Optimal Cost of Null Controllability

• Optimal cost of null controllability is defined as the minimal norm of controls needed to drive system states to zero, characterized via sharp analytic inequalities and precise geometric conditions.
• It relies on geometric criteria like the flushing condition and thickness, which determine whether the control cost remains uniformly bounded or exhibits exponential blow-up.
• Analytical methods such as Carleman estimates, Agmon inequalities, and spectral techniques underpin the asymptotic analysis, guiding control design and sensor placement in various PDE systems.

The optimal cost of null controllability quantifies the minimal norm of admissible controls that steer the solution of a given evolution equation—typically parabolic, transport-diffusion, or Stokes-type systems—to zero at a prescribed time. This cost, typically expressed as a function of system parameters such as diffusivity, domain geometry, or horizon time, is governed by sharp analytic inequalities (Carleman, Agmon, spectral) and precise geometric conditions on the control region. Its asymptotic behavior under time or viscosity limits encodes the “difficulty” of exerting null controllability and is intricately linked to control theory, unique continuation, optimal design, and the microlocal structure of PDEs.

1. Problem Setting and Definitions

Consider a general linear evolution equation in a bounded domain $\Omega \subset \mathbb{R}^d$ with a smooth boundary and an internal or boundary control supported on a measurable subset $\omega$ over time interval $[0,T]$. A common setting is the transport-diffusion equation: $$\begin{cases} \partial_t y - \varepsilon \Delta y + B(x,t)\cdot \nabla y = u(x,t)\mathbf{1}_\omega(x), & (x,t)\in\Omega\times(0,T), \ \text{(Boundary conditions)}, \ y(\cdot,0)=y_0 \in L^2(\Omega), \end{cases}$$ with control $u \in L^2(\omega\times(0,T))$, diffusivity $\varepsilon>0$, and drift $B$ tangential to the boundary.

The null-controllability cost $C(\varepsilon,T)$ is defined as the smallest constant such that every initial datum $y_0$ can be driven to zero at time $T$ by some control $u$ with

$$\|u\|_{L^2(\omega\times(0,T))} \leq C(\varepsilon,T)\|y_0\|_{L^2(\Omega)}.$$

Alternatively, via duality and the Hilbert Uniqueness Method (HUM), the same constant arises as the optimal observability constant for the corresponding adjoint problem (Et-tahri et al., 2023).

2. Geometric Conditions and the Flushing Paradigm

The scaling and boundedness properties of the optimal cost are governed by characteristic geometric conditions linking system trajectories and the control region.

Flushing Condition: There exist $T_0 \in (0,T)$ and $r_0>0$ such that every backward trajectory of the flow associated to $B$ starting from any point in $\bar\Omega$ and time $[T_0,T]$ enters $\omega$ within a time at most $T_0$, even under perturbations of the starting point of size $r_0$. This ensures that no trajectory avoids the region of effect of the control for too long—a necessary feature for uniform controllability as diffusivity vanishes (Et-tahri et al., 2023).

Thickness and Integral Thickness: For heat-like equations (including those on unbounded domains), null-controllability is characterized by the thickness of $\omega$: every sufficiently large box must contain a uniform proportion of $\omega$ (“thick with respect to $\Omega$”) (Egidi, 2018, Bombach et al., 2020). For non-autonomous or hypoelliptic problems (e.g., the Ornstein-Uhlenbeck flows), an “integral thickness” (averaged in time and space under the flow) is the sharp threshold (Alphonse et al., 2022).

The following table summarizes optimal cost behaviors under geometric conditions:

System Type	Geometric Condition	Optimal Cost Behavior	Reference
Transport-diffusion, $\varepsilon\to0$	Flushing condition satisfied	Uniformly bounded as $\varepsilon\to0$ if $T\gg T_0$	(Et-tahri et al., 2023)
	Flushing fails	$C(\varepsilon,T) \gtrsim \exp(c/\varepsilon)$	(Et-tahri et al., 2023)
Heat on $\mathbb{R}^d$ or strip	Thickness (of $\omega$)	$C(T) \leq A\exp(B/T)$	(Egidi, 2018)

3. Asymptotic Behavior of the Cost: Phase Transitions and Dichotomies

The optimal cost for null controllability typically exhibits a sharp dichotomy or phase transition determined by time or viscosity parameters relative to the geometric structure.

Vanishing Viscosity / Small-Time Asymptotics:

• If the control region $\omega$ satisfies the appropriate geometric (flushing/thickness) condition and $T \gg T_0$, then

$$\sup_{0 < \varepsilon < \varepsilon_0} C(\varepsilon, T) < \infty.$$

• If even one characteristic trajectory fails to reach $\omega$ within $[0,T]$,

$$C(\varepsilon, T) \gtrsim \exp(c/\varepsilon) \quad (\varepsilon \to 0),$$

i.e., the cost exhibits exponential blow-up as diffusivity vanishes, independently of control time unless $T$ is large (Et-tahri et al., 2023).

