Quantitative rapid stabilization for parabolic equations via the linear quadratic theory

Technical Summary: Quantitative Rapid Stabilization for Parabolic Equations via Linear Quadratic Theory

Problem Statement
This paper addresses the problem of quantitative rapid stabilization for parabolic equations, specifically focusing on non-self-adjoint systems in multidimensional domains. While the qualitative rapid stabilization of such systems (achieving arbitrary decay rates) is known, the dependence of the stabilization cost on the prescribed decay rate $\lambda$ remains a critical open issue. The authors aim to construct feedback laws that stabilize the system with a decay rate $\lambda$ while providing explicit estimates on the stabilization cost $M_\lambda$. A primary challenge is that existing quantitative methods (e.g., backstepping, spectral inequalities) often rely on self-adjoint structures or specific kernel bounds that do not readily extend to non-self-adjoint parabolic equations with convection terms.

Methodology
The authors develop a framework based on Linear Quadratic (LQ) optimal control theory combined with quantitative observability inequalities. The methodology proceeds through the following steps:

1. Weighted LQ Formulation: Instead of standard LQ problems, the authors introduce a weighted cost functional:
   $$J(y, u; y_0) = \frac{1}{2} \int_0^\infty e^{2\lambda t} \left( \|(-\Delta)^\alpha y(t)\|_{L^2}^2 + \|u(t)\|_{L^2}^2 \right) dt$$
   where $\alpha \in [0, 1]$. By selecting $\alpha = 1/2$, they ensure the cost functional captures the $L^2$ norm of the state, which is essential for deriving explicit lower bounds.

2. Transformation and Equivalence: Through the transformation $z(t) = e^{\lambda t}y(t)$ and $v(t) = e^{\lambda t}u(t)$, the rapid stabilization problem for the original system is converted into a standard exponential stabilization problem for a transformed system with a shifted operator.

3. Quantitative Bounds via Duality:

   • Upper Bound: The authors utilize a quantitative observability inequality for the adjoint system. By the classical duality principle between observability and null controllability, they construct a feasible control with an explicit cost estimate. This provides an upper bound for the optimal value of the LQ problem.
   • Lower Bound: Using energy estimates and the specific structure of the cost functional (with $\alpha=1/2$), they derive a coercive lower bound for the optimal cost.

4. Riccati Equation Solution: The existence of an optimal pair implies the existence of a solution $P_\lambda$ to an operator Riccati equation. The quantitative bounds on the optimal cost translate directly into bounds on the operator norm $\|P_\lambda\|$, which determines the magnitude of the feedback control $u = -B^*P_\lambda y$.

Key Contributions and Results

1. Quantitative Rapid Stabilization for Non-Self-Adjoint Systems (Theorem 1.1):
   The paper establishes the first quantitative rapid stabilization result for multidimensional non-self-adjoint parabolic equations (e.g., with convection terms $b \cdot \nabla y$).

   • Result: For any decay rate $\lambda > 1$, there exists a stationary feedback law $K_\lambda$ such that the closed-loop system satisfies:
     $$\|u(t)\| + \|y(t)\| \leq e^{\tilde{C}\sqrt{\lambda}} e^{-\lambda t} \|y_0\|$$
   • Significance: The stabilization cost grows at most like $e^{C\sqrt{\lambda}}$, which is crucial for ensuring robustness against perturbations and feasibility over short time intervals.

2. Abstract Equivalence Theorem (Theorem 1.5):
   The authors prove a general equivalence between quantitative rapid stabilization and quantitative observability for abstract linear systems.

   • Result: A system is $e^{O(\lambda^{\frac{p}{p+1}})}$-rapidly stabilizable if and only if it is $e^{O(T^{-p})}$-observable.
   • Significance: This bridges the gap between control cost estimates and observability inequalities, clarifying the parameter dependencies that were previously unclear in the literature.

3. Application to Navier–Stokes Equations (Theorem 1.7):
   The framework is extended to the nonlinear Navier–Stokes equations around non-trivial steady states.

   • Result: By linearizing around a steady state $y_e$ and utilizing the quantitative observability of the associated Oseen system, the authors construct a feedback law that achieves local quantitative rapid stabilization.
   • Condition: The initial perturbation must be sufficiently small, specifically $\|y_0 - y_e\| \leq e^{-C\sqrt{\lambda}}$.

4. Finite-Dimensional Extension for Self-Adjoint Cases (Theorem 5.2):
   For the heat equation (self-adjoint case), the authors demonstrate that the quantitative LQ framework can be adapted to finite-dimensional feedback laws.

   • Method: They employ the frequency-Lyapunov method, separating low and high frequencies. The low-frequency dynamics are controlled via a finite-dimensional Riccati equation, while high frequencies are damped by the natural dissipation of the operator.
   • Result: This recovers and refines previous quantitative results (e.g., Xiang [43]) using the LQ perspective.

Significance and Claims
The paper claims that its approach offers a distinct alternative to explicit constructive methods like backstepping or spectral inequalities.

• Indirect Derivation: Unlike methods that rely on explicit kernel bounds or spectral decompositions, this approach derives quantitative estimates indirectly from global observability inequalities via the LQ framework. This makes the method applicable to non-self-adjoint systems where spectral properties are not well-behaved.
• Robustness: The explicit dependence of the cost on $\lambda$ ($e^{C\sqrt{\lambda}}$) provides necessary information for assessing the robustness of the closed-loop system against bounded perturbations.
• Generality: The abstract equivalence theorem (Theorem 1.5) suggests that the quantitative rapid stabilization problem can be solved for any system where a quantitative observability inequality is available, including time-dependent potentials and boundary control problems (noted as future work).

The authors note that while the finite-dimensional extension works for self-adjoint cases, establishing a finite-dimensional feedback law for the general non-self-adjoint parabolic equation remains an open problem, as it would require spectral estimates that are currently unavailable for such systems.