Control of the Hyperbolic Ill-posed Cauchy Problem by Controllability

The main purpose of this paper is the control of the hyperbolic ill-posed Cauchy problem. To do this, we adapt to the present case the controllability method previously introduced in the stationary case (Guel and Nakoulima 2023). So we interpret the problem as an inverse problem, and therefore a controllability problem. This point of view induces a regularization method that makes it possible, on the one hand, to characterize the existence of a regular solution to the problem. On the other hand, this method makes it possible to obtain a singular optimality system for the optimal control, without using any additional assumption, such as that of non-vacuity of the interior of the sets of admissible controls, an assumption that many analyses have had to use.


Introduction
In this paper, we are interested in the control of an ill-posed system relating to the Cauchy problem for an hyperbolic operator.It is a model example of a singular distributed system that occurs in several physical applications.It is the case in gravimetry, for what concerns the stationary case.But also in questions of the transport of electrical energy (Hadamard 1923), passing through the control of enzymatic reactions (Kernevez 1980 and the bibliography of this work) and the form of plasmas, for the evolution cases.
To introduce the problem, let Ω ⊂ R n be a bounded and regular domain of class C 2 , with boundary Γ = Γ 0 ∪ Γ 1 , where Γ 0 and Γ 1 are disjointed, regular and with superficial positive measures.For T ∈ R + \ {0}, we denote by We consider in Q, the boundary value problem where v = (v 0 , v 1 ) is given in L 2 (Σ 0 ) 2 .
The problem (1) is the ill-posed Cauchy problem for the wave operator.It is well known (Lions 1983 andHadamard 1923) that this problem is ill-posed in Hadamard's sense, that is to say, for a given vector v = (v 0 , v 1 ), the problem does not always admits a solution, and it may lead to instability of the solution when it exists.
We therefore consider a priori pairs (v, z) such as and satisfying (1).It is said that such pairs constitute the set of control-state pairs.
Remark 1 (see.Hadamard 1923).It is important to note that, when it exists, the solution to the ill-posed Cauchy problem is unique.
U 0 ad and U 1 ad being two non-empty closed convex subsets of L 2 (Σ 0 ), a control-state pair (v, z) will be said admissible if with (v, z) satisfying (1).We use the notation (v, z) ∈ A to say that A is the set of admissible control-state pairs and assume (an example is given below) Now we introduce the cost function where The non-vacuity assumption (4), the structure of the set of admissible control-state pairs (it is not difficult to show that and that of J easily show that the problem (6) admits a unique solution, the optimal control-state pair (u, y).
The cost function J being differentiable, the first order Euler-Lagrange conditions make it possible to establish that the optimal control-state pair (u, y) satisfies the optimality condition: Remains to characterize the optimal control-state pair (u, y) through a singular optimality system.
According to the literature on this problem, the origins of the Cauchy system control problem can be traced back to Lions 1983.Indeed in this book, presenting solutions to the main difficulties encountered in the enterprise of controlling singular distributed systems, J. L. Lions deals with the Cauchy system for an elliptic operator, considering a desired state of the trace on the boundary Γ 1 .In order to obtain a decoupled singular distributed system, J. L. Lions uses the penalization method in the particular cases If the strong convergence of the process is then obtained in the first case, the second requires recourse to the additional Slater type assumption that the interior of U 0 ad is non-empty in L 2 (Γ 0 ) .
However, J. L. Lions conjectures that one should be able to solve the problem only with the usual assumptions of nonvacuity, convexity and closure of the control sets U 0 ad and U 1 ad , without resorting to the Slater type assumption (8).
Many authors have studied the control of the ill-posed Cauchy problem.One of the first to take an interest in it was O.
Nakoulima who, in Nakoulima 1994, showed that one could indeed do without the Slater type assumption.In this article quoted above, considering the distributed observation problem (as it is the case in the present paper) in the elliptic case, O. Nakoulima effectively manages well, via a regularization-penalization method, to do without that assumption, the sets of controls then considered being of empty interior.The approach adopted considers the control problem as a "singular" limit of a sequence of well-posed control problems.These results did not, however, exhaust the problem, since they only concerned a particular case of constraints on controls.
A little later, G. Mophou and O. Nakoulima propose a new approach in Mophou 2009.The authors use a regularization method (without penalization), called elliptic-elliptic regularization, and managed to obtain strong convergence of the process, but resort for that to the Slater type assumption.
Still in the elliptic case, one of the latest results to our knowledge concerns the work of A. Berhail and A. Omrane (cf. Berhail 2015).Thanks to which, using the low and no-regret control notion, the authors managed to characterize the optimal solution through a strong and decoupled singular optimality system, but this in the particular unconstrained case In the evolution cases, the bibliography refers to Barry 2013 andBarry 2014, in which the authors, M. Barry, G. B. Ndiaye and O. Nakoulima take up the idea of penalization method proposed in the elliptic case by J. L. Lions.The authors then obtained results similar to those obtained in the stationary case.
Nevertheless, in general, the problem remains open.Indeed, as shown by the literature review above, almost all of the work carried out concerns only specific cases of controls (v 0 , v 1 ), such as the following: , the "unconstrained" case; • or with the additional Slater type assumption that the interiors of U 0 ad and/or U 1 ad are non-empty in L 2 (Γ 0 ) .
In this paper, we adapt to the hyperbolic case, an original method recently proposed in Bylli 2023.The point of view adopted consists in interpreting the initial problem (1) as an inverse problem, and therefore a controllability problem.
This approach induces a regularization method that makes it possible, on the one hand, to characterize the existence of a regular solution to the problem.On the other hand, this method makes it possible to obtain a strong and decoupled singular optimality system for the optimal control, without using any additional assumption, such as that of non-vacuity of the interior of the sets of admissible controls, an assumption that many analyses have had to use.
The rest of the paper is organized as follows.Section 2 is devoted to interpreting the initial problem as an inverse problem.
In Section 3, we return to the control problem, starting by regularizing it via the controllability results previously obtained.
After establishing the convergence of the process in Section 3.2, then the approached optimality system in Section 3.3, we end in Section 3.4 with the singular optimality system for the initial problem.
Example 1 (Non-vacuity of the set of admissible control-state pairs).Suppose that Then, the set of admissible control-state pairs is non-empty.Indeed, given What defines, and that in a unique way Thus, the control-state pair ξ| Σ 0 , v 1 , ξ is admissible.

