Lydia Ouaili | Search arXiv

arXiv2024

Sharp estimates of the one-dimensional boundary control cost for parabolic systems

In this work we present new results on the cost of the boundary controllability of parabolic systems at time $T > 0$. In particular, we will study optimal estimates of the control cost at time $T$ ($T$ small enough) when the eigenvalues of the generator of the $C_0$ semigroup accumulate and do not satisfy a gap condition. The main ingredient we will use is the moment method combined with sharp estimates of the $L^2(0,T; \mathbb C)$-norm of the elements of biorthogonal families to complex exponentials.

math.OC↗

arXiv2024

Sharp estimates for biorthogonal families to exponential functions associated to complex sequences without gap conditions

The general goal of this work is to obtain upper and lower bounds for the $L^2$-norm of biorthogonal families to complex exponential functions associated to sequences $\{ Λ_k \}_{k \ge 1} \subset \mathbb C$ which satisfy appropriate assumptions but without imposing a gap condition on the elements of the sequence. As a consequence, we also present new results on the cost of the boundary null controllability of two parabolic systems at time $T > 0$: a phase-field system and a parabolic system whose generator has eigenvalues that accumulate. In the latter case, the behavior of the control cost when $T$ goes to zero depends strongly on the accumulation parameter of the eigenvalue sequence.

math.OC↗