We consider the average controllability problem for control systems depending on the parameters (either discretely or continuously), in order to find a control, independent of the unknown parameters, so that the average of the states is controlled. We do so in the context of conservative models, both in an abstract setting and by analyzing specific examples of the wave and Schr¨odinger equations. As we shall see, the average controllability properties will depend significantly on the nature of the average measurement.
This article is devoted to answering these questions, both in the abstract version (1.1), in which the generator of the semi-group Aζ is skew-adjoint, and in the abstract version (1.1), in which the generator of the semi- group Aζ is skew-adjoint. To address the mean controllability problem, consider a probability measure of the form η= (1−θ)δζ0+θ˜η, where ˜η is a probability measure for Randθ∈[0,1], a small parameter such that : in practice we are dealing with a small perturbation of an atomic measure concentrated at ζ0. For the specific example of the string equation in (1.4) holds (see Appendix A)ς(ζ) =ζand Lζis affine with respect toζ(see (A.2)).
Analyzing the goodness of fit of the control system under consideration also requires the mean admissibility inequality. 31, 37] for some of the most important existing results on the control and stabilization of networks of 1d wave equations. The proof of the duality result, between average controllability and average observability, is given in §2.2.
These results are applied in §3.2 (compare to [24]) to the average controllability of wave equations, where coefficients depend smoothly on the space variable and are easily measurable with respect to the unknown parameter, and also in the context of non-harmonic Fourier series (see §3.3).
Averaged admissibility, controllability and observabil- ity
In the following paragraphs, following this general abstract path, we prove admissibility and exact average observability results for the corresponding adjoint systems. Using the admissibility condition given in the previous section, one can easily develop a perturbation argument that leads to average controllability.
Perturbation argument
This result can be applied to many examples such as the wave, Sch¨odinger and plate equations, etc. However, the proof, which is quite straightforward, is based on a triviality argument and, therefore, does not cover the results in [24 ] for the average controllability of two wave equations with internal control, or those in [40] for additive overlap of the wave and heat equations. In Theorem 3.1, we assume that the perturbation measure η˜ is a probability measure so that η= (1−θ)δζ0+θη˜is a probability measure for every θ∈[0,1].
Of course, a similar study could be conducted using improbability measures, but we consider probability measures to ensure that we are dealing with "averages". For example, we could think of problems depending on the parameter d using the probability space (Rd,B(Rd), θ˜η+(1−θ)δζ0), with θ∈[0,1]and ζ0∈Rd. Using Proposition 2.1, Assumptions 1 and 2 provide the average admissibility of (Bζ)ζ for (Tζ)ζ with respect to the measure ˜η.
This together with the admissibility of Bζ0 for Tζ0 (assumption 3) provides the average admissibility of (Bζ)ζ for (Tζ)ζ with respect to the measure η given by (3.2) for each θ∈[0,1].
Averaged control of parameter depending Schr¨ odinger and wave systems
Let us briefly explain how the parameter-dependent control system (3.5) fits into our abstract setting. According to the previous remarks, in order to apply Theorem 3.1, we only need to prove the observability inequality for ζ = 1. But from [3], the geometric control condition for the control system indexed by ζ = 1 ensures that this system is time-accurately controllable.
This result applies in the special caseηθ= (1−θ)δ1+θδ2where two wave equations with different propagation velocities are averaged. This case was treated in [24, Theorem 2.1], where it was proved that the system satisfies the average control property for each θ∈[0,1), provided,. 3.6) The proof of this result is based on micro-local defect measures and the fact that the characteristic manifolds of the two wave equations involved are disjoint.
This example shows that the smallness condition we impose on the perturbations is not harsh.
Perturbation of Ingham inequalities
Let us now apply our perturbation argument developed in §3.1 to these non-harmonic Fourier series. Furthermore, assume that ζ 7→ Lζ and ς are measurable, ς(ζ) 6= 0 for almost every ζ∈R with respect to the measure η, and˜. The condition T > 1/(|ς(ζ0)|γ) is only required in light of the fact that we have used the classical formulation of Ingham's inequality.
But, for example, if the sequence (λn)n∈N∗ is non-decreasing and satisfies the asymptotic gap condition lim inf.
Averaged control of parameter depending string sys- tems
In §3.3 we used a perturbation argument to prove, roughly speaking, the stability of Ingham inequalities when the targetη is a Dirac mass plus a small enough perturbation. In this section we look at another interesting case where a limited number of equations are involved. In other words, we address the case where the unknown parameter varies on a finite set.
