Main results
Sketch of the proof, structure of the article
Hilbert Uniqueness method, which proves the equivalence between the null controllability of the 2d system (1.1) and the observability of its 2d continuous system. The proof of the positive controllability results (theorems 1.2 and 1.3 i)), uses the equivalence between the observability of the 2djoint system and the observability of the 1d heat equations solved by the Fourier modes, uniformly with respect to the Fourier frequencies∈N∗. For Theorem 1.2, this uniform1observability is proved thanks to a global Carleman estimate. Observing up to {x = 0} gives more latitude in the construction of the weight functions than in the proof of Theorem 1.1 in [6].
By the lateral propagation of the energy method on the resulting 1dwave equations, we optimally obtain observability constants exponential in the Fourier frequency. These observability constants are sharper in this paper than those proven by Carleman estimates in [6]. The proof of Theorem 1.4 also uses the optimality of the observability constants obtained above.
The exponential dependence on the observable constant is balanced by the analytical regularity of the initial data.
Comments and conjectures
The positive result can be proved by a refinement of Haraux's lateral energy method and the negative result can be proved by Agmon's estimates, as in Allibert's paper [2] on the boundary control of the wave equation on the cylindrical surface of a barrel.
Bibliographical comments
- Null controllability of the heat equation
- Boundary-degenerate parabolic equations
- Parabolic equations degenerating inside the domain
- Other occurences of minimal times in a parabolic context 7
- Agmon distance, analytic control
Recently, the existence and characterization of the minimal time has been obtained by Ammar Khodja, Benabdallah, Gonzalez-Burgos and De Teresa, for null-controllability of systems of parabolic equations. The proof developed in this paper uses the transmutation strategy and precise estimates of the resulting wave equations with respect to the potential. The estimates of the resulting wave equations rely on a lateral (or 'sideways') propagation of energy (this is where the role of time and space variables are interchanged) as used by Haraux [23] and Zuazua [35] or more recently by Haraux, Liard and Privat [24].
In [26], Lebeau studies the boundary control of the wave equation, when unique continuation holds, but the geometric control condition does not hold. He proves a quantification of the analyticity regularity required for the initial conditions to be zero controllable. His strategy relies on Fourier series expansion (with respect to the angular variable), as in the present paper.
In this section, we recall the regularity results for the Grushin equation (1.1) in Section 2.1 and the Hilbert uniqueness method in Section 2.2. In Section 2.3, we justify the expansion of the solutions of the 2d system into a Fourier series and reduce the 2d observability problem to the observability of the 1dheat equations, uniform with respect to the Fourier parameter.
Null controllability and observability
Fourier expansion and uniform observability
System (2.5) is uniformly observable from ωx at time T if it is observable from ωx at time T with exponential cost d= 0. In all that follows, we will study zero controllability of 2dsystems through the uniform observability of a connected sequence of 1dproblems. The following proposition relates the null controllability of the analytic initial conditions (1.1) and the cost of observability (if not uniform) (2.5).
Using a classical duality argument (see [16, 27] by Dolecki, Russell and Lions for pioneering works and [33] by Tucsnak and Weiss for a complete overview), if (2.5) is observable from ωxin time T with exponential priceα, then linear map.
Dissipation speed
By assumption, there exists an extraction (nk)k∈N∗ such that kUT ,nkk> keαnk for everyk∈N∗. We are interested in the asymptotic behavior (as n → +∞) of λn,γ, which quantifies the dissipation rate of the solution of (2.5). Then it is sufficient to prove observability from(0,1)×(0,1), which relies on a global Carleman estimate for solutions of (2,5), given in Sect.
The property given in the assumption can be proved by the strategy developed in this article: the fact that the interval x∈(−1, b) is nonsymmetric does not play any role; the only important thing is that we check to the limit. With uniform parabolism in casej = 1 (see for example [21]) and with assumption and reevaluation in case j = 2, there exist uj ∈L2((0, T)×Ωj) such that the solutions of.
A global Carleman estimate
Uniform observability
The proof of this theorem is very similar to that of theorem 3.1 (except that here there are two strips in the control domain) and is left to the reader. Then Theorem 1.3i) is a consequence of the following result. Using an integral transformation, the transmutation strategy induces observability properties of the heat equation (2.5) from studying the following wave equation. 4.1) The integral transformation is based on the following kernel in which existence is proved [18, Theorem 3.1]. Once this kernel is defined, the transmutation of a solution of the parabolic equation (2.5) to a solution of the hyperbolic equation (4.1) is given by the following theorem (see [18, Theorem 2.1] for the proof).
Study of wave equations
Upper bound on the minimal time
Using the definitions of M and q˜ given in Proposition 4.5, it follows that M does not depend on n and thus we obtain. For a fixed n ∈N∗ we denote by (λ(n)j )j∈N∗ and (ϕ(n)j )j∈N∗ the non-decreasing sequence of eigenvalues and the corresponding sequence of L2normalized eigenvectors of the operator − ∂2xx+ (nπ)2x2 z domainH2∩H01(−1,1) i.e. 4.21) Then simple calculations lead to. 4.23). In (4.19), we see that the term in the win on the right-hand side of the observability-like inequality is easy to handle.
Both are of the same order when γ = 1, and thus the dissipation only compensates the cost of observability for T > a22.
Lower bound on the minimal time
We want to prove that system (2.5) is not observable by ωx at time T with exponential cost. We want to prove that system (2.5) is observable by ωx at time T with any exponential cost > πaγ+1γ+1. We want to prove that system (2.5) is not observable by ωx in time T at exponential cost.
In this paper, we have studied the zero-controllability of degenerate parabolic equations, of the Grushin type, on a rectangle domain, with strip-shaped control supports. We have proved that when the control acts on a strip touching the degeneracy line {x= 0}, then the zero controllability holds for any time T >0 and for any degeneracy γ >0. In this setting, it was already known that a positive minimum time is required for zero controllability when γ = 1 [6].
Here we give the exact value of this minimum time and its interpretation in terms of the Agmon distance between the control region and the degeneracy line. It was also known that, in special configurations (γ >1or[γ= 1, small T]), the null controllability does not hold. In these cases, we characterize initial conditions that are null controllable: their regularity depends on the control region ω, the time T and the degeneracy parameter γ.
The results of this paper suggest that, for given degeneracy parameter γ, control area ω and time T, the regularity of initial conditions that are zero-controllable depends on (γ, ω, T). This characterization is an open problem if the control region does not consist of two symmetrical strips (even if it consists of one strip). When the control region ω is not striped, the existence of a critical parameter γ for which the null control requires a positive minimum time is also an open problem.
The key point of this proof is the first inequality in the above formula: beyond the control support, the only term that depends positively on niche and can be neglected. Let us prove that the second term of the right-hand side can be dominated by terms similar to the third one. Minimal time for the zero controllability of parabolic systems: the effect of the condensation index of complex series.J.
On the optimality of the observability inequalities for parabolic and hyperbolic systems with potential. Zero controllability of the heat equation as singular limit of the exact controllability of dissipative wave equation under the Bardos-Lebeau-Rauch geometrical control condition.
