Exact controllability of the heat equation with bilinear control

This paper deals with exact controllability of bilinear heat equation. Namely, given the initial state, we would like to provide a class of target states that can be achieved through the heat equation at a finite time by applying multiplicative controls. For this end, an explicit control strategy is constructed. Simulations are provided. Copyright © 2015 John Wiley & Sons, Ltd.     

Null controllability of a pseudo-parabolic equation with moving control

In this paper, we study the internal controllability of the pseudo-parabolic equation on the one-dimensional torus. Our control function is acting on a moving small interval with a constant velocity. With this moving distributed control, we obtain the system is null controllable for given data with certain regularity.     

Null controllability for the heat equation with singular inverse-square potentials

We prove the null controllability of the heat equation perturbed by a singular inverse-square potential arising in quantum mechanics and combustion theory. This is done within the range of subcritical coefficients of the singular potential, provided the control acts on an annular set around the singularity. Our proof uses a splitting argument on the domain, decomposition in spherical harmonics, new Carleman inequalities and refined Hardy inequalities.     

Exact controllability of the wave equation with Neumann boundary control

We consider the wave equation defined on a smooth bounded domainR ⁿ with boundary =₀₁, with ₀ possibly empty and ₁ nonempty and relatively open in . The control action is exercised in the Neumann boundary conditions only on ₁, while homogeneous boundary conditions of Dirichlet type are imposed on the complementary part ₀. We study by a direct method (i.e., without passing through uniform stabilization) the problem of exact controllability on some finite time interval [0,T] for initial data on some preassigned spaceZ=Z ₁ ×Z ₂ based on and with control functions in some preassigned space based on ₁ and [0,T]. We consider several choices of pairs [Z, ] of spaces, and others may be likewise studied by similar methods. Our main results are exact controllability results in the following cases: (i) and and , both under suitable geometrical conditions on the triplet {, ₀, ₁} expressed in terms of a general vector field; (iii)Z = L ² ()×[H ¹ ()] in the Neumann case ₀=Ø in the absence of geometrical conditions on , but with a special classV of controls, larger thanL ² (). The key technical issues are, in all cases, lower bounds on theL ² ( ₁)-norm of appropriate traces of the solution to the corresponding homogeneous problem. These are obtained by multiplier techniques.This paper was presented by the second named author at the IFIP WG 7.2 Conference on Boundary Control and Boundary Variations held at the Department de Mathématiques, Universite de Nice, France, June 10–13, 1986 (a preliminary version will appear as Lectures Notes in Control Sciences, edited by J. P. Zolezio [T4]); at the International Conference on Control of Distributed Parameter Systems, held at Vorau, Austria, July 6–12, 1986; at the Second Workshop on Control of Systems Governed by Partial Differential Equations, held at Val David, Quebec, Canada, October 5–9, 1986; and at the 26th IEEE Conference on Decision and Control, held at Los Angeles, CA, December 9–11, 1987 [LT6]. This research was partially supported by the National Science Foundation under Grant No. NSF 8301668 and by the Air Force Office of Scientific Research under Grant No. AFOSR-84-0365.     

On the controllability of the heat equation with nonlinear boundary Fourier conditions

In this paper we analyze the approximate and null controllability of the classical heat equation with nonlinear boundary conditions of the form and distributed controls, with support in a small set. We show that, when the function f is globally Lipschitz-continuous, the system is approximately controllable. We also show that the system is locally null controllable and null controllable for large time when f is regular enough and f(0)=0. For the proofs of these assertions, we use controllability results for similar linear problems and appropriate fixed point arguments. In the case of the local and large time null controllability results, the arguments are rather technical, since they need (among other things) Hölder estimates for the control and the state.     

Analytic transient solutions of a cylindrical heat equation with a heat source

The method of separation of variables is applied in order to investigate the analytical solutions of a certain two-dimensional cylindrical heat equation. In the analysis presented here, the partial differential equation is directly transformed into ordinary differential equations. The closed-form transient temperature distributions and heat transfer rates are generalized for a linear combination of the products of Fourier-Bessel series of the exponential type. Relevant connections with some other closely-related recent works are also indicated.     

On the treatment of free boundary problems with the heat equation via optimal control

In this paper we consider a free boundary problem of general type for the heat equation in one space dimension. We formulate this problem as an optimal control problem and derive necessary conditions for a solution of it. In order to compute a solution of the control problem, we apply the projection-gradient-method. Simple numerical examples illustrate the results.     