This asymptotic dichotomy completely resolves the question posed by Guerrero and Lebeau (2007), establishing that the vanishing viscosity limit is benign (uniform cost) precisely under the geometric flushing condition, and pathological (exponential cost) otherwise (Et-tahri et al., 2023).

Classical Parabolic Problems:

For scalar heat, Stokes systems, degenerate/singular operators, and wide classes of parabolic equations, the cost as $T \to 0$ universally blows up as

$C(T) \sim \exp(C/T),$

with sharp constants dictated by domain, operator, and control geometry (Egidi, 2018, Chaves-Silva, 2013, Biccari et al., 2020, Galo-Mendoza, 13 Mar 2024). This scaling is sharp (cannot be improved), and holds for scalar and vectorial systems, with or without drift, as long as geometric control conditions are met (Chaves-Silva et al., 4 Dec 2025, Balc'h et al., 6 Feb 2025, Cornilleau et al., 2012).

4. Analytical Methods: Carleman, Agmon, and Spectral Estimates

The derivation of sharp cost bounds is based on quantitative unique continuation and dissipation inequalities applied to the adjoint (backward) equations.

• Carleman Estimates: Weighted energy inequalities with exponential weights tailored to the geometry and degeneracy of the system. In the presence of a drift or transport term, weights are constructed along flows and characteristics; for vectorial/parabolic systems, advanced Carleman inequalities with pseudodifferential weights and suitable boundary conditions are employed (Et-tahri et al., 2023, Balc'h et al., 6 Feb 2025, Chaves-Silva et al., 4 Dec 2025).
• Agmon Inequalities: Dissipation estimates for exponential-weighted solutions, especially powerful in the vanishing viscosity regime, allowing control of the mass of solutions (adjoint variables) away from the control region (Et-tahri et al., 2023).
• Spectral (Moment) Methods: For 1D, degenerate, or singular problems, the cost bounds are constructed via explicit biorthogonal families to the exponential modes of the controlled operator (Fattorini–Russell moment method). These yield precise exponential upper and lower bounds on the cost (Biccari et al., 2020, Galo-Mendoza et al., 2023, Galo-Mendoza, 13 Mar 2024).

The interplay of these analytic tools, together with geometric covering arguments (e.g., for "thickness"), yields optimal cost estimates both above and below.

5. Specific Systems and Generalizations

• Heat and Fractional Heat: In domains with measurable control sets, null-controllability cost is governed by the thickness property, with $C(T) \leq A\exp(B/T)$, with explicit dependence on the measure and distribution of $\omega$ (Egidi, 2018, Bombach et al., 2020).
• Degenerate and Singular Parabolic Operators: Null controllability is possible under suitable parameter regimes, with cost scaling as $\exp(c/T)$ and matching lower bounds constructed through moment and representation-theoretic arguments (Biccari et al., 2020, Galo-Mendoza et al., 2023, Galo-Mendoza, 13 Mar 2024).
• Stokes and Coupled Stokes Systems: For the classical Stokes system and its coupled generalizations, the cost behaves as $\exp(C/T)$; the Kalman rank condition is both necessary and sufficient for uniform null-controllability, and this extends to reduced-component controls (e.g., $n-1$ scalar controls in $\mathbb{R}^n$) with no additional singularity in the exponent (Chaves-Silva, 2013, Balc'h et al., 6 Feb 2025, Chaves-Silva et al., 4 Dec 2025).
• Transport-Diffusion with Drift: The vanishing viscosity limit brings about a geometric phase transition for the cost, as established by both Carleman and Agmon arguments (Et-tahri et al., 2023, Et-tahri et al., 21 Dec 2024).

6. Broader Impact and Open Directions

The precise quantification of the optimal cost of null controllability has implications for control design, sensor/actuator placement, time-critical operations, and stabilization strategies across deterministic, stochastic, and coupled PDE systems. The identification of geometric thresholds (flushing, thickness, Kalman condition) and the matching of exponential asymptotics above and below ensure that practical implementations are informed by non-improvable analytic bounds.

Open directions include:

• Refinement of constants and structural dependencies in the exponential cost, particularly in high-dimensional and mixed-type systems.
• Lower bounds for non-self-adjoint systems and for controls acting only on subsets of system components.
• Characterization of cost under more general (nonlinear, stochastic, or nonlocal) dynamics, and the effect of rough or time-dependent control regions.

The canonical dichotomy between uniform boundedness and exponential blow-up—driven by geometric control conditions and propagated through Carleman-type analysis—forms the backbone of the modern theory of null-controllability cost (Et-tahri et al., 2023, Chaves-Silva et al., 4 Dec 2025, Balc'h et al., 6 Feb 2025).