Delay and Memory Null Controllability

• Delay and memory-type null controllability is defined as the simultaneous nullification of the state, memory functional, and delayed history to prevent post-control reactivation.
• The approach employs analytic techniques such as Carleman estimates, spectral decomposition, and moment problem methods under strict geometric control conditions.
• This concept extends classical control theory, proving essential for practical applications in viscoelasticity, hereditary materials, and systems with after-effects.

Delay and memory-type null controllability addresses the problem of steering systems governed by partial differential equations (PDEs) or finite-dimensional ordinary differential equations (ODEs), which are perturbed by both memory (hereditary) and delay effects, to the equilibrium in a way that prevents any subsequent reactivation due to accumulated memory or delayed influence. This strengthened notion of controllability, motivated by the persistent impact of the past trajectory in memory and delay terms, requires simultaneous annihilation of the state, the memory functional, and—when delays are present—the delayed state history at a prescribed terminal time. The subject has developed rapidly in both finite and infinite-dimensional settings and is foundational for control theory in viscoelasticity, hereditary materials, and systems with after-effects.

1. Problem Formulation and Core Definitions

Let $y$ denote the state variable governed by either a PDE or ODE including memory (typically of Volterra type) and discrete or distributed delay terms. A canonical model in the parabolic setting is

$$\begin{cases} y_t(t,x) - \Delta y(t,x) + \int_0^t M(t-s)y(s,x)\,ds + A_1 y(t-h,x) = u(t,x)\chi_{\omega(t)}(x), \ y(0,x) = y^0(x),\quad y(t,x) = 0\ \text{on}\ \partial\Omega, \ y(t,x) = \varphi(t,x),\quad t\in[-h,0], \end{cases}$$

where $M(\cdot)$ is a memory kernel, $A_1$ is a bounded operator, $h>0$ is the delay, and $u$ acts on a possibly time-dependent support $\omega(t)\subset\Omega$ (Jha et al., 27 Jun 2025, Jha et al., 15 Dec 2025).

Delay and memory-type null controllability is defined as the existence, for any admissible initial state and delay history, of a control $u$ such that:

• $y(T) = 0$ (the state is zero at final time),
• $\int_0^T \widetilde{M}(T-s) y(s)\,ds = 0$ (memory functional vanishes at $T$ for a possibly distinct kernel $\widetilde{M}$),
• $y(t) = 0$ for $t\in[T-h,T]$ (the state’s delayed trace on the final delay window is zero) (Jha et al., 27 Jun 2025, Jha et al., 14 Jan 2026).

For equations with only memory and no delays, the third condition is omitted (Chaves-Silva et al., 2017, Jha et al., 15 Dec 2025).

2. Structural Consequences of Memory and Delay

The presence of memory and delay fundamentally alters classical controllability. If only the state is brought to rest at $T$, the dynamical system retains accumulated memory and/or delayed state dependencies, leading to post-control reactivation. For memory-type null controllability, enforcing both $y(T) = 0$ and vanishing accumulated memory at $T$ ensures that for $t > T$, the homogeneous system remains at equilibrium (Chaves-Silva et al., 2017, Jha et al., 15 Dec 2025).

When delay terms are included, the control must further kill the delayed history, as any nontrivial state in $(T-h, T)$ will act as a latent source for $t > T$ (Jha et al., 27 Jun 2025, Jha et al., 14 Jan 2026). This yields a notion strictly stronger than classical null controllability.

A sharp dichotomy emerges in generic parabolic models with memory:

• Approximate controllability (dense reachability of final states) is typically inherited from the memoryless system, under mild regularity of the memory kernel (Pandolfi et al., 2014).
• Null controllability is generically lost for nontrivial memory kernels unless the memory term can be entirely eliminated by the control, which is only possible in special cases or with appropriate moving control strategies and geometric conditions (Pandolfi et al., 2014, Fernández-Cara et al., 2018, Jha et al., 15 Dec 2025).

3. Main Results: Criteria and Methods

3.1. Finite-Dimensional Systems

For ODE systems of the form

$$\dot y(t) = Ay(t) + \int_0^t M(t-s)y(s)\,ds + D y(t-h) + Bu(t),$$

delay and memory-type null controllability is characterized by an augmented observability estimate for the adjoint system, together with an algebraic rank condition generalizing the Kalman criterion. Explicitly,

$$\operatorname{rank} \left[ B, AB, \ldots, A^{n-1}B,\, M_0, \ldots, M_{n-1},\, D, AD, \ldots, A^{n-1}D \right] = n$$

ensures that the control can kill the state, memory functional, and history at $T$ (Jha et al., 14 Jan 2026, Jha et al., 27 Jun 2025). Similar recursive rank-type conditions are available for just memory (Silverman–Meadows–type) (Chaves-Silva et al., 2017).

