ISSN:

2156-8472

eISSN:

2156-8499

## Mathematical Control & Related Fields

2016 , Volume 6 , Issue 1

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2016, 6(1): 1-25
doi: 10.3934/mcrf.2016.6.1

*+*[Abstract](409)*+*[PDF](557.5KB)**Abstract:**

We study a damped semi-linear wave equation in a bounded domain of $\mathbb{R}^3$ with smooth boundary. It is proved that any $H^2$-smooth solution can be stabilised locally by a finite-dimensional feedback control supported by a given open subset satisfying a geometric condition. The proof is based on an investigation of the linearised equation, for which we construct a stabilising control satisfying the required properties. We next prove that the same control stabilises locally the non-linear problem.

2016, 6(1): 27-52
doi: 10.3934/mcrf.2016.6.27

*+*[Abstract](435)*+*[PDF](1221.7KB)**Abstract:**

The purpose of this work is to establish stability estimates for the unique continuation property of the nonstationary Stokes problem. These estimates hold without prescribing boundary conditions and are of logarithmic type. They are obtained thanks to Carleman estimates for parabolic and elliptic equations. Then, these estimates are applied to an inverse problem where we want to identify a Robin coefficient defined on some part of the boundary from measurements available on another part of the boundary.

2016, 6(1): 53-94
doi: 10.3934/mcrf.2016.6.53

*+*[Abstract](371)*+*[PDF](686.4KB)**Abstract:**

In this paper, we derive a version of the Pontryagin maximum principle for general finite-dimensional nonlinear optimal sampled-data control problems. Our framework is actually much more general, and we treat optimal control problems for which the state variable evolves on a given time scale (arbitrary non-empty closed subset of $\mathbb{R}$), and the control variable evolves on a smaller time scale. Sampled-data systems are then a particular case. Our proof is based on the construction of appropriate needle-like variations and on the Ekeland variational principle.

2016, 6(1): 95-112
doi: 10.3934/mcrf.2016.6.95

*+*[Abstract](536)*+*[PDF](451.9KB)**Abstract:**

In this paper we study a distributed control problem for a phase field system of Caginalp type with logarithmic potential. The main aim of this work would be to force the location of the diffuse interface to be as close as possible to a prescribed set. However, due to the discontinuous character of the cost functional, we have to approximate it by a regular one and, in this case, we solve the associated control problem and derive the related first order necessary optimality conditions.

2016, 6(1): 113-141
doi: 10.3934/mcrf.2016.6.113

*+*[Abstract](372)*+*[PDF](585.4KB)**Abstract:**

In this paper we consider a state constrained differential inclusion $\dot x\in \mathbb A x+ F(t,x)$, with $\mathbb A$ generator of a strongly continuous semigroup in an infinite dimensional separable Banach space. Under an ``inward pointing condition'' we prove a relaxation result stating that the set of trajectories lying in the interior of the constraint is dense in the set of constrained trajectories of the convexified inclusion $\dot x\in \mathbb A x+ \overline{\textrm{co}}F(t,x)$. Some applications to control problems involving PDEs are given.

2016, 6(1): 143-165
doi: 10.3934/mcrf.2016.6.143

*+*[Abstract](381)*+*[PDF](535.8KB)**Abstract:**

In this paper, we study controllability for a parabolic system of chemotaxis. With one control only, the local exact controllability to positive trajectory of the system is obtained by applying Kakutani's fixed point theorem and the null controllability of associated linearized parabolic system. The positivity of the state is shown to be remained in the state space. The control function is shown to be in $L^\infty(Q)$, which is estimated by using the methods of maximal regularity and $L^p$-$L^q$ estimate for parabolic equations.

2016, 6(1): 167-183
doi: 10.3934/mcrf.2016.6.167

*+*[Abstract](499)*+*[PDF](422.2KB)**Abstract:**

In this paper we give results on the counting function associated with the interior transmission eigenvalues. For a complex refraction index we estimate of the counting function by $Ct^{n}$. In the case where the refraction index is positive we give an equivalent of the counting function.

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