L₂(Σ)‐regularity of the boundary to boundary operator B^∗L for hyperbolic and Petrowski PDEs

Abstract

This paper takes up and thoroughly analyzes a technical mathematical issue in PDE theory, while-as a by-pass product-making a larger case. The technical issue is the L 2 (Σ)-regularity of the boundary → boundary operator B * L for (multidimensional) hyperbolic and Petrowski-type mixed PDEs problems, where L is the boundary input → interior solution operator and B is the control operator from the boundary. Both positive and negative classes of distinctive PDE illustrations are exhibited and proved. The larger case to be made is that hard analysis PDE energy methods are the tools of the trade-not soft analysis methods. This holds true not only to analyze B * L but also to establish three inter-related cardinal results: optimal PDE regularity, exact controllability, and uniform stabilization. Thus, the paper takes a critical view on a spate of "abstract" results in "infinite-dimensional systems theory," generated by unnecessarily complicated and highly limited "soft" methods, with no apparent awareness of the high degree of restriction of the abstract assumptions made-far from necessary-as well as on how to verify them in the case of multidimensional dynamical systems such as PDEs.1. A historical overview: hard analysis beats soft analysis on regularity, exact controllability, and uniform stabilization of hyperbolic and Petrowski-type PDEs under boundary control At first, naturally, PDEs boundary control theory for evolution equations tackled the most established PDE classes-parabolic PDEs-whose Hilbert space theory for mixed problems was already available in a form close to an optimal book form [51,56,57,58] since the early 1970s. Next, in the early 1980s, when the study of boundary control problems for (linear) PDEs began to address hyperbolic and Petrowski-type systems on a multidimensional bounded domain [10,26] (see [5,6, 35,44,45] Hard analysis energy methods. A happy and quite challenging exception was the optimal-both interior and boundary-regularity theory for mixed, nonsymmetric, noncharacteristic first-order hyperbolic systems culminated through repeated efforts in the early 1970s [16,63,64]. Its final, full success required eventually the use of pseudodifferential energy methods (Kreiss' symmetrizer). Apart from this isolated case, mathematical knowledge of global optimal regularity theory of hyperbolic and Petrowski-type mixed problems was scarce, save for some trivial one-dimensional cases. Thus, in this gloomy scenario, one may say that optimal control theory [10, 26, 51] provided a forceful impetus in seeking to attain an optimal global regularity theory for these classes of mixed PDEs problems. To this end, PDEs (hard analysis) energy methods-both in differential and pseudodifferential form-were introduced and brought to bear on these problems. The case of second-order hyperbolic equations under Dirichlet boundary control was tackled first. The resulting theory that emerged turns out to be optimal and does not depend on the space dimension [22,24,25,43,52]. It was best achieved by the use of energ...

Improving Convergence in Numerical Analysis Using Observers — The Wave-Like Equation Case

“…Concerning other types of boundary measurements such as pertaining to velocity or boundary-normal derivatives, as discussed in Ref. 19 they generally correspond to cases for which proving stabilization by the observability condition is no longer the adequate strategy -although uniform stabilization can frequently be obtained by simple dissipative feedbacks -hence this lies beyond the scope of the present paper.…”

Exponential stabilization of well-posed systems by colocated feedback

“…If one examines these examples carefully, one can recognize that they fit into our framework (the assumptions ESAD and COL are satisfied) and they use the feedback u = −κy +v for stabilization. There are also examples in the literature where the open-loop system is not well posed, but application of the static feedback results in a well-posed exponentially stable closedloop system, see Rebarber [27], Rebarber and Townley [28], Weiss [49], Lasiecka and Triggiani [20]. These examples are not covered by the theory in this paper (except for some partial results in Remarks 5.3 and 5.4).…”

“…The last two assumptions (and also the fact that Q = 0) are more restrictive than in our framework, but on the other hand, they do not require the open-loop system to be well posed. We note that our examples in sections 5 and 6 are not covered by the theory in [20], because they do…”

“…The recent paper [20] is interesting because it also gives an abstract framework for treating exponential stabilization by colocated feedback. The assumptions are ESAD (with Q = 0), COL, 0 ∈ ρ(A) and A − 1 2 B ∈ L(U, X).…”

“…The main thrust of [20] is to give examples of nonwell-posed systems satisfying the assumptions mentioned earlier, for which A − BB * is exponentially stable. (See also the corrections to [20] in [21].…”