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Open research questions in Stability and Controllability of Differential Equations

103 gap statements mined from Stability and Controllability of Differential Equations papers in our 4.5M-paper local library, which holds 1,038 papers on the topic — drawn mostly from each paper's own stated research gap, future-work, challenge and limitation notes, and its abstract. The ones listed below are a selection still marked open; each names the study that raised it, with a DOI link where the paper has one.

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What the literature leaves open

  • Future research should address the case where the control constraints are not convex. Future research should address the application of the paper's results to specific fields, such as engineering and economics. Future research should investigate the use of other methods, such as numerical methods, to solve optimal control problems.

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

  • The class of memory kernel considered in this paper was not analyzed in previous studies. There was a lack of explicit estimates of the connection between the kernel function's decay constant and the energy's decay constant.

    Energy decay for evolution equations with glassy type memory · 2026 · DOI

  • The paper identifies the need to study the stability of the one-dimensional linear truncated thermoelastic Bresse system. The paper mentions the lack of exponential stability results for the system.

    On a Shear Thermoelastic Bresse System: Well Posedness and Stability · 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 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

  • 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

  • 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

  • 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

  • Future research can study the null controllability of degenerate parabolic equations with internal control applied on a non-measurable subset. Future research can study the behavior of degenerate parabolic equations in various fields. Future research can design control systems for degenerate parabolic equations.

    Null Controllability for Degenerate Parabolic Equations with Internal Control Applied on a Measurable Subset · 2026

  • Future research could focus on providing a direct proof of the result using functional calculus. The authors suggest that the conclusion of Theorem 1.2 holds with β = α, maybe under more restrictive assumptions.

    Periodic solutions for weakly damped systems · 2026

  • 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 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 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

  • Future research should focus on extending the method to other types of systems, including nonlinear systems. Future research should investigate the application of the method to various practical problems. Future research should explore the use of other control methods, including model predictive control.

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

  • The lack of a framework for stability analysis of non-autonomous infinite-dimensional systems with superlinear terms. The need for Lyapunov analytical tools for the LiISS analysis of such systems.

    Local integral input-to-state stability for non-autonomous infinite-dimensional systems · 2026 · DOI

  • We have considered an approximating lattice dynamical system for a non-autonomous parabolic equation which describes the behavior of concentrated suspensions. The existence of a compact uniform attractor for the family of semi-processes generated by solutions of the lattice system in the phase space l1,w has been established.

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

  • The paper assumes that the control constraints are given by a closed, convex, and bounded subset of L^2(ω). The paper does not address the case where the control constraints are not convex.

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

  • 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

  • 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

  • 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 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 identifies a gap in the existing literature on the long-time dynamics of semilinear wave equations with dissipative boundary conditions. The paper also identifies a gap in the understanding of the set of equilibria for such equations.

    Non-convergence and convergence of bounded solutions to semilinear wave equation with dissipative boundary condition · 2026 · DOI

Most-cited papers in Stability and Controllability of Differential Equations

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Open questions raised by papers listed on this page include: (1) Future research should address the case where the control constraints are not convex (“Infinite Horizon Control Problems for Semilinear Parabolic Equations with…”, 2026); (2) There was a lack of explicit estimates of the connection between the kernel function's decay constant and the energy's decay constant (“Energy decay for evolution equations with glassy type memory”, 2026); (3) The paper mentions the lack of exponential stability results for the system (“On a Shear Thermoelastic Bresse System: Well Posedness and Stability”, 2026); (4) The gap in the existing literature is the lack of a suitable method for solving the quadratic regulator problem (“The Regulator Problem for the Wave Equation with High Internal Damping…”, 2026); (5) The lack of a proof for null controllability of the 1D heat equation with an interior inverse square potential in arbitrary small time (“Null controllability of the 1D heat equation with interior inverse square…”, 2026). 103 gap statements have been mined from Stability and Controllability of Differential Equations papers in our 4.5M-paper local library, which holds 1,038 papers on the topic; the gaps come from whichever of those papers state one. They are mostly the research gaps the authors state and the papers' abstracts, plus future-work, limitations and challenges passages. The ones listed below are a selection still marked open; each names the study that raised it, with a DOI link where the paper has one, so you can read the original claim in context.

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