# Observability and controllability for systems in Banach spaces

### Working Groups: Chair Applied Analysis

### Collaborators (MAT): Dr.Â habil. Christian Seifert, Fabian Gabel, M. Sc.

### Collaborators (External): Clemens Bombach, Michela Egidi, Dennis Gallaun, Jan Meichsner, Martin Tautenhahn

## Description

Let \(X,Y\) be Banach spaces, \((S_t)_{t \geq 0}\) a \(C_0\)-semigroup on \(X\), \(-A\) the corresponding infinitesimal generator on \(X\), \(C\) a bounded operator from \(X\) to \(Y\) and \(T>0\). We consider systems of the form \[\begin{equation} \begin{aligned} \dot{x}(t) & = -Ax(t), \quad & &t\in (0,T] ,\quad x(0) = x_0 \in X, \\ y(t) &= Cx(t), \quad & & t\in [0,T] \end{aligned} \end{equation}\]

and study the question whether one can reconstruct the final state \(x(T)\) from the measurements \(y(t)\) for \(t \in (0,T)\). This relates to a final state observability estimate, that is, there exists \(C_{\mathrm{obs}} > 0\) such that for all \(x_0 \in X\) we have \(\lVert x (T) \rVert_X \leq C_{\mathrm{obs}} \lVert y \rVert_{L_r ((0,T) ; Y)}\) for some \(r\in [1,\infty]\).

The most studied example is the self-adjoint SchrÃ¶dinger operator \(A = \Delta - V\) in \(L_2 (\Omega)\) with bounded potential \(V\), and \(C = \mathrm{1}_E\) for \(E \subset \Omega \subset \mathbb{R}^d\). One possible approach to show an observability estimate is the so-called Lebeau-Robbiano method [1], that is, to prove a quantitative *uncertainty relation* for spectral projectors. This is an inequality of the type \[\begin{equation*}
\forall \lambda > 0 \ \forall \psi \in L_2 (\Omega) \colon \quad
\lVert P (\lambda) \psi \rVert_{L_2 (\Omega)}
\leq
d_0 \mathrm{e}^{d_1 \lambda^{\gamma}}
\lVert \mathrm{1}_E P (\lambda) \psi \rVert_{L_2 (E)} ,
\end{equation*}\]

where \(\gamma \in (0,1)\), \(d_0,d_1 > 0\), and where \(P (\lambda)\) denotes the projector to the spectral subspace of \(-\Delta + V\) below \(\lambda\). Subsequently, this strategy is generalized to semigroups in abstract Hilbert spaces. In particular the \(P(\lambda)\) are allowed to be arbitrary projectors (onto semigroup invariant subspaces) by assuming additionally a so-called *dissipation estimate*, that is, a decay estimate of the semigroup on the orthogonal complement of the range of \(P(\lambda)\).

A natural setup to ask for observability estimates is the context of Banach spaces and \(C_0\)-semigroups. In [2], we extend the above-mentioned strategy to the Banach space setting. In particular, we show in the general framework of Banach spaces that an uncertainty relation together with a dissipation estimate implies a final state observability estimate. Our observability constant \(C_{\mathrm{obs}}\) is given explicitly with respect to the parameters coming from the uncertainty relation and the dissipation estimate and, in addition, is sharp in the dependence on \(T\).

Using the well-known relation between observability and null-controllability of the predual system one can proof controllability results by the adapted Lebeau-Robbiano method. In [3] we consider parabolic control systems on \(L_p(\mathbb{R}^d)\), \(p\in [1,\infty)\), of the form \[\begin{equation} \dot{x}(t) = -A_p x(t) + \mathrm{1}_E u(t),\quad t\in (0,T],\quad x(0) = x_0\in L_p(\mathbb{R}^d), \end{equation}\]

where \(-A_p\) is a strongly elliptic differential operator with constant coefficients. Assuming that \(E\subset \mathbb{R}^d\) is a so-called thick set, we proof (approximate) null-controllability, i.e.Â for all \(x_0\in L_p(\mathbb{R}^d)\) there is \(u \in L_r ((0,T);L_p (E))\) which steers the mild solution at time \(T\) (approximately) to zero.

A weaker concept than null-controllability is stabilizability. In the work [4], the uncertainty relation and the dissipation estimate in the Lebeau-Robbiano method are weakened to prove stabilizabilty for systems which are not null-controllable. The work [5] extends the Lebeau-Robbiano strategy to non-autonomous Cauchy problems and studies final-state observability for observation systems with moving sensor sets.

## References

[1] G. Lebeau and L. Robbiano. ContrÃ´le exact de lâ€™Ã©quation de la chaleur, Comm. Partial Differential Equations, 20(1â€“2):335â€“356, 1995.

[2] D. Gallaun, C. Seifert, and M. Tautenhahn. Sufficient criteria and sharp geometric conditions for observability in Banach spaces, SIAM Journal of Control and Optimization, 58(4):2639â€“2657, 2020. doi:10.1137/19M1266769

[3] C. Bombach, D. Gallaun, C. Seifert, and M. Tautenhahn. Observability and null-controllability for parabolic equations in \(L_p\)-spaces, Mathematical Control and Related Fields, 2022. doi:10.3934/mcrf.2022046

[4] M. Egidi, D. Gallaun, C. Seifert, and M. Tautenhahn. Sufficient criteria for stabilization properties in Banach spaces. arXiv:2108.09028.

[5] C. Bombach, F. Gabel, C. Seifert, and M. Tautenhahn. Observability for non-autonomous systems, to appear in SIAM Journal of Control and Optimization, 2022. arXiv:2203.08469