Controllability for the Hyperbolic Ill-posed Cauchy Problem
We adapt here to the hyperbolic case the idea of controllability, previously introduced in Bylli 2023.Which consists in interpreting the ill-posed Cauchy problem (1) as a system of inverse problems.
We establish, as in the stationary case, that when it exists, the solution of the hyperbolic ill-posed Cauchy problem coincides with the common solution of the system of inverse problems mentioned above and we also manage to characterize the existence of a regular solution to the hyperbolic Cauchy system.
Starting by the initial problem (1), we consider the systems and in Ω, for which we set ourselves the objective of observing Having, for each of the problems ( 11) and ( 12), a datum and an "observation" on the border Σ 0 and no information on the border Σ 1 , we look at the whole (11)(12)(13) as a system of inverse problems, setting ourselves the problem of knowing how to complete ( 11) and ( 12) on the border Σ 1 , with "dummy controls" w 1 and w 2 , respectively, which can guarantee to make the observations (13).
More precisely, we consider the following problem, said inverse problem (also said exact controllability problem): given 2 such that, if y 1 and y 2 are respective solutions of and then y 1 and y 2 further satisfy the conditions (13).
Remark 2. The symmetric character of the roles played by y 1 and y 2 in the formulation of the controllability problem is obvious.Consequently, one could very well be satisfied with only one of these states in the definition of the problem, thus considering one or the other of problems ( 14) and (15) with the corresponding observation objective in (13).This is evidenced by the first part of the proof of Theorem 1.
As far as the present analysis is concerned, it is precisely this symmetric nature of the roles of y 1 and y 2 that motivates their simultaneous use (which facilitates, perhaps for a short time, the continuation of the analysis), but also the wish to remain faithful to the framework of Cauchy's problem.
Remark 3 (Well-defined nature of the controllability problem, see for instance Lions and Magenes 1968).
Thus, seeking, within the framework of controllability problems, functions of L 2 (Σ 1 ) making it possible to reach, or if not, approaching, the targets fixed still in L 2 (Σ 0 ), it is necessary that the accessible states y 1 and y 2 are in H 2,2 (Q).
Hence the necessity, within the framework of the problem of optimal control, to consider that it is, beyond the non-vacuity assumption of A ∅, the set which is non-empty.
Then, we approach the problem (17) by density, establishing for this purpose the following proposition.
Proposition 1.Let us denote by the sets of zero and one orders traces, on Σ 0 , of the reachable states y 1 and y 2 , respectively.
Then, we have that sets E 1 and E 2 are dense in L 2 (Σ 0 ) , and then we speak of the approached controllability of the system (y 1 (0, w 1 ) , y 2 (0, w 2 )).
Proof.It is clear that E 1 and E 2 constitute vector subspaces of L 2 (Σ 0 ).Hence, by the Hahn-Banach Theorem, E 1 and E 2 are dense in L 2 (Σ 0 ) if and only if their orthogonal E ⊥ 1 and E ⊥ 2 are reduced to {0}.
But, by definition of y 1 (0, w 1 ), we have it comes that that is to say As taking w 1 = ϕ on Σ 1 , we have So therefore, it comes from ( 20), that ϕ satisfies the ill-posed Cauchy problem But then, due to the uniqueness, when it exists, of the solution of such problem, we obtain that ϕ ≡ 0, and consequently, that This last equality being valid for all w 1 ∈ L 2 (Σ 1 ), we deduce that: Analogeously, one conclude at the same result for E 2 .
