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Exact Controllability of the Heat Equation with Hyperbolic Memory Kernel

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... Several "controllability" results have been recently proved for the heat equation with memory, see for example [2,12,[16][17][18]20]. These papers study control problems for equations of the following form (here t > 0 and x ∈ (0, π), but also the case x ∈ R n has been considered): ...
... In fact, as seen in Lemma 1 below, a powerful obstruction to interpolation follows from the following completeness condition. If (20) holds then the sequence of exponentials {e λ n t } is complete in L 2 (0, T ) for every T > 0. ...
... See [19, p. 105, "complement" to Remark 2] for an even more general formulation. Note that the condition in Obstruction 1 implies condition (20). We see in Lemma 1 below that when the condition in Obstruction 2 holds then our interpolation/moment problem cannot be solved if ξ has to be an arbitrary initial condition in L 2 (0, π). ...
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We prove that the one-dimensional heat equation with memory cannot be controlled to rest for large classes of memory kernels and controls. The approach is based on the application of the theory of interpolation in Paley–Wiener spaces.
... The case that the space variable belongs to IR n has been considered in [23], still using Carleman estimates, and in [19]. The proof in this last paper (see also [20]) is based on Baire theorem and compactness results, and it is not constructive. ...
... We need to recall the known controllability results, from [5] (under more restrictive assumptions) and from [23], under the assumption in this paper (and in fact more general, see below). These papers study controllability under distributed control. ...
... The results on controllability in [5] concerns one dimensional space variable and require that N (t) is continuous for t ≥ 0 and completely monotonic, i.e. of class C ∞ with derivatives of alternating sign; in particular, N (t) ≥ 0. The result in [23] considers the case of elliptic operators with variable coefficients in a region Ω of IR n . Also the kernel N can depend on x, N = N (t, x). ...
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In this paper we give a semigroup-based definition of the solution of the Gurtin-Pipkin equation with Dirichlet boundary conditions. It turns out that the dominant term of the input-to-state map is the control to displacement operator of the wave equation. This operator is surjective if the time interval is long enough. We use this observation in order to prove exact controllability in finite time of the Gurtin-Pipkin equation. Heat equation with memory–Exacty controllability–Cosine operators
... The solution rests on Carleman estimates and it can't be considered constructive. The case that the space variable belongs to IR n has been considered in [23], still using Carleman estimates, and in [19]. The proof in this last paper (see also [20]) is based on Baire theorem and compactness results, and it is not constructive. ...
... The required conditions are smoothness conditions on ν(t, x), satisfied in a dense subset of L 2 ((0, T ) × (−, 0)). We need to recall the known controllability results, from [5] (under more restrictive assumptions) and from [23], under the assumption in this paper (and in fact more general, see below). These papers study controllability under distributed control. ...
... The results on controllability in [5] concerns one dimensional space variable and require that N (t) is continuous for t ≥ 0 and completely monotonic, i.e. of class C ∞ with derivatives of alternating sign; in particular, N (t) ≥ 0. The result in [23] considers the case of elliptic operators with variable coefficients in a region Ω of IR n . Also the kernel N can depend on x, N = N (t, x). ...
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The main result we derive is the proof that a particular set of functions related to the controllability of the heat equation with memory and finite signal speed, with suitable kernel, is a Riesz system. Riesz systems are important tools in applied mathematics, for example for the solution of inverse problems. In this paper we shows that the Riesz system we identify can be used to give a constructive method for the computation of the control steering a given initial condition to a prescribed target.
... It is worthy of mentioning that, based on Laplace transform and cosine operator approach, respectively, [1] and [20] studied the controllability problem for (1.2) when (a ij ) n×n = I , the identity matrix, and the memory kernel b does not depend on x. On the other hand, by means of Carleman estimate, exact controllability result for (1.2) with (a ij ) n×n = I was given in [26]. Recently, further related results have been presented in [9,21], especially an interesting negative controllability result can be found in [9]. ...
... The key observation in [26] is that, due to the special structure of system (1.4) with (a ij ) n×n = I , a modified Carleman inequality can be employed to derive the observability estimate for it. By combining and modifying carefully the Carleman estimate developed in [26] and [7], the first concern of this paper is to establish the observability estimate (1.5) for system (1.4) with general thermal conductivity matrix (a ij ) n×n under some further assumptions on the controller/observer ω and the waiting time T . ...
... The key observation in [26] is that, due to the special structure of system (1.4) with (a ij ) n×n = I , a modified Carleman inequality can be employed to derive the observability estimate for it. By combining and modifying carefully the Carleman estimate developed in [26] and [7], the first concern of this paper is to establish the observability estimate (1.5) for system (1.4) with general thermal conductivity matrix (a ij ) n×n under some further assumptions on the controller/observer ω and the waiting time T . ...
