Optimal bounds for the cost of fast controls of a KdV system

TL;DR

The paper tackles the problem of determining optimal bounds on the cost of fast controls for linearized and locally nonlinear KdV systems under a right Neumann boundary control, focusing on noncritical lengths $L \notin \mathcal{N}$. It circumvents difficulties from the non-self-adjoint generator by relating the original system to a skew-adjoint KdV problem and applying the moment method to obtain sharp upper and lower bounds on the control cost, with an exponential blow-up rate of $e^{\frac{c}{T^{1/2}}}$. For the linearized system, the bounds are established for small times $0<T< T_0$ and extend to the nonlinear system for small initial data, demonstrating that the minimal control cost scales like $e^{\frac{c}{T^{1/2}}}$ times the initial energy. The results provide precise, implementable cost estimates and support finite-time stabilization arguments for KdV-type controls.

Abstract

We study the cost of fast controls for a linearized KdV system and a nonlinear KdV system locally, using right Neumann boundary control for non-critical lengths. Since the operator associated with the linearized system is neither self-adjoint nor skew-adjoint, its (known) spectral properties are not directly amenable to the moment method, leaving optimal cost bounds an open problem. We address this difficulty by shifting attention to a related KdV system and deriving the optimal bounds from the new one.

Theorems & Definitions (13)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• Proposition 2.1
• Lemma 2.2
• Lemma 2.3
• Remark 3.1
• Remark 3.2
• Remark 3.3
• Lemma A.1