Control Of Parabolic Equations Research Articles

Observability and control of parabolic equations on networks with loops

Network theory can be useful for studying complex systems such as those that arise, for example, in physical sciences, engineering, economics and sociology. In this paper, we prove the observability of parabolic equations on networks with loops. By using a novel Carleman inequality, we find that the observability of the entire network can be achieved under certain hypothesis about the position of the observation domain. The main difficulty we tackle, due to the existence of loops, is to avoid entering into a circular fallacy, notably in the construction of the auxiliary function for the Carleman inequality. The difficulty is overcome with a careful treatment of the boundary terms on the junctions. Finally, we use the observability to prove the null controllability of the network and to obtain the Lipschitz stability for an inverse problem consisting on retrieving a stationary potential in the parabolic equation from measurements on the observation domain.

Cost of null controllability for parabolic equations with vanishing diffusivity and a transport term

In this paper we consider the heat equation with Neumann, Robin and mixed boundary conditions (with coefficients on the boundary which depend on the space variable). The main results concern the behaviour of the cost of the null controllability with respect to the diffusivity when the control acts in the interior. First, we prove that if we almost have Dirichlet boundary conditions in the part of the boundary in which the flux of the transport enters, the cost of the controllability decays for a time T sufficiently large. Next, we show some examples of Neumann and mixed boundary conditions in which for any time T > 0 the cost explodes exponentially as the diffusivity vanishes. Finally, we study the cost of the problem with Neumann boundary conditions when the control is localized in the whole domain.

Optimal control of parabolic equations by spectral decomposition

This paper deals with an optimal control problem of Bolza type for a class of parabolic equations. It consists in finding the initial datum that minimizes a cost functional, which comprises an energy term on the control and a regulation term given by a distributed cost on the state, and such that the final state lies within a prescribed distance to a given target. Beyond the particularities of the chosen problem, our aim here is to propose a new methodology based on a spectral decomposition of the operator governing the evolution of the system, describe its implementation for this kind of problem, and finally perform some preliminary numerical experiments to assess its behaviour.

Optimal Wentzell Boundary Control of Parabolic Equations

This paper deals with a class of optimal control problems governed by an initial-boundary value problem of a parabolic equation. The case of semi-linear boundary control is studied where the control is applied to the system via the Wentzell boundary condition. The differentiability of the state variable with respect to the control is established and hence a necessary condition is derived for the optimal solution in the case of both unconstrained and constrained problems. The condition is also sufficient for the unconstrained convex problems. A second order condition is also derived.

On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations

Local and global Carleman estimates play a central role in the study of some partial differential equations regarding questions such as unique continuation and controllability. We survey and prove such estimates in the case of elliptic and parabolic operators by means of semi-classical microlocal techniques. Optimality results for these estimates and some of their consequences are presented. We point out the connexion of these optimality results to the local phase-space geometry after conjugation with the weight function. Firstly, we introduce local Carleman estimates for elliptic operators and deduce unique continuation properties as well as interpolation inequalities. These latter inequalities yield a remarkable spectral inequality and the null controllability of the heat equation. Secondly, we prove Carleman estimates for parabolic operators. We state them locally in space at first, and patch them together to obtain a global estimate. This second approach also yields the null controllability of the heat equation.

Carleman Inequality and Null Controllability for Parabolic Equations

Carleman Inequality and Null Controllability for Parabolic Equations

Controllability for parabolic equations with nonlinear memory

This paper is concerned with the controllability of a parabolic system with nonlinear memory. Based on the localized estimate of the solution, we prove that the system with a superlinear growth memory is not controllable. Furthermore, two controllability results for some initial data and targets are given as well.

A global Carleman inequality and exact controllability of parabolic equations with mixed boundary conditions

This paper establishes a global Carleman inequality of parabolic equations with mixed boundary conditions and an estimate of the solution. Further, we prove exact controllability of the equation by controls acting on an arbitrarily given subdomain or subboundary.

Quadratic Pareto optimal control of parabolic equation with state-control constraints and an infinite number of variables

Quadratic Pareto optimal control of parabolic equation with state-control constraints and an infinite number of variables

Existence, uniqueness and controllability for parabolic equations in non-cylindrical domains

Existence, uniqueness and controllability for parabolic equations in non-cylindrical domains

Controllability of parabolic equations

This paper studies two problems of exact controllability of semilinear parabolic equations. In the first, the control is on the right-hand side of the parabolic equation and distributed over an arbitrary subdomain of the domain . In the second, the control is contained in the boundary conditions and distributed over a subdomain of the boundary . If the original data satisfy certain conditions, then both problems are solvable.

Controllability for parabolic equations with uniformly bounded nonlinear terms

We consider a control system for a parabolic equation in a Banach space with uniformly bounded nonlinear termF, $$dz(t)/dt + Az(t) = F(t, z(t)) + Bf(t), t > 0.$$ Here,Bf(t) corresponds to a finite-dimensional control. We prove the equivalence of approximate controllability for the above nonlinear system and that for the linear system $$dz(t)/dt + Az(t) = Bf(t), t > 0.$$ Our method is based on results on approximate controllability for linear parabolic systems and estimates of solutions. A practical example is also given for a diffusion and reaction model.