178 research outputs found
Sparse Stabilization and Control of Alignment Models
From a mathematical point of view self-organization can be described as
patterns to which certain dynamical systems modeling social dynamics tend
spontaneously to be attracted. In this paper we explore situations beyond
self-organization, in particular how to externally control such dynamical
systems in order to eventually enforce pattern formation also in those
situations where this wished phenomenon does not result from spontaneous
convergence. Our focus is on dynamical systems of Cucker-Smale type, modeling
consensus emergence, and we question the existence of stabilization and optimal
control strategies which require the minimal amount of external intervention
for nevertheless inducing consensus in a group of interacting agents. We
provide a variational criterion to explicitly design feedback controls that are
componentwise sparse, i.e. with at most one nonzero component at every instant
of time. Controls sharing this sparsity feature are very realistic and
convenient for practical issues. Moreover, the maximally sparse ones are
instantaneously optimal in terms of the decay rate of a suitably designed
Lyapunov functional, measuring the distance from consensus. As a consequence we
provide a mathematical justification to the general principle according to
which "sparse is better" in the sense that a policy maker, who is not allowed
to predict future developments, should always consider more favorable to
intervene with stronger action on the fewest possible instantaneous optimal
leaders rather than trying to control more agents with minor strength in order
to achieve group consensus. We then establish local and global sparse
controllability properties to consensus and, finally, we analyze the sparsity
of solutions of the finite time optimal control problem where the minimization
criterion is a combination of the distance from consensus and of the l1-norm of
the control.Comment: 33 pages, 5 figure
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Challenges in Optimal Control of Nonlinear PDE-Systems
The workshop focussed on various aspects of optimal control problems for systems of nonlinear partial differential equations. In particular, discussions around keynote presentations in the areas of optimal control of nonlinear/non-smooth systems, optimal control of systems involving nonlocal operators, shape and topology optimization, feedback control and stabilization, sparse control, and associated numerical analysis as well as design and analysis of solution algorithms were promoted. Moreover, also aspects of control of fluid structure interaction problems as well as problems arising in the optimal control of quantum systems were considered
Control and stabilization of waves on 1-d networks
We present some recent results on control and stabilization of waves on 1-d networks.The fine time-evolution of solutions of wave equations on networks and, consequently, their control theoretical properties, depend in a subtle manner on the topology of the network under consideration and also on the number theoretical properties of the lengths of the strings entering in it. Therefore, the overall picture is quite complex.In this paper we summarize some of the existing results on the problem of controllability that, by classical duality arguments in control theory, can be reduced to that of observability of the adjoint uncontrolled system. The problem of observability refers to that of recovering the total energy of solutions by means of measurements made on some internal or external nodes of the network. They lead, by duality, to controllability results guaranteeing that L 2-controls located on those nodes may drive sufficiently smooth solutions to equilibrium at a final time. Most of our results in this context, obtained in collaboration with R. Dáger, refer to the problem of controlling the network from one single external node. It is, to some extent, the most complex situation since, obviously, increasing the number of controllers enhances the controllability properties of the system. Our methods of proof combine sidewise energy estimates (that in the particular case under consideration can be derived by simply applying the classical d'Alembert's formula), Fourier series representations, non-harmonic Fourier analysis, and number theoretical tools.These control results belong to the class of the so-called open-loop control systems.We then discuss the problem of closed-loop control or stabilization by feedback. We present a recent result, obtained in collaboration with J. Valein, showing that the observability results previously derived, regardless of the method of proof employed, can also be recast a posteriori in the context of stabilization, so to derive explicit decay rates (as) for the energy of smooth solutions. The decay rate depends in a very sensitive manner on the topology of the network and the number theoretical properties of the lengths of the strings entering in it.In the end of the article we also present some challenging open problems
Explicit approximate controllability of the Schr\"odinger equation with a polarizability term
We consider a controlled Schr\"odinger equation with a dipolar and a
polarizability term, used when the dipolar approximation is not valid. The
control is the amplitude of the external electric field, it acts non linearly
on the state. We extend in this infinite dimensional framework previous
techniques used by Coron, Grigoriu, Lefter and Turinici for stabilization in
finite dimension. We consider a highly oscillating control and prove the
semi-global weak stabilization of the averaged system using a Lyapunov
function introduced by Nersesyan. Then it is proved that the solutions of the
Schr\"odinger equation and of the averaged equation stay close on every finite
time horizon provided that the control is oscillating enough. Combining these
two results, we get approximate controllability to the ground state for the
polarizability system
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