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The exact controllability and observability for a heat equation with hyperbolic memory kernel in anisotropic and nonhomogeneous media are considered. Due to the appearance of such a kind of memory, the speed of propagation for solutions to the heat equation is finite and the corresponding controllability property has a certain nature similar to hyperbolic equations, and is significantly different from that of the usual parabolic equations. By means of Carleman estimate, we establish a positive controllability and observability result under some geometric condition. On the other hand, by a careful construction of highly concentrated approximate solutions to hyperbolic equations with memory, we present a negative controllability and observability result when the geometric condition is not satisfied.
... In [41], a multi-dimensional problem with more general memory kernel is considered ...
... The paper [42] represents a generalization of [41] for anisotropic and nonhomogeneous medium, i.e. the equation ...
... is considered. The similar results on exact controllability and observability are proved under the geometric conditions, which are more complex than in [41]. On the other hand, if the geometric conditions are not satisfied, a negative controllability and observability result is presented. ...
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We are motivated by the problem of control for a non-homogeneous elastic string with memory. We reduce the problem of controllability to a non-standard moment problem. The solution of the latter problem is based on an auxiliary Riesz basis property result for a family of functions quadratically close to the nonharmonic exponentials. This result requires the detailed analysis of an integro-differential equation and is of interest in itself for Function Theory. Controllability of the string implies observability of a dual system.
... We consider the integrals at the lines (20)- (21). The integral at the line (20) can be integrated by parts twice, and it is seen to be of the order of 1/β 2 n . Instead, a second integration by parts of the integral at the line (21) gives a term of the order 1/β 2 n plus the terms N 0 te αt 2β n sin β n t and the integral below, which can be estimated by a n (t) β n , +∞ n=1 |a n (t)| 2 < M . ...
... Although we believe that the arguments in [19,20] can be easily adapted to the case under study here (essentially, to the case b(x) = 0) we gave an independent proof of the property of ω-independence, which will be usefull also in different contexts. ...
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In this paper we reduce observability and controllability of a heat equation with memory to the solution of a moment problem. We prove that this moment problem is solvable by proving that a suitable sequence of functions, associated with the heat equation with memory, is a Riesz system.
... See [23] for different basis associated to Eq. (1) (when the kernel N (t) has a special form) and see [2,7,10,13,14,17,18,24] for previous papers which study controllability properties of equations of the type considered in this paper. ...
... Instead, it was deduced from the already known fact that system (1) with boundary control (in the special case there considered) is approximately controllable in time π. Although we believe that the arguments in [7,24] can be adapted to the case under study here, we gave an independent proof of the property of ω-independence, which has its own interest and which has already been used in different contexts, see [3]. ...
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In this paper we consider an integrodifferential system, which governs the vibration of a viscoelastic one-dimensional object. We assume that we can act on the system at the boundary and we prove that it is possible to control both the position and the velocity at every point of the body and at a certain time $ T $, large enough. We shall prove this result using moment theory and we shall prove that the solution of this problem leads to identify a Riesz sequence which solves controllability and observability. So, the result as presented here are constructive and can lead to simple numerical algorithms.
... See [16] for an application of this property. See [19] for different basis associated to Eq. (1) and see [2, 8, 12, 13, 14, 15, 20, 21] for previous papers which study controllability properties of equations of the type considered in this paper. Finally we note that methods based on suitable basis and nonharmonic Fourier series are widely used in the study of controllability/observability problems, mostly thanks to the pioneering researches of D. Russel, see [17]. ...
... Instead, it was deduced from the already known fact that system (1) with boundary control (in the special case α = 0, a(x) = 1, b(x) = 0 there considered) is approximately controllable in time π. Although we believe that the arguments in [20, 21] can be adapted to the case under study here (essentially, to the case b(x) = 0) we gave an independent proof of the property of ω-independence, which will be usefull also in different contexts. ...
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In this paper we study the equation of linear viscoelasticity and we prove that two sequences of functions, naturally associated with this equation, are Riesz systems. These sequences appear naturally when observability and controllability problems are reformulated in terms of suitable interpolation/moment problems. The key contribution of the paper is to be found in the way used to prove that the two sequences are Riesz systems, an idea already applied to the study of different control problems.
... In [27,33] the authors consider a nonlinear and non degenerate version of (1.1) in the case that y is independent of a and proved that the problem is null controllable assuming that the memory kernel is sufficiently smooth and vanishes at the neighborhood of initial and final times. This assumption has been relaxed by Q. Tao and H. Gao in [35]; for related results on this subject, we refer to [30] for wave equation, to [6] for viscoelasticity equation, to [32] for thermoelastic system and to [37] in the case of heat equation with hyperbolic memory kernel (see also the bibliography therein). ...
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In this paper we consider the null controllability for a population model depending on time, on space and on age. Moreover, the diffusion coefficient degenerate at the boundary of the space domain. The novelty of this paper is that for the first time we consider the presence of a memory term, which makes the computations more difficult. However, under a suitable condition on the kernel we deduce a null controllability result for the original problem via new Carleman estimates for the adjoint problem associated to a suitable nonhomogeneous parabolic equation.
