equilibrium point, we establish the maximum principle as a necessary condition and derive the verification theorem as a sufficient condition. Applying the theoretical results and stochastic filtering theory, we obtain the explicit investment strategy of a partial information financial problem.

On a logarithmic stability estimate for an inverse heat conduction problem
Aymen Jbalia
2019, 9(2): 277-287 doi: 10.3934/mcrf.2019014 +[Abstract](1229) +[HTML](551) +[PDF](399.46KB)
Abstract:

We are concerned with an inverse problem arising in thermal imaging in a bounded domain $Ω\subset \mathbb{R}^n$, $n=2, 3$. This inverse problem consists in the determination of the heat exchange coefficient $q(x)$ appearing in the boundary of a heat equation with Robin boundary condition.

Application of the boundary control method to partial data Borg-Levinson inverse spectral problem
Yavar Kian, Morgan Morancey and Lauri Oksanen
2019, 9(2): 289-312 doi: 10.3934/mcrf.2019015 +[Abstract](1344) +[HTML](477) +[PDF](611.59KB)
Abstract:

We consider the multidimensional Borg-Levinson problem of determining a potential \begin{document}$q$\end{document}, appearing in the Dirichlet realization of the Schrödinger operator \begin{document}$A_q = -\Delta+q$\end{document} on a bounded domain \begin{document}$\Omega\subset\mathbb{R}^n$\end{document}, \begin{document}$n\geq2$\end{document}, from the boundary spectral data of \begin{document}$A_q$\end{document} on an arbitrary portion of \begin{document}$\partial\Omega$\end{document}. More precisely, for \begin{document}$\gamma$\end{document} an open and non-empty subset of \begin{document}$\partial\Omega$\end{document}, we consider the boundary spectral data on \begin{document}$\gamma$\end{document} given by \begin{document}${\rm BSD}(q, \gamma): = \{(\lambda_{k}, {\partial_\nu \varphi_{k}}_{|\gamma}):\ k \geq1\}$\end{document}, where \begin{document}$\{ \lambda_k:\ k \geq1\}$\end{document} is the non-decreasing sequence of eigenvalues of \begin{document}$A_q$\end{document}, \begin{document}$\{ \varphi_k:\ k \geq1 \}$\end{document} an associated orthonormal basis of eigenfunctions, and \begin{document}$\nu$\end{document} is the unit outward normal vector to \begin{document}$\partial\Omega$\end{document}. Our main result consists of determining a bounded potential \begin{document}$q\in L^\infty(\Omega)$\end{document} from the data \begin{document}${\rm BSD}(q, \gamma)$\end{document}. Previous uniqueness results, with arbitrarily small \begin{document}$\gamma$\end{document}, assume that \begin{document}$q$\end{document} is smooth. Our approach is based on the Boundary Control method, and we give a self-contained presentation of the method, focusing on the analytic rather than geometric aspects of the method.

The generalised singular perturbation approximation for bounded real and positive real control systems
Chris Guiver
2019, 9(2): 313-350 doi: 10.3934/mcrf.2019016 +[Abstract](886) +[HTML](438) +[PDF](778.2KB)
Abstract:

The generalised singular perturbation approximation (GSPA) is considered as a model reduction scheme for bounded real and positive real linear control systems. The GSPA is a state-space approach to truncation with the defining property that the transfer function of the approximation interpolates the original transfer function at a prescribed point in the closed right half complex plane. Both familiar balanced truncation and singular perturbation approximation are known to be special cases of the GSPA, interpolating at infinity and at zero, respectively. Suitably modified, we show that the GSPA preserves classical dissipativity properties of the truncations, and existing a priori error bounds for these balanced truncation schemes are satisfied as well.

Nonlinear Schrödinger equations on a finite interval with point dissipation
Jing Cui and Shu-Ming Sun
2019, 9(2): 351-384 doi: 10.3934/mcrf.2019017 +[Abstract](1241) +[HTML](482) +[PDF](546.51KB)
Abstract:

The paper considers the initial value problem of a general type of nonlinear Schrödinger equations

posed on a finite domain \begin{document}$ x\in [0, L] $\end{document} with an \begin{document}$ L^2 $\end{document}-stabilizing feedback control law \begin{document}$ u(0, t) = \beta u(L, t), \beta u_x(0, t)-u_x(L, t) = i\alpha u(0, t), $\end{document} where \begin{document}$ L>0 $\end{document}, \begin{document}$ \alpha, \beta $\end{document} are real constants with \begin{document}$ \alpha\beta<0 $\end{document} and \begin{document}$ \beta\neq \pm 1 $\end{document}, and \begin{document}$ f(u) $\end{document} is a smooth function from \begin{document}$ \mathbb{C} $\end{document} to \begin{document}$ \mathbb{C} $\end{document} satisfying some growth conditions. It is shown that for \begin{document}$ s \in \left ( \frac12, 1\right ] $\end{document} and \begin{document}$ w_0 (x) \in H^s(0, L ) $\end{document} with the boundary conditions described above, the problem is locally well-posed for \begin{document}$ u \in C([0, T]; H^s (0, L )) $\end{document}. Moreover, the solution with small initial condition exists globally and approaches to 0 as \begin{document}$ t \rightarrow + \infty $\end{document}.

Characterizations of equilibrium controls in time inconsistent mean-field stochastic linear quadratic problems. I
Tianxiao Wang
2019, 9(2): 385-409 doi: 10.3934/mcrf.2019018 +[Abstract](1227) +[HTML](455) +[PDF](458.19KB)
Abstract:

In this paper, a class of time inconsistent linear quadratic optimal control problems for mean-field stochastic differential equations (SDEs) are considered under Markovian framework. Open-loop equilibrium controls and their particular closed-loop representations are introduced and characterized via variational ideas. Several interesting features are revealed and a system of coupled Riccati equations is derived. In contrast with the analogue optimal control problems of SDEs, the mean-field terms in state equation, which is another reason of time inconsistency, prompts us to define the above two notions in new manners. An interesting result, which is almost trivial in the counterpart problems of SDEs, is given and plays significant role in the previous characterizations. As application, the uniqueness of open-loop equilibrium controls is discussed.

2018  Impact Factor: 1.292

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