Annales de l'Institut Henri Poincaré C, Analyse non linéaire
Exact controllability of semilinear heat equations in spaces of analytic functions
Introduction
The null controllability of nonlinear parabolic equations is well understood since the nineties. It was derived in [6] in dimension one by solving some “ill-posed problem” with Cauchy data in some Gevrey spaces (see also [13]), and in [4], [5] in any dimension and for any control region by using some “parabolic Carleman estimates”.
The null controllability was actually extended to the controllability to trajectories in [5]. However, it is a quite hard task to decide whether a given state is the value at some time of a trajectory of the system without control (free evolution). In practice, the only known examples of such states are the steady states.
As noticed in [17], in the linear case, the steady states are Gevrey functions of order 1/2 in x (and thus analytic over ) for which infinitely many traces vanish at the boundary, a fact which is also a very conservative condition leading to exclude e.g. all the nontrivial polynomial functions.
The recent paper [17] used the flatness approach and a Borel theorem to provide an explicit set of reachable states composed of states that can be extended as analytic functions on a ball B. It was also noticed in [17] that any reachable state could be extended as an analytic function on a square included in the ball B. We refer the reader to [1], [7] for new sets of reachable states for the linear 1D heat equation, with control inputs chosen in . We notice that the flatness approach applied to the control of PDEs, first developed in [12], [3], [19], [24], was revisited recently to recover the null controllability of (i) the heat equation in cylinders [15]; (ii) a family of parabolic equations with unsmooth coefficients [16]; (iii) the Schrödinger equation [18]; (iv) the Korteweg-de Vries equation with a control at the left endpoint [14]. One of the main features of the flatness approach is that it provides control inputs developed as explicit series, which leads to very efficient numerical schemes.
The aim of the present paper is to extend the results of [17] to semilinear heat equations. Roughly, we shall prove that a reachable state for the linear heat equation is also reachable for the semilinear one, provided that its magnitude is not too large and its poles and those of the nonlinear term are sufficiently far from the origin. The method of proof is inspired by [6] where a Cauchy problem in the variable x is investigated. The main novelty is that we prove an exact controllability result (and not only a null controllability result as in [6]), and we need to investigate the influence of the nonlinear terms on the jets of the time derivatives of two traces at . Here, we do not use some series expansions of the control inputs as in the flatness approach, but we still use some Borel theorem as in [22], [17]. It is unclear whether the same results could be obtained by the classical approach using the exact controllability of the linearized system and a fixed-point argument.
To be more precise, we are concerned with the exact controllability of the following semilinear heat equation where is analytic with respect to all its arguments in a neighborhood of . More precisely, we assume that and that with for some constants Note that for all by (1.5). For let We infer from (1.6) and (1.7) that Among the many physically relevant instances of (1.1) satisfying (1.5)-(1.8), we quote:
- (1)
the heat equation with an analytic potential: where , with for all and some constants , .
- (2)
the Allen-Cahn equation
- (3)
the viscous Burgers' equation Note that our controllability result is still valid when the nonlinear term in (1.9) is replaced by a term like with φ as in (1), and .
The main result in this paper is the following exact controllability result.
Theorem 1.1 Let be as in (1.5)-(1.8) with . Let and . Then there exists some number such that for all , there exists such that the solution y of (1.1)-(1.4) is defined for all and satisfies for all . Furthermore, we have that .
It is likely that using the smoothing effect, a similar result could be obtained with a less regular initial data (e.g. in ) as for the linear Korteweg-de Vries equation in [14, Corollary 1.1]. This would however require to estimate carefully the domain of analyticity in x of the solution.
A similar result with only one control can be derived assuming that f is odd w.r.t. . Consider the control system
Corollary 1.2 Let be as in (1.5)-(1.8) with , and assume that Let and . Then there exists some number such that for all with for all , there exists such that the solution y of (1.10)-(1.13) is defined for all and satisfies for all . Furthermore, we have that .
Corollary 1.2 can be applied e.g. to (i) the heat equation with an even analytic potential; (ii) the Allen-Cahn equation; (iii) the viscous Burgers' equation.
The constant is probably not optimal, but our main aim was to provide an explicit (reasonable) constant. It is expected that the optimal constant is , with a diamond-shaped domain of analyticity as in [1] and [7] for the linear heat equation.