Of course, our perturbation argument can be applied in this case (see §4.1), but this argument required some smallness assumptions. We will see in §4.2 that some mean Ingham inequalities are still valid without this smallness assumption. To deal with this case and prove the necessary average Ingham inequalities, we use a different argument.
This is the case for the 1d wave and Schrödinger equations with Dirichlet boundary conditions, for which the solutions are time periodic. Since we are averaging a finite number of parameters, the inequality (3.8) guarantees for any T >0 the existence of a constant C(T)>0 such that:
Application of the perturbation result
We have shown that mean versions of Ingham's inequalities hold under a suitable smallness condition on the perturbative measures. On the one hand, it is necessary in a certain sense as the example below shows, but on the other hand, with some more assumptions, this assumption is not necessary as we will see in §4.2.
A time-periodicity argument
Letting N go to infinity, we see that no Ingham inequality can hold whatever T > 0 is. 139] which can be applied to simultaneous control of finitely many strings (see §5.8.2 of that book). First of all, by changing γς(ζk) to ζk, we can assume without loss of generality that γ= 1 and ς(ζ) =ζ.
With a few more conditions on the parameters ς(ζk), the following unique continuation property can be easily obtained from (4.4). However, with some more restrictive conditions on the parametersς(ζk), we can obtain an observability inequality. For every ε > 0 there exists a set Bε ⊂ R such that the Lebesgue measure of R\ Bε is equal to zero, and a constantρε>0 for which, ifζ∈Bε then,.
Assume that the conditions of Corollary 4.2 hold and furthermore that for every α >0 there existsΛζ0,α>0 such that (4.9) holds. In Corollary 4.1 no irrationality condition is needed and the observability inequality holds in the classical `2-norm, while in Corollaries 4.3 and 4.4 an irrationality condition is required and the observability inequality is only valid for coefficients that are in some subspace of `2. In addition, the minimal observation time required for the observability inequality in Corollaries 4.3 and 4.4 is larger than the one required in Corollary 4.1.
But to obtain the observation inequality of Corollary 4.1, we need a weight θk0 that is close enough to 1. Finally, Corollaries 4.3 and 4.4 become relevant when none of the weights θk is close enough to 1.
Application to the string equation
This result ensures that all parameter-dependent trajectories, and hence their average, can be directed to a prescribed target with an input that is independent of the parameter. As expected, the assumption (4.10) needed to obtain mean controllability is weaker than (4.12), the assumption needed for simultaneous controllability. The aim of this paper was to provide a systematic result, based on perturbation arguments, on the mean controllability and observability of parameter-dependent families of equations, mainly in the context of time-reversible groups of isometries.
In the case Lζ = Id and ς(ζ) = ζ, the issues we discussed in the previous paragraph on the averages of non-harmonic Fourier series can be reformulated in terms of the stability properties of the Riesz sequence of the family {t7 →η(−λˆ nt )}n (ˆη is the Fourier-Stieltjes transform of the probability density η), in the closed subspace of L2(0, T) they generate. Then, (ηε)ε>0 converges in the sense of measures to the Dirac measure δ1 when ε goes to 0. When dealing with the control system (3.5), in [24], the condition (3.6) was required to ensure the mean controllability.
This result still remains in the general case where (λn)n satisfies (3.7) and assuming that the values ς(ζk)λn6=ς(ζl)λmfork6=l orn6=m. The analysis of all these examples can contribute to the achievement of sharp results for the average controllability of many equations of finite sequences. The goal is to find parameter-independent controls that perform well for all parameter values.
For this purpose, it was a first and logical choice to check the average of the parameter-dependent outputs. Of course, the best we can expect is a control, independent of the values of the unknown parameters, that directs all parameter-dependent trajectories towards a common fixed goal, i.e. there is a natural connection between the control of the mean and the stronger idea of simultaneous control .
But as κ increases, the control, in addition to guaranteeing the average controllability property, also forces the reduction of the variance of the output. Let us briefly describe how the string equation with Dirichlet boundary check (1.4) enters the abstract formalism introduced in Section 2. So the observation operator is: . ζ∈R, α∈R), (A.3) These last inequalities ensure that for everyζ ∈R∗, and every α >0,Lζ is a linear continuous operator bounded from below in one of the spaces.
The reason for the introduction of this real parameterα and these spaces`2−α, will make sense in corollaries 4.3 and 4.4 and in Proposition 4.3, in particular, regarding the definition of the spaces Xα, given by relation (4.11) . Estimates of the constants in generalized Ingham's inequality and applications to the governing wave equation.