Numerical inversions for a nonlinear source term in the heat equation by optimal perturbation algorithm

This paper deals with an inverse problem of identifying a nonlinear source term g=g(u) in the heat equation u_t-u_xx=a(x)g(u). By data compatibility analysis, the forward problem is proved to have a unique positive solution with a maximum of M>0, with which an optimal perturbation algorithm is applied to determine the source function g(u) on u∈[0,M]. Numerical inversions are carried out for g(u) with functional forms of polynomial, trigonometric and index functions. The inversion reconstruction sources basically coincide with the true source solution showing that the optimal perturbation algorithm is efficient to the inverse source problem here. By the computations we find that the inversion results are better for polynomial sources than those of trigonometric and index sources. The inversion algorithm seems to be very sharp if the solution’s maximum M of the forward problem is relatively small; otherwise, the deviations in the source solutions become large especially near the endpoint of u=M.     

Uniform null controllability of the heat equation with rapidly oscillating periodic density

We consider the heat equation with fast oscillating periodic density, and an interior control in a bounded domain. First, we prove sharp convergence estimates depending explicitly on the initial data for the corresponding uncontrolled equation; these estimates are new, and their proof relies on a judicious smoothing of the initial data. Then we use those estimates to prove that the original equation is uniformly null controllable, provided a carefully chosen extra vanishing interior control is added to that equation. This uniform controllability result is the first in the multidimensional setting for the heat equation with oscillating density. Finally, we prove that the sequence of null controls converges to the optimal null control of the limit equation when the period tends to zero. To cite this article: L. Tebou, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Investigation of a nonlinear heat equation with a quadratic source

We consider a particular case of the nonlinear heat equation on a straight line. A family of exact solutions of the form p(t) + q(t) cos (x/ ) is constructed, where p(t) and q(t) satisfy some dynamical system. A detailed analysis of the system is given. The existence of blowup solutions as well as solutions that decay to a nonzero background is proved for the Cauchy problem for the given equation. Part of the solutions from this family are close in a certain sense to the analytical solution of the nonlinear equation with power nonlinearities evolving in the S-regime. Profiles of various solutions are constructed and localization is investigated numerically. __________ Translated from Prikladnaya Matematika i Informatika, No. 24, pp. 5–23, 2006.     

Identification of a source factor in a control problem for the heat equation with a boundary memory term

We consider a material with thermal memory occupying a bounded region Ω with boundary Γ. The evolution of the temperature u(t,x) is described by an integrodifferential parabolic equation containing a heat source of the form f(t)z₀(x). We formulate an initial and boundary value control problem based on a feedback device located on Γ and prescribed by means of a quite general memory operator. Assuming both u and the source factor f are unknown, we study the corresponding inverse and control problem on account of an additional information. We prove a result of existence and uniqueness of the solution (u,f). Copyright © 2015 John Wiley & Sons, Ltd.     

Bang-bang property for time optimal control of semilinear heat equation

This paper studies the bang-bang property for time optimal controls governed by semilinear heat equation in a bounded domain with control acting locally in a subset. Also, we present the null controllability cost for semilinear heat equation and an observability estimate from a positive measurable set in time for the linear heat equation with potential.     

Local controllability and non-controllability for a 1D wave equation with bilinear control

We consider a linear wave equation, on the interval (0,1), with bilinear control and Neumann boundary conditions. We study the controllability of this nonlinear control system, locally around a constant reference trajectory. We prove that the following results hold generically.

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For every T>2, this system is locally controllable in H³×H², in time T, with controls in L²((0,T),R).

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For T=2, this system is locally controllable up to codimension one in H³×H², in time T, with controls in L²((0,T),R): the reachable set is (locally) a non-flat submanifold of H³×H² with codimension one.

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For every T<2, this system is not locally controllable, more precisely, the reachable set, with controls in L²((0,T),R), is contained in a non-flat submanifold of H³×H², with infinite codimension.

The proof of these results relies on the inverse mapping theorem and second order expansions.     

Note on the cost of the approximate controllability for the heat equation with potential

We prove the approximate controllability for the heat equation with potential with a cost of order e^c/ε when the target is in with a precision in L2(Ω) norm. Also a quantification estimate of the unique continuation for initial data in L2(Ω) of the heat equation with potential is established.     

Numerical null controllability of a semi-linear heat equation via a least squares method

This Note deals with the computation of distributed null controls for a semi-linear 1D heat equation, in the sublinear and slightly superlinear cases. Under sharp growth assumptions, the existence of controls has been obtained in [E. Fernández-Cara, E. Zuazua, Null and approximate controllability for weakly blowing up semi-linear heat equation, Ann. Inst. Henri Poincaré Analyse non linéaire 17 (5) (2000) 583] via a fixed point reformulation; see also [V. Barbu, Exact controllability of the superlinear heat equation, Appl. Math. Optim. Optimization, Theory and Applications 42 (1) (2000) 73]. More precisely, Carleman estimates and Kakutani?s theorem together ensure the existence of fixed points for a corresponding linearized control mapping. In practice, the difficulty is to extract from the Picard iterates a convergent (sub)sequence. We introduce and analyze a least squares reformulation of the problem; we show that this strategy leads to an effective and constructive way to compute fixed points.