... For related results on this subject, we refer to Lu et al 15 for wave equation, Barbu and Iannelli 16 for viscoelasticity equation, Muñoz and Naso 17 for thermoelastic system, and Yong and Zhang 18 in the case of heat equation with hyperbolic memory kernel (see also the bibliography therein). ...
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... In [24,28] the authors consider a nonlinear and non degenerate version of (1.1) in the case that y is independent of a and proved that the problem is null controllable assuming that the memory kernel is sufficiently smooth and vanishes at the neighborhood of initial and final times. This assumption has been relaxed by Q. Tao and H. Gao in [30]; for related results on this subject, we refer to [35] for wave equation, [6] for viscoelasticity equation, [27] for thermoelastic system and [32] in the case of heat equation with hyperbolic memory kernel (see also the bibliography therein). ...
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In this paper we consider the null controllability for a population model depending on time, on space and on age. Moreover, the diffusion coefficient degenerate at the boundary of the space domain. The novelty of this paper is that for the first time we consider the presence of a memory term, which makes the computations more difficult. However, under a suitable condition on the kernel we deduce a null controllability result for the original problem via new Carleman estimates for the adjoint problem associated to a suitable nonhomogeneous parabolic equation.
... The proof relies on Carleman estimates and a fixed point method. This assumption has been relaxed by Q. Tao and H. Gao in [24], where the authors showed that null controllability holds provided b fulfills the restriction For related results on this subject, we refer to [4] for viscoelasticity equation, [22] for thermoelastic system and [27] in the case of heat equation with hyperbolic memory kernel (see also the bibliography therein). ...
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... A paper which do not use these ideas is [40] (and also [74,77,127] discussed below). The higly technical paper [40] studies the case that A is a uniformly elliptic operator is a region Ω and the control is a distributed control in Ω (possibly localized close to a suitable part of the boundary). ...
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... A paper which do not use these ideas is [40] (and also [74,77,127] discussed below). The higly technical paper [40] studies the case that A is a uniformly elliptic operator is a region Ω and the control is a distributed control in Ω (possibly localized close to a suitable part of the boundary). ...
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... Confining ourselves to cite the oldest ones, see for example [3,16,22,38]. In more recent times, this equation has been studied from the point of view of the stability theory and controllability, see for example [2,4,8,33,45,47,55,57]. A recent approach based on moment theory is in [49,50]. ...
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... Yong and Zhang [20] considered the heat equation with hyperbolic memory kernel ...
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... Although we believe that the arguments in [21,22] can be easily adapted to the case under study here (essentially, to the case b(x) = 0) we gave an independent proof of the property of ω-independence, which will be usefull also in different contexts. ...
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... where Ω is a bounded interval in R, has been discussed by Beceanu [8] by establishing a Carleman estimate for appropriate linearized equation. Besides, approximate controllability of the linear heat equations with memory kernels has been analyzed by Barbu and Iannelli [6] under some technical conditions on the memory kernels by means of Laplace transform and Yong and Zhang [28] studied the controllability of the heat equation with hyperbolic memory kernel. The paper by Sakthivel et al. [22] discusses the exact controllability of some nonlinear parabolic integrodifferential equations with periodic boundary conditions when the nonlinearity is of globally Lipschitz and the control u ∈ L 2 ((0, T ) × ω). ...
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Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://project.dml.cz In this paper, we prove the exact null controllability of certain diffusion system by rewriting it as an equivalent nonlinear parabolic integrodifferential equation with variable coefficients in a bounded interval of R with a distributed control acting on a subinterval. We first prove a global null controllability result of an associated linearized integrodif-ferential equation by establishing a suitable observability estimate for adjoint system with appropriate assumptions on the coefficients. Then this result is successfully used with some estimates for parabolic equation in L k spaces together with classical fixed point theorem, to prove the null controllability of the nonlinear model.
... To the authors' knowledge, the determination of the source term by establishing a Carleman estimate for parabolic equations with memory is not available in the literature. However, the problem of exact null controllability of parabolic equations with memory has been studied by establishing global Carleman estimates for associated adjoint linear problems in Sakthivel et al. [23,22,24] and in [28], an exact controllability problem is considered for a related system with the kernel of the form k(x, t, τ ). In general, stability estimate is derived from a Carleman estimate by the method of Bukhgeim and Klibanov [7] and the proof of Theorem 3.1 follows some ideas used in [11]. ...
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... Also, a similar idea played a key role in establishing the observability estimate for the wave equations with Neumann boundary conditions in [24] (which should be compared with [19]). We refer the reader to [12, 13] for further application of Theorem 4.1 and its generalization, and to [7] for related work. Proof of Theorem 4.1. ...
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