Controllability of Evolution Equations with Memory

Chaves-Silva, Felipe W.; Zhang, Xu; Zuazua, Enrique

doi:10.1137/151004239

Cited by 59 publications

(49 citation statements)

References 26 publications

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“…For completeness, we have to mention that spectral accumulation phenomena similar to the ones observed for (4.2) arise in other classes of models, such as PDEs with memory terms ( [8,19,55] Let us consider the following interaction graph in Figure 7 and let…”

Section: The Summentioning

confidence: 84%

Dynamics and control for multi-agent networked systems: A finite-difference approach

Biccari

Zuazua

2019

Math. Models Methods Appl. Sci.

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We analyze the dynamics of multi-agent collective behavior models and its control theoretical properties. We first derive a large population limit to parabolic diffusive equations. We also show that the non-local transport equations commonly derived as the mean-field limit, are subordinated to the first one. In other words, the solution of the non-local transport model can be obtained by a suitable averaging of the diffusive one.We then address the control problem in the linear setting, linking the multi-agent model with the spatial semi-discretization of parabolic equations. This allows us to use the existing techniques for parabolic control problems in the present setting and derive explicit estimates on the cost of controlling these systems as the number of agents tends to infinity. We obtain precise estimates on the time of control and the size of the controls needed to drive the system to consensus, depending on the size of the population considered.Our approach, inspired on the existing results for parabolic equations, possibly of fractional type, and in several space dimensions, shows that the formation of consensus may be understood in terms of the underlying diffusion process described by the heat semi-group. In this way, we are able to give precise estimates on the cost of controllability for these systems as the number of agents increases, both in what concerns the needed control time-horizon and the size of the controls.2010 Mathematics Subject Classification. 34D20, 35B36, 65M06, 92D25, 93B05.

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Section: The Summentioning

confidence: 84%

Dynamics and control for multi-agent networked systems: A finite-difference approach

Biccari

Zuazua

2019

Math. Models Methods Appl. Sci.

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“…On the other hand, the introduction of a memory term may produce accumulation effects that affect the stability of the system. For these reasons, in some recent papers (see, e.g., [2,4,17]), the classical notion of controllability for a wave equation, requiring the state and its velocity to vanish at time T , has been extended with the additional imposition that the control shall shut down also the memory effects, which in our case corresponds to the condition This special notion of controllability is generally called memory-type null controllability. The aim of the present article is to completely analyze the controllability properties of (1.1) in the above mentioned framework.…”

Section: Moreover With (−Dmentioning

confidence: 99%

Null-controllability properties of a fractional wave equation with a memory term

Biccari¹,

Warma²

2020

Evolution Equations &Amp; Control Theory

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We study the null-controllability properties of a one-dimensional wave equation with memory associated with the fractional Laplace operator. The goal is not only to drive the displacement and the velocity to rest at some time-instant but also to require the memory term to vanish at the same time, ensuring that the whole process reaches the equilibrium. The problem being equivalent to a coupled nonlocal PDE-ODE system, in which the ODE component has zero velocity of propagation, we are required to use a moving control strategy. Assuming that the control is acting on an open subset ω(t) which is moving with a constant velocity c ∈ R, the main result of the paper states that the equation is null controllable in a sufficiently large time T and for initial data belonging to suitable fractional order Sobolev spaces. The proof will use a careful analysis of the spectrum of the operator associated with the system and an application of a classical moment method.2010 Mathematics Subject Classification. 30E05, 35L05, 35R11, 45K05, 93B05, 93B60, 93C05.

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“…It is worth mentioning that the idea of making the control to move in order make the system become controllable has been used for many different problems in the past few years, see [6,10,11,15]. Here we will use the approach introduced in [5], based on Carleman inequalities, which allows us to treat multi-dimensional problems.…”

Section: Introductionmentioning

confidence: 99%