ISSN:

2156-8472

eISSN:

2156-8499

## Mathematical Control & Related Fields

December 2012 , Volume 2 , Issue 4

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2012, 2(4): 331-359
doi: 10.3934/mcrf.2012.2.331

*+*[Abstract](1235)*+*[PDF](514.6KB)**Abstract:**

We consider the Cauchy problem associated to the heat equation firstly in a plane domain with a reentrant corner, then in a cracked domain. By constructing a weight function, we prove a Carleman inequality and we deduce a result of controllability.

2012, 2(4): 361-382
doi: 10.3934/mcrf.2012.2.361

*+*[Abstract](937)*+*[PDF](416.1KB)**Abstract:**

In this paper we deal with the local exact controllability to a particular class of trajectories of the $N$-dimensional Boussinesq system with internal controls having $2$ vanishing components. The main novelty of this work is that no condition is imposed on the control domain.

2012, 2(4): 383-398
doi: 10.3934/mcrf.2012.2.383

*+*[Abstract](1155)*+*[PDF](539.6KB)**Abstract:**

The use of combined therapies to treat cancer is common nowadays and some papers already addressed the relative optimization problems. In particular, it is natural to have state constraints, which usually correspond to bounds on feasible amounts of drugs to be used. The application of Pontryagin Maximum Principle is particularly difficult in such case. Therefore, we resort to sufficient conditions for optimality to achieve results more easily applicable to systems biology models. The approach is developed both for candidate value functions and optimal syntheses. Then it is shown at work on some specific problems in combined cancer therapy.

2012, 2(4): 399-427
doi: 10.3934/mcrf.2012.2.399

*+*[Abstract](954)*+*[PDF](519.5KB)**Abstract:**

In this paper we study absolute minimizers and the Aronsson equation for a noncoercive Hamiltonian. We extend the definition of absolutely minimizing functions (in a viscosity sense) for the minimization of the $L^\infty$ norm of a Hamiltonian, within a class of locally Lipschitz continuous functions with respect to possibly noneuclidian metrics. The metric structure is naturally associated to the Hamiltonian and it is related to the a-priori regularity of the family of subsolutions of the Hamilton-Jacobi equation. A special but relevant case contained in our framework is that of Hamiltonians with a Carnot-Carathéodory metric structure determined by a family of vector fields (CC for short in the following), in particular the eikonal Hamiltonian and the corresponding anisotropic infinity-Laplace equation. In this case, the definition of absolute minimizer can be written in an almost classical way, by the theory of Sobolev spaces in a CC setting. In general open domains and with a prescribed continuous Dirichlet boundary condition, we prove the existence of an absolute minimizer which satisfies the Aronsson equation as a viscosity solution. The proof is based on Perron's method and relies on a-priori continuity estimates for absolute minimizers.

2012, 2(4): 429-455
doi: 10.3934/mcrf.2012.2.429

*+*[Abstract](1641)*+*[PDF](494.7KB)**Abstract:**

In this article we study a controllability problem for a parabolic and a hyperbolic partial differential equations in which the control is the shape of the domain where the equation holds. The quantity to be controlled is the trace of the solution into an open subdomain and at a given time, when the right hand side source term is known. The mapping that associates this trace to the shape of the domain is nonlinear. We show (i) an approximate controllability property for the linearized parabolic problem and (ii) an exact local controllability property for the linearized and the nonlinear equations in the hyperbolic case. We then address the same questions in the context of a finite difference spatial semi-discretization in both the parabolic and hyperbolic problems. In this discretized case again we prove a local controllability result for the parabolic problem, and an exact controllability for the hyperbolic case, applying a local surjectivity theorem together with a unique continuation property of the underlying adjoint discrete system.

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