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Mathematical Control & Related Fields

2016 , Volume 6 , Issue 2

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Partial null controllability of parabolic linear systems
Farid Ammar Khodja, Franz Chouly and Michel Duprez
2016, 6(2): 185-216 doi: 10.3934/mcrf.2016001 +[Abstract](276) +[PDF](592.1KB)
Abstract:
This paper is devoted to the partial null controllability issue of parabolic linear systems with $n$ equations. Given a bounded domain $\Omega$ in $\mathbb{R}^N$ ($N\in \mathbb{N}^*$), we study the effect of $m$ localized controls in a nonempty open subset $\omega$ only controlling $p$ components of the solution ($p,m \le n$). The first main result of this paper is a necessary and sufficient condition when the coupling and control matrices are constant. The second result provides, in a first step, a sufficient condition of partial null controllability when the matrices only depend on time. In a second step, through an example of partially controlled $2\times2$ parabolic system, we will provide positive and negative results on partial null controllability when the coefficients are space dependent.
Accessibility conditions of MIMO nonlinear control systems on homogeneous time scales
Zbigniew Bartosiewicz, Ülle Kotta, Maris Tőnso and Małgorzata Wyrwas
2016, 6(2): 217-250 doi: 10.3934/mcrf.2016002 +[Abstract](240) +[PDF](454.4KB)
Abstract:
A necessary and sufficient accessibility condition for the set of nonlinear higher order input-output (i/o) delta differential equations is presented. The accessibility definition is based on the concept of an autonomous element that is specified to the multi-input multi-output systems. The condition is presented in terms of the greatest common left divisor of two left differential polynomial matrices associated with the system of the i/o delta-differential equations defined on a homogenous time scale which serves as a model of time and unifies the continuous and discrete time. We associate the subspace $\mathcal{H}_{\infty}$ of the vector space of differential one-forms with the considered system. This subspace is invariant with respect to taking delta derivatives. The relation between $\mathcal{H}_\infty$ and the element of a left free module over the ring of left differential polynomials is presented. The presented accessibility condition provides a basis for system reduction, i.e. for finding the transfer equivalent minimal accessible representation of the set of the i/o equations which is a suitable starting point for constructing an observable and accessible state space realization. Moreover, the condition allows to check the transfer equivalence of nonlinear systems, defined on homogeneous time scales.
Determination of time dependent factors of coefficients in fractional diffusion equations
Kenichi Fujishiro and Yavar Kian
2016, 6(2): 251-269 doi: 10.3934/mcrf.2016003 +[Abstract](372) +[PDF](448.5KB)
Abstract:
In the present paper, we consider initial-boundary value problems for partial differential equations with time-fractional derivatives which evolve in $Q=\Omega\times(0,T)$ where $\Omega$ is a bounded domain of $\mathbb{R}^d$ and $T>0$. We study the stability of the inverse problems of determining the time-dependent parameter in a source term or a coefficient of zero-th order term from observations of the solution at a point $x_0\in\overline{\Omega}$ for all $t\in(0,T)$.
Exponential stabilization of Timoshenko beam with input and output delays
Xiu-Fang Liu and Gen-Qi Xu
2016, 6(2): 271-292 doi: 10.3934/mcrf.2016004 +[Abstract](296) +[PDF](461.7KB)
Abstract:
In this paper, we consider the exponential stabilization issue of Timoshenko beam with input and output delays. By using the Luenberger observer and Smith predictor we obtain an estimate of the state of the system, and by the partial state predictor we transform the delayed system into a without delay system, and then by the collocated feedback of the without delay system to obtain the control signal. We prove that under the control signal, the Timoshenko beam with output and input delays can be stabilized exponentially.
An optimal control approach to ciliary locomotion
Jorge San Martín, Takéo Takahashi and Marius Tucsnak
2016, 6(2): 293-334 doi: 10.3934/mcrf.2016005 +[Abstract](269) +[PDF](754.8KB)
Abstract:
We consider a class of low Reynolds number swimmers, of prolate spheroidal shape, which can be seen as simplified models of ciliated microorganisms. Within this model, the form of the swimmer does not change, the propelling mechanism consisting in tangential displacements of the material points of swimmer's boundary. Using explicit formulas for the solution of the Stokes equations at the exterior of a translating prolate spheroid the governing equations reduce to a system of ODE's with the control acting in some of its coefficients (bilinear control system). The main theoretical result asserts the exact controllability of the prolate spheroidal swimmer. In the same geometrical situation, we consider the optimal control problem of maximizing the efficiency during a stroke and we prove the existence of a maximum. We also provide a method to compute an approximation of the efficiency by using explicit formulas for the Stokes system at the exterior of a prolate spheroid, with some particular tangential velocities at the fluid-solid interface. We analyze the sensitivity of this efficiency with respect to the eccentricity of the considered spheroid and show that for small positive eccentricity, the efficiency of a prolate spheroid is better than the efficiency of a sphere. Finally, we use numerical optimization tools to investigate the dependence of the efficiency on the number of inputs and on the eccentricity of the spheroid. The ``best'' numerical result obtained yields an efficiency of $30.66\%$ with $13$ scalar inputs. In the limiting case of a sphere our best numerically obtained efficiency is of $30.4\%$, whereas the best computed efficiency previously reported in the literature is of $22\%$.
Optimal control of a two-phase flow model with state constraints
Theodore Tachim-Medjo
2016, 6(2): 335-362 doi: 10.3934/mcrf.2016006 +[Abstract](352) +[PDF](488.5KB)
Abstract:
We investigate in this article the Pontryagin's maximum principle for a class of control problems associated with a two-phase flow model in a two dimensional bounded domain. The model consists of the Navier-Stokes equations for the velocity $v, $ coupled with a convective Allen-Cahn model for the order (phase) parameter $\phi. $ The optimal problems involve a state constraint similar to that considered in [18]. We derive the Pontryagin's maximum principle for the control problems assuming that a solution exists. Let us note that the coupling between the Navier-Stokes and the Allen-Cahn systems makes the analysis of the control problem more involved. In particular, the associated adjoint systems have less regularity than the one derived in [18].

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