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Mathematical Control & Related Fields

2014 , Volume 4 , Issue 2

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Local controllability of 1D Schrödinger equations with bilinear control and minimal time
Karine Beauchard and  Morgan Morancey
2014, 4(2): 125-160 doi: 10.3934/mcrf.2014.4.125 +[Abstract](48) +[PDF](657.7KB)
We consider a linear Schrödinger equation, on a bounded interval, with bilinear control.
    In [10], Beauchard and Laurent prove that, under an appropriate non degeneracy assumption, this system is controllable, locally around the ground state, in arbitrary time. In [18], Coron proves that a positive minimal time is required for this controllability result, on a particular degenerate example.
    In this article, we propose a general context for the local controllability to hold in large time, but not in small time. The existence of a positive minimal time is closely related to the behaviour of the second order term, in the power series expansion of the solution.
Internal control of the Schrödinger equation
Camille Laurent
2014, 4(2): 161-186 doi: 10.3934/mcrf.2014.4.161 +[Abstract](54) +[PDF](498.4KB)
In this paper, we intend to present some already known results about the internal controllability of the linear and nonlinear Schrödinger equations.
    After presenting the basic properties of the equation, we give a self contained proof of the controllability in dimension $1$ using some propagation results. We then discuss how to obtain some similar results on a compact manifold where the zone of control satisfies the Geometric Control Condition. We also discuss some known results and open questions when this condition is not satisfied. Then, we present the links between the controllability and some resolvent estimates. Finally, we discuss the additional difficulties when we consider the nonlinear Schrödinger equation.
Optimal insurance in a changing economy
Jingzhen Liu , Ka-Fai Cedric Yiu , Tak Kuen Siu and  Wai-Ki Ching
2014, 4(2): 187-202 doi: 10.3934/mcrf.2014.4.187 +[Abstract](50) +[PDF](392.0KB)
We discuss a general problem of optimal strategies for insurance, consumption and investment in a changing economic environment described by a continuous-time regime switching model. We consider the situation of a random investment horizon which depends on the force of mortality of an economic agent. The objective of the agent is to maximize the expected discounted utility of consumption and terminal wealth over a random future lifetime. A verification theorem for the Hamilton-Jacobi-Bellman (HJB) solution related to the optimal consumption, investment and insurance is provided. In the cases of a power utility and an exponential utility, we derive analytical solutions to the optimal strategies. Numerical results are given to illustrate the proposed model and to document the impact of switching regimes on the optimal strategies.
Carleman estimates for semi-discrete parabolic operators with a discontinuous diffusion coefficient and applications to controllability
Thuy N. T. Nguyen
2014, 4(2): 203-259 doi: 10.3934/mcrf.2014.4.203 +[Abstract](51) +[PDF](785.6KB)
In the discrete setting of one-dimensional finite-differences we prove a Carleman estimate for a semi-discretization of the parabolic operator $\partial_t-\partial_x (c\partial_x )$ where the diffusion coefficient $c$ has a jump. As a consequence of this Carleman estimate, we deduce consistent null-controllability results for classes of semi-linear parabolic equations.
Errata: Controllability of the cubic Schroedinger equation via a low-dimensional source term
Andrey Sarychev
2014, 4(2): 261-261 doi: 10.3934/mcrf.2014.4.261 +[Abstract](43) +[PDF](163.0KB)

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