Our method of proof combines two steps.
- •
The first one is the analysis of a Cauchy problem in the spatial variable. We prove the existence of global solutions of the semilinear heat equation defined for x in the full interval associated with two initial data , for . To do that, we refine the method developed in [20], [21] which gives solely local solutions in x. The solution is completely defined in terms of the initial data and (flatness property) but, in contrast to [17], there is no representation of the solution as an explicit series.
- •
The second one is a “jets analysis” which investigate the relationship between the jet and the jets and . This step is needed to reach a given state in the reachable space.
We notice that the method of proof in the linear case (see [17]) was also based on the same two steps, the computations being however easier and explicit. We note also that our approach does not follow the classical “linearization + fixed point-argument” approach which is widely used to deal with the controllability of nonlinear PDEs. Among the advantages of our method, we could mention (i) its robustness, in the sense that it can be adapted to many other PDEs (see [11] for an extension to PDEs of the form ) with ) (ii) its possible use to elaborate efficient numerical schemes (see [15] in the linear case). For the restrictions of the method, we should say that (i) the constants are not optiomal and (ii) it is (to date) only applicable in dimension one.
The paper is organized as follows. Section 2 is concerned with the wellposedness of the Cauchy problem in the x-variable (Theorem 2.1). The relationship between the jet of space derivatives and the jet of time derivatives at some point (jet analysis) for a solution of (1.1) is studied in Section 3. In particular, we show that the semilinear heat equation (1.1) can be (locally) solved forward and backward if the initial data can be extended as an analytic function in some ball of (Proposition 3.6). Finally, the proofs of Theorem 1.1 and Corollary 1.2 are displayed in Section 4.
Section snippets
Statement of the global wellposedness result
Let be as in (1.5)-(1.8). We are concerned with the wellposedness of the Cauchy problem in the variable x: for some given functions . The aim of this section is to prove the following result.
Theorem 2.1 Let be as in (1.5)-(1.8). Let and . Then there exists some number such that for all with there exist
Correspondence between the space derivatives and the time derivatives
We are concerned with the relationship between the time derivatives and the space derivatives of any solution of a general semilinear heat equation where is of class on .
When , then the jet is nothing but the reunion of the jets and , for When f is no longer assumed to be 0, then the relations (3.2)-(3.3) do not hold anymore. Nevertheless, there is still a one-to-one
Proofs of the main results
Let us start with the proof of Theorem 1.1. Let and let be (for the moment) the constant given by Proposition 3.6. Pick any . We infer from Proposition 3.6 applied with and (resp. ) the existence of two functions satisfying (2.1) and such that Let be such that and . Let
Declaration of Competing Interest
No competing interest.
Acknowledgements
The authors thank an anonymous referee for carefully reading the manuscript and making several useful remarks. The first author would like to thank Jean-Michel Coron for introducing him to the use of Gevrey functions in control. The first author was supported by ANR project ISDEEC (ANR-16-CE40-0013). The second author was supported by the ANR project Finite4SoS (ANR-15-CE23-0007).
References (25)
- et al.
Null controllability of the heat equation using flatness
Automatica
(2014) - et al.
Controllability of the 1D Schrödinger equation using flatness
Automatica
(2018) - et al.
On the reachable set for the one-dimensional heat equation
SIAM J. Control Optim.
(2018) - et al.
An abstract Cauchy-Kowaleska theorem in scales of Gevrey classes
- et al.
Motion planning for a nonlinear Stefan problem
ESAIM Control Optim. Calc. Var.
(2003) Controllability of parabolic equations
Mat. Sb.
(1995)- et al.
Controllability of Evolution Equations
(1996) - et al.
Null boundary controllability for semilinear heat equations
Appl. Math. Optim.
(1995) - et al.
From the reachable space of the heat equation to Hilbert spaces of homorphic functions
- et al.
Sur les ondes de surfaces de l'eau avec une justification mathématique des équations des ondes en eau peu profonde
J. Math. Kyoto Univ.
(1979)
On the Cauchy problem for non linear PDEs in the Gevrey class with shrinkings
J. Math. Soc. Jpn.
Analyticity at the boundary of solutions of nonlinear second-order parabolic equations
Commun. Pure Appl. Math.
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