191 research outputs found
Linear Operator Inequality and Null Controllability with Vanishing Energy for unbounded control systems
We consider linear systems on a separable Hilbert space , which are null
controllable at some time under the action of a point or boundary
control. Parabolic and hyperbolic control systems usually studied in
applications are special cases. To every initial state we
associate the minimal "energy" needed to transfer to in a time ("energy" of a control being the square of its norm). We
give both necessary and sufficient conditions under which the minimal energy
converges to for . This extends to boundary control
systems the concept of null controllability with vanishing energy introduced by
Priola and Zabczyk (Siam J. Control Optim. 42 (2003)) for distributed systems.
The proofs in Priola-Zabczyk paper depend on properties of the associated
Riccati equation, which are not available in the present, general setting. Here
we base our results on new properties of the quadratic regulator problem with
stability and the Linear Operator Inequality.Comment: In this version we have also added a section on examples and
applications of our main results. This version is similar to the one which
will be published on "SIAM Journal on Control and Optimization" (SIAM
On the cost of null-control of an artificial advection-diffusion problem
In this paper we study the null-controllability of an artificial
advection-diffusion system in dimension . Using a spectral method, we prove
that the control cost goes to zero exponentially when the viscosity vanishes
and the control time is large enough. On the other hand, we prove that the
control cost tends to infinity exponentially when the viscosity vanishes and
the control time is small enough.Comment: 16 page
On Some Rigidity Properties in PDEs
This thesis is dedicated to the study of three rigidity properties arising in different partial differential equations: (1) the backward uniqueness property of the heat equation in two-dimensional conical domains, (2) the weak and strong unique continuation principles for fractional Schrödinger equations with rough potentials and (3) the rigidity and non-rigidity of exactly stress-free configurations of a differential inclusion describing the cubic-to-orthorhombic phase transition in the geometrically linearized theory of elasticity
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
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