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Open research questions in Stability and Controllability of Differential Equations

59 unresolved questions extracted from the limitations and future-work sections of 368 Stability and Controllability of Differential Equations papers in our library. Each links back to the study that raised it.

What the literature leaves open

  • Further research can be conducted to extend the results to more complex systems. The study can be extended to consider other types of uncertainties and disturbances.

    Guaranteed Cost Boundary Control for Reaction–Diffusion Systems · 2026 · DOI

  • Prior work on guaranteed cost control for partial differential systems did not take into account the effects of time delay and uncertain parameters. There is a need to investigate the guaranteed cost boundary control for uncertain reaction-diffusion systems with time delays.

    Guaranteed Cost Boundary Control for Reaction–Diffusion Systems · 2026 · DOI

  • Future research should focus on extending the results to other types of noise or perturbations. Future research should focus on developing numerical methods for simulating and analyzing the stochastic Nernst-Planck-Navier-Stokes system. Future research should focus on applying the results to real-world problems and applications.

    Random attractors for the stochastic Nernst-Planck-Navier-Stokes system with multiplicative white noise · 2026 · DOI

  • The problem is ill-posed due to the presence of a term that can cause the solution to blow up. The function J γ ε is not convex, which leads to non-uniqueness of the low-regret control. The state of the system is unknown on the boundary of the domain.

    Low-regret control of a nonlinear parabolic problem with missing data · 2026 · DOI

  • The paper identifies a gap in the study of autoignition fronts in reactive co-flow jets. The paper addresses the need for a dynamical system approach to analyze the model.

    Existence and uniqueness of traveling fronts for a free interface model of autoignition in reactive jets · 2026 · DOI

  • The paper identifies a gap in the literature on the global existence and nonexistence of solutions to the viscoelastic wave equation with combined power-type nonlinearities. The paper addresses the need for a more comprehensive understanding of the behavior of materials with memory.

    Finite time blow-up and global solutions for the viscoelastic wave equation with combined power-type nonlinearities · 2026 · DOI

  • The stochastic Nernst-Planck-Navier-Stokes system remains poorly understood, particularly with regards to the existence and properties of random attractors. Prior work has focused on the deterministic Nernst-Planck-Navier-Stokes system, but the stochastic system requires new techniques and approaches.

    Random attractors for the stochastic Nernst-Planck-Navier-Stokes system with multiplicative white noise · 2026 · DOI

  • The system has competing viscoelastic and frictional dissipative effects. The energy function needs to be estimated using mathematical techniques.

    Exponential Stability for the Coupled Klein–Gordon–Schrödinger Equations with Competing Viscoelastic and Frictional Dissipative Effects · 2026 · DOI

  • We have considered an approximating lattice dynamical system for a non-autono- mous parabolic equation which describes the behavior of concentrated suspensions. The existence of a compact uniform attractor for the family of semi-processes gener- ated by solutions of the lattice system in the phase space l1,w has been established. For the future, we plan to study issues related to the upper semicontinuity of the global attractors for finite-dimensional approximations of this model. We also plan to investigate the relationship between the attractors of the lattice dynamical system and the original problem. In the autonomous case, this issue has been investigated by Kapustyan et al. [16], but for the non-autonomous case, it remains open. Finally, we intend to conduct numerical studies using splitting-based methods to further analyze the behavior of the model under various shear flows. Author Contributions N.V.G. and B.R. contributed equally to the formulation, mathematical analysis, theoretical results, and preparation of the manuscript. All authors reviewed the manuscript and approved the final version. Funding Open Access funding enabled and organized by Projekt DEAL. This research received no exter- nal funding. Data Availability No datasets were generated or analysed during the current study. Code Availability Not applicable.

    Asymptotic Behavior of Solutions for a Lattice System Generated by Non-Autonomous Complex Flows Modeling Concentrated Suspensions · 2026 · DOI

  • The proof of Theorem 4.3 for n=1 relies on a Taylor expansion argument (equation 4.51-4.52) that requires precise estimates of the Hessian convergence near ū. The paper does not discuss whether these estimates can be made uniform across problem classes or whether they degrade under perturbations in the Lagrangian coefficients or nonlinear semilinearity terms.

