Mohamed Fadili | Searcharxiv

arXiv2026

A noncommunicative Kalman condition for null controllability of backward stochastic parabolic systems

We study null controllability of backward stochastic heat systems with constant couplings in both the state and the martingale integrand, common scalar diffusion, and a localized drift control. We prove that null controllability is equivalent to a noncommutative word-rank condition on the two coupling matrices and the control matrix. The proof combines a factorization of the dual dynamics, positivity and small-time coercivity of a finite-dimensional stochastic Gramian, and a Lebeau--Robbiano spectral argument. Failure of the rank condition yields an invariant unobservable subspace.

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arXiv2026

Null controllability of degenerate parabolic equations with a time-varying delay

This paper is devoted to the null controllability of a one-dimensional linear parabolic equation with a time-varying state delay and a diffusion coefficient that degenerates at one endpoint. Both the weakly and strongly degenerate cases are considered. When the delay map $g(t)=t-τ(t)$ is strictly increasing, the adjoint equation contains an advanced term weighted by the Jacobian associated with $g^{-1}$. For a distributed control acting on an interior subinterval, we combine a degenerate Carleman estimate on the final delay-free interval with a monotonicity functional whose upper integration limit moves in time. The resulting observability inequality gives null controllability under a terminal decay condition on the delayed coefficient outside the control region. We then obtain Dirichlet boundary controllability at the nondegenerate endpoint by extending the equation in space. The constant-delay equation is recovered as a special case.

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