3.2. Parabolic and Evolutionary PDEs

For parabolic equations with memory and delay, delay and memory-type null controllability is established via a duality framework: controllability is equivalent to an observability inequality for an augmented adjoint system comprising coupled PDE–ODE (memory) and potentially delay components (Jha et al., 27 Jun 2025, Jha et al., 15 Dec 2025, Allal et al., 2021). The key analytic tools are:

• Global Carleman estimates tailored to moving control regions, enabling weighted energy inequalities that can absorb the nonlocal-in-time memory and delay terms (Chaves-Silva et al., 2017, Jha et al., 15 Dec 2025, Allal et al., 2021).
• Reduction to a moment problem or Hilbert Uniqueness Method (HUM) minimization, leveraging biorthogonal families or flow-adapted weights (Biccari et al., 2019, Jha et al., 15 Dec 2025).

A central result is that memory-type null controllability holds under a Memory Geometric Control Condition (MGCC): every spatial point of the domain must be visited by the moving control region within the control horizon (Jha et al., 15 Dec 2025, Jha et al., 27 Jun 2025).

3.3. Spectral and Moment Problem Techniques in Fractional and Hyperbolic Systems

For fractional or hyperbolic equations with memory, the coupled PDE–ODE system is analyzed via spectral decomposition. The controllability problem reduces to a non-degenerate moment problem for a biorthogonal family associated to the shifted spectrum of the controlled operator (Biccari et al., 2019). This requires detailed gap conditions in the spectrum to construct entire functions of the Paley–Wiener–type with controlled exponential type.

4. Geometric and Analytic Conditions

The necessity of geometric control conditions is pronounced in systems with memory and/or delay. For heat or parabolic equations with memory kernels admitting a finite exponential expansion, the MGCC criterion (the moving control region sweeps the spatial domain) suffices for memory-type null controllability (Jha et al., 15 Dec 2025). When this condition is violated, one can construct solutions concentrated in "blind" regions never reached by the control, demonstrating failure of observability and hence of controllability.

The analytic regularity of reachable final states restricts null controllability further: for parabolic equations with control restricted to a proper subset, all reachable states at final time are analytic on the uncontrolled region. This excess regularity obstructs the possibility to steer arbitrary $L^2$ initial data to zero (Pandolfi et al., 2014).

5. Limitations and Obstructions

There exist significant obstructions to delay and memory-type null controllability in the generic case:

• For classical heat or Stokes equations with memory (Coleman–Gurtin or viscoelastic-type), even with full boundary or distributed control, null controllability fails for nontrivial kernels (Pandolfi et al., 2014, Fernández-Cara et al., 2018).
• The infinite-dimensional structure of Volterra or delay operators introduces non-compact memory effects that prevent exact steering to zero.
• For degenerate parabolic systems, null controllability with memory requires extra decay or structural assumptions on the memory kernel to be compatible with Carleman weighting (Allal et al., 2021).

The dichotomy between approximate controllability and true null controllability thus persists: memory effects block full erasure of past influence, surface as lack of unique continuation for the adjoint, and preclude global null controllability except under strong geometric or analytic conditions.

6. Extensions, Generalizations, and Open Problems

Current research addresses several generalizations and unsolved issues:

• Delay and Memory in Higher Dimensions: The extension of the geometric MGCC and spectral gap techniques to higher-dimensional and non-rectangular domains relies on the specificity of the spectrum and the flow structure, remaining unresolved for general operators (Jha et al., 15 Dec 2025, Biccari et al., 2019).
• Non-analytic or Non-exponential Kernels: For memory kernels without an explicit exponential form, the Carleman/HUM duality program is not yet available, and the necessity of MGCC or moment method constructions remains open (Jha et al., 15 Dec 2025).
• Minimal Time and Control Cost: Quantitative control cost estimates and minimal null-controllability times under delay and memory remain to be established, especially in the presence of slow memory decay or small delays (Biccari et al., 2019, Jha et al., 27 Jun 2025).
• Nonlinear Systems and Applications: The role of nonlinearities, robust control strategies, and practical implementation for viscoelastic, hereditary, and biological systems with degenerate diffusion or age-structure remains a focus (Allal et al., 2021).
• Relation to Other Notions of Controllability: The distinction between approximate, exact, and null controllability in hereditary systems is intricate due to the regularity and analytic nature of reachable states (Pandolfi et al., 2014).

7. Numerical Verification and Illustrative Examples

Numerical simulations validate the theoretical predictions in delay and memory-type controllability:

• Discrete schemes (finite differences or backward Euler) are adapted to handle both delay and convolution memory integrals (Jha et al., 27 Jun 2025).
• Examples confirm that, under moving control regions covering the domain, the state, accumulated memory, and delayed history are simultaneously annihilated at final time, establishing practical reachability under the established theoretical criteria.

In summary, delay and memory-type null controllability is a modern and active direction in control theory of evolutionary systems with after-effects, sitting at the intersection of geometric control analysis, spectral theory, Carleman inequalities, and functional analysis of nonlocal and hereditary dynamics. The field is shaped by sharp negative results in classical settings and by positive criteria under geometric covering and fast memory kernels, with ongoing work on sharp necessary and sufficient conditions, analytic extension barriers, and numerical realization.

Key references: (Pandolfi et al., 2014, Chaves-Silva et al., 2017, Fernández-Cara et al., 2018, Biccari et al., 2019, Allal et al., 2021, Jha et al., 27 Jun 2025, Jha et al., 15 Dec 2025, Jha et al., 14 Jan 2026).