Remark 7. The approximate controllability problem (19) expresses the following idea: failing to find w 1 , w 2 ∈ L 2 (Σ 1 ) making it possible to reach the targets ∂y 1 ∂ν Σ 0 = 0 and y 2 | Σ 0 = 0 fixed by the exact controllability problem (17), one can obtain sequences by the through which the fixed targets can be approached to ε close, and this, for all ε > 0.
The two corollaries which follow specify this result, first for the exact controllability problem (17) then, by linearity of the problem (cf.Remark (6)), for the exact controllability problem (14)(15)( 16).
1. Let ε > 0. According to Corollary 2, it exists w 1ε , w 2ε ∈ L 2 (Σ 1 ), such that there exist solutions of ( 27),( 28) and ( 29).Then, we generate Assuming that the sequence (w 1ε ) ε is bounded in L 2 (Σ 1 ), it follows, the mixed Dirichlet-Neumann problem ( 27) being well posed in the Hadamard's sense, that the sequence (y 1 (v 0 , w 1ε )) ε is bounded in H 2,2 (Q), and therefore again in L 2 (Q), by continuity of the canonical injection of H 2,2 (Q) in L 2 (Q).Then, we deduce that we can extract, from (w 1ε ) ε and (y 1 (v 0 , w 1ε )) ε respectively, subsequences, again denote in the same way, which converge in L 2 (Σ 1 ) and H 2,2 (Q), respectively.Thus, there exist such that, when ε → 0, w 1ε −→ w 1 weakly in L 2 (Σ 1 ) , But then, we have on the one hand that, when ε → 0, involve, by continuity of the trace operator γ 1 : Lions 1968, p. 21), that On the other hand, for all ϕ ∈ C ∞ Q , we have: that is to say and so By passing to the limit, it comes that which is equivalent to L 2 (Ω) L 2 (Ω) This last equality being valid for all ϕ ∈ C ∞ Q , it follows that Then, (31) and (32) give in particular that Symmetrically, the above shows that, assuming that (w 2ε ) ε is bounded in L 2 (Σ 1 ), we likewise obtain that there exist where y 2 ∈ H 2,2 (Q) is also solution to the ill-posed Cauchy problem (30).
Then, we have So that, for all ε > 0, we can easily define to obtain existence of sequences bounded in L 2 (Σ 1 ) since constants; from where the result.

The Optimal Control problem
Let us start by recalling that we are interested here in the control of the hyperbolic ill-posed Cauchy problem.Starting by the following problem we consider, for all control-state pair (v, z), the cost function being interested in the control problem More precisely, it is here about the characterization of the optimal control-state pair (u, y), via a singular strong and decoupled optimality system.
To do this, we propose in the rest of this section, a regularization method, called controllability method, based on the results of the previous section.This last, recently introduced in the stationary case (cf.Bylli 2023), approaches the initial control problem by a sequence of approached control problems relating to the well-posed problems ( 27) and ( 28).The control problems then considered being regular, the classical theory of optimal control easily apply to lead to the expected result.

Remark 5 .
Problems (14) and (15), mixed Dirichlet-Neumann problems for the wave operator are then two well-posed problems in the sense of Hadamard.
solution of the ill-posed Cauchy problem (30), a regular solution, due to the well-posed nature of (27).