# American Institute of Mathematical Sciences

ISSN:
2156-8472

eISSN:
2156-8499

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## Mathematical Control & Related Fields

December 2020 , Volume 10 , Issue 4

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2020, 10(4): 669-698 doi: 10.3934/mcrf.2020015 +[Abstract](1223) +[HTML](518) +[PDF](544.59KB)
Abstract:

In this paper, we treat the problem of uniform exact boundary controllability for the finite-difference space semi-discretization of the \begin{document}$1$\end{document}-\begin{document}$d$\end{document} coupled wave equations with a control acting only in one equation. First, we show how, after filtering the high frequencies of the discrete initial data in an appropriate way, we can construct a sequence of uniformly (with respect to the mesh size) bounded controls. Thus, we prove that the weak limit of the aforementioned sequence is a control for the continuous system. The proof of our results is based on the moment method and on the construction of an explicit biorthogonal sequence.

2020, 10(4): 699-714 doi: 10.3934/mcrf.2020016 +[Abstract](1295) +[HTML](528) +[PDF](360.04KB)
Abstract:

We establish Strichartz estimates for the regularized Schrödinger equation on a two dimensional compact Riemannian manifold without boundary. As a consequence we deduce global existence and uniqueness results for the Cauchy problem for the nonlinear regularized Schrödinger equation and we prove under the geometric control condition the Kato smoothing effect for solutions of this equation in this particular geometries.

2020, 10(4): 715-734 doi: 10.3934/mcrf.2020017 +[Abstract](1147) +[HTML](463) +[PDF](506.79KB)
Abstract:

In 2013, Mao initiated the study of stabilization of continuous-time hybrid stochastic differential equations (SDEs) by feedback control based on discrete-time state observations. In recent years, this study has been further developed while using a constant observation interval. However, time-varying observation frequencies have not been discussed for this study. Particularly for non-autonomous periodic systems, it's more sensible to consider the time-varying property and observe the system at periodic time-varying frequencies, in terms of control efficiency. This paper introduces a periodic observation interval sequence, and investigates how to stabilize a periodic SDE by feedback control based on periodic observations, in the sense that, the controlled system achieves \begin{document}$L^p$\end{document}-stability for \begin{document}$p>1$\end{document}, almost sure asymptotic stability and \begin{document}$p$\end{document}th moment asymptotic stability for \begin{document}$p \ge 2$\end{document}. This paper uses the Lyapunov method and inequalities to derive the theory. We also verify the existence of the observation interval sequence and explain how to calculate it. Finally, an illustrative example is given after a useful corollary. By considering the time-varying property of the system, we reduce the observation frequency dramatically and hence reduce the observational cost for control.

2020, 10(4): 735-759 doi: 10.3934/mcrf.2020018 +[Abstract](1151) +[HTML](373) +[PDF](2569.33KB)
Abstract:

We consider variational discretization [18] of a parabolic optimal control problem governed by space-time measure controls. For the state discretization we use a Petrov-Galerkin method employing piecewise constant states and piecewise linear and continuous test functions in time. For the space discretization we use piecewise linear and continuous functions. As a result the controls are composed of Dirac measures in space-time, centered at points on the discrete space-time grid. We prove that the optimal discrete states and controls converge strongly in \begin{document}$L^q$\end{document} and weakly-\begin{document}$*$\end{document} in \begin{document}$\mathcal{M}$\end{document}, respectively, to their smooth counterparts, where \begin{document}$q \in (1,\min\{2,1+2/d\}]$\end{document} is the spatial dimension. Furthermore, we compare our approach to [8], where the corresponding control problem is discretized employing a discontinuous Galerkin method for the state discretization and where the discrete controls are piecewise constant in time and Dirac measures in space. Numerical experiments highlight the features of our discrete approach.

2020, 10(4): 761-783 doi: 10.3934/mcrf.2020019 +[Abstract](973) +[HTML](536) +[PDF](1326.38KB)
Abstract:

This paper studies the portfolio management problem for an individual with a non-exponential discount function and habit formation in finite time. The investor receives a deterministic income, invests in risky assets, buys insurance and consumes continuously. The objective is to maximize the utility of excessive consumption, heritage and terminal wealth. The non-exponential discounting makes the optimal strategy adopted by a naive person time-inconsistent. The equilibrium for a sophisticated person is Nash subgame perfect equilibrium, and the sophisticated person is time-consistent. We calculate the analytical solution for both the naive strategy and equilibrium strategy in the CRRA case and compare the results of the two strategies. By numerical simulation, we find that the sophisticated individual will spend less on consumption and insurance and save more than the naive person. The difference in the strategies of the naive and sophisticated person decreases over time. Furthermore, if an individual of either type is more patient in the future or has a greater tendency toward habit formation, he/she will consume less and buy less insurance, and the degree of inconsistency will also be increased. The sophisticated person's consumption and habit level are initially lower than those of a naive person but are higher in later periods.

2020, 10(4): 785-826 doi: 10.3934/mcrf.2020020 +[Abstract](1541) +[HTML](467) +[PDF](547.12KB)
Abstract:

In this paper, we investigate a class of time-inconsistent stochastic control problems for stochastic differential equations with deterministic coefficients. We study these problems within the game theoretic framework, and look for open-loop Nash equilibrium controls. Under suitable conditions, we derive a verification theorem for equilibrium controls via a flow of forward-backward stochastic partial differential equations. To illustrate our results, we discuss a mean-variance problem with a state-dependent trade-off between the mean and the variance.

2020, 10(4): 827-854 doi: 10.3934/mcrf.2020021 +[Abstract](926) +[HTML](364) +[PDF](502.43KB)
Abstract:

We study an optimal control problem for a quasi-linear elliptic equation with anisotropic p-Laplace operator in its principal part and \begin{document}$L^1$\end{document}-control in coefficient of the low-order term. We assume that the matrix of anisotropy belongs to BMO-space. Since we cannot expect to have a solution of the state equation in the classical Sobolev space, we introduce a suitable functional class in which we look for solutions and prove existence of optimal pairs using an approximation procedure and compactness arguments in variable spaces.

2020, 10(4): 855-875 doi: 10.3934/mcrf.2020022 +[Abstract](927) +[HTML](429) +[PDF](458.25KB)
Abstract:

We consider stochastic impulse control problems when the impulses cost functions depend on \begin{document}$t$\end{document} and \begin{document}$x$\end{document}. We use the approximation scheme and viscosity solutions approach to show that the value function is a unique viscosity solution for the associated Hamilton-Jacobi-Bellman equation (HJB) partial differential equation (PDE) of stochastic impulse control problems.

2020, 10(4): 877-911 doi: 10.3934/mcrf.2020023 +[Abstract](806) +[HTML](354) +[PDF](902.3KB)
Abstract:

We study one-parametric perturbations of finite dimensional real Hamiltonians depending on two controls, and we show that generically in the space of Hamiltonians, conical intersections of eigenvalues can degenerate into semi-conical intersections of eigenvalues. Then, through the use of normal forms, we study the problem of ensemble controllability between the eigenstates of a generic Hamiltonian.

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