    Infinite Horizon Control Problems for Semilinear Parabolic Equations with Pointwise State Constraints · 2026 · DOI

  • The infinite horizon control problem with horizon parameter T̄u appears implicitly in the formulation (equations 4.35-4.39), but the paper does not analyze how the choice of T̄u affects the optimality conditions or whether there exists a minimum horizon length required for the second-order sufficient conditions to hold in the parabolic control setting.

    Infinite Horizon Control Problems for Semilinear Parabolic Equations with Pointwise State Constraints · 2026 · DOI

  • The gap in the existing literature is the lack of a suitable method for solving the quadratic regulator problem. The prior work uses a different method that assumes u ∈ H 1 and decouples u(0) from u(•).

    The Regulator Problem for the Wave Equation with High Internal Damping Controlled on the Boundary: A New Look via Systems with Memory · 2026 · DOI

  • The Lp-theory of the evolutionary Stokes system is more difficult than the Hilbert L2-setting. The paper must establish the existence of an analytic semigroup and optimal regularity estimates. The authors need to treat the case of a half-space.

    On the $$L^p$$-semigroups for Stokes equations with dynamic slip boundary conditions in the half-space · 2026 · DOI

  • The paper only considers linear quadratic optimal control problems. The results are limited to the case of homogeneous Dirichlet or Neumann boundary conditions.

    Constrained Optimal Control Problems for Parabolic PDEs with Initial Data in Besov Spaces · 2026 · DOI

  • There is a gap in the existing literature regarding the regularity of adjoint states for optimal control problems with non-homogeneous boundary conditions. The paper aims to fill this gap.

    Constrained Optimal Control Problems for Parabolic PDEs with Initial Data in Besov Spaces · 2026 · DOI

  • Further study of the null controllability of the 1D heat equation with other types of singular potentials. Application of the results to other fields such as combustion theory and quantum mechanics.

    Null controllability of the 1D heat equation with interior inverse square potential · 2026 · DOI

  • The lack of a proof for null controllability of the 1D heat equation with an interior inverse square potential in arbitrary small time. The need for a precise spectral study of the singular operator.

    Null controllability of the 1D heat equation with interior inverse square potential · 2026 · DOI

  • The paper assumes a bounded domain in R^d, d ∈ {2, 3}, with smooth boundary. The approach may not be applicable to systems with more complex boundary conditions. The paper does not address the implementation of the controller in practice.

    Boundary stabilizability of generalized Burgers–Huxley equation with memory · 2026 · DOI

  • Future research can focus on implementing the controller in practice. The approach can be applied to systems with similar dynamics. Further research can investigate the stabilization of more complex systems.

    Boundary stabilizability of generalized Burgers–Huxley equation with memory · 2026 · DOI

  • The study assumes a specific form for the memory term and fractional derivative damping order. The results are limited to the specific system studied.

    Existence and stabilization of the wave equation with past history and fractional damping controls · 2026 · DOI

  • The study identifies a gap in the existing literature on wave equations with memory term and fractional derivative damping order. The paper aims to fill this gap by establishing the well-posedness and strong stability of the system.

    Existence and stabilization of the wave equation with past history and fractional damping controls · 2026 · DOI

  • The paper does not prove the full coupled (B, X, h)-Lyapunov structure of Conjecture L.D.13. The paper only considers the floor sector and does not address the full dynamics.

    Admissibility of the Regulated Blockade Floor: Cone–Invariance and Eventual Positivity for Symmetrized Recovery Dynamics · 2026 · DOI

  • Future research should address the full coupled (B, X, h)-Lyapunov structure of Conjecture L.D.13. Future research should explore the implications of the results for the study of symmetrized recovery dynamics.

    Admissibility of the Regulated Blockade Floor: Cone–Invariance and Eventual Positivity for Symmetrized Recovery Dynamics · 2026 · DOI

  • The lack of a characterization of the optimal control for nonlinear parabolic problems with missing data. The need for a new approach to handle the non-convexity of the function J γ ε.

    Low-regret control of a nonlinear parabolic problem with missing data · 2026 · DOI

  • The method is limited to the case of stochastic perturbations. The method requires the solution of LMIs, which can be computationally expensive. The method is not applicable to the case of non-stochastic perturbations.

    Constructive boundary observer-based control of high-dimensional semilinear heat equations · 2026 · DOI

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59 open questions have been extracted from the limitations and future-work passages of 368 Stability and Controllability of Differential Equations papers in our library. Each one below links back to the study that raised it, so you can read the original claim in context.

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