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Mathematical Control & Related Fields

September 2012 , Volume 2 , Issue 3

A special issue
Dedicated to Professor Helmut Maurer on the occasion of his 65th birthday

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Numerical null controllability of semi-linear 1-D heat equations: Fixed point, least squares and Newton methods
Enrique Fernández-Cara and Arnaud Münch
2012, 2(3): 217-246 doi: 10.3934/mcrf.2012.2.217 +[Abstract](1779) +[PDF](2080.8KB)
Abstract:
This paper deals with the numerical computation of distributed null controls for semi-linear 1D heat equations, in the sublinear and slightly superlinear cases. Under sharp growth assumptions, the existence of controls has been obtained in [Fernandez-Cara $\&$ Zuazua, Null and approximate controllability for weakly blowing up semi-linear heat equation, 2000] via a fixed point reformulation; see also [Barbu, Exact controllability of the superlinear heat equation, 2000]. More precisely, Carleman estimates and Kakutani's Theorem together ensure the existence of solutions to fixed points for an equivalent fixed point reformulated problem. A nontrivial difficulty appears when we want to extract from the associated Picard iterates a convergent (sub)sequence. In this paper, we introduce and analyze a least squares reformulation of the problem; we show that this strategy leads to an effective and constructive way to compute fixed points. We also formulate and apply a Newton-Raphson algorithm in this context. Several numerical experiments that make it possible to test and compare these methods are performed.
Controllability of the cubic Schroedinger equation via a low-dimensional source term
Andrey Sarychev
2012, 2(3): 247-270 doi: 10.3934/mcrf.2012.2.247 +[Abstract](1612) +[PDF](519.2KB)
Abstract:
We study controllability of $d$-dimensional defocusing cubic Schroe-din-ger equation under periodic boundary conditions. The control is applied additively, via a source term, which is a linear combination of few complex exponentials (modes) with time-variant coefficients - controls. We manage to prove that controlling $2^d$ modes one can achieve controllability of the equation in each finite-dimensional projection of the evolution space $H^{s}(\mathbb{T}^d), \ s>d/2$, as well as approximate controllability in $H^{s}(\mathbb{T}^d)$. We also present a negative result regarding exact controllability of cubic Schroedinger equation via a finite-dimensional source term.
Time-inconsistent optimal control problems and the equilibrium HJB equation
Jiongmin Yong
2012, 2(3): 271-329 doi: 10.3934/mcrf.2012.2.271 +[Abstract](2612) +[PDF](637.9KB)
Abstract:
A general time-inconsistent optimal control problem is considered for stochastic differential equations with deterministic coefficients. Under suitable conditions, a Hamilton-Jacobi-Bellman type equation is derived for the equilibrium value function of the problem. Well-posedness such an equation is studied, and time-consistent equilibrium strategies are constructed. As special cases, the linear-quadratic problem and a generalized Merton's portfolio problem are investigated.

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