Logarithmic convexity and impulsive controllability for the 1-D heat equation with dynamic boundary conditions

TL;DR

This paper establishes a logarithmic convexity property and impulsive controllability for a 1-D heat equation with dynamic boundary conditions, providing explicit control bounds and a constructive algorithm validated by numerical tests.

Contribution

It introduces a novel logarithmic convexity estimate and develops an explicit, minimal-norm impulsive control algorithm for the heat equation with dynamic boundaries.

Findings

Proved a logarithmic convexity estimate for the heat equation.

Established impulsive approximate controllability with explicit control bounds.

Designed and validated a constructive minimal-norm impulsive control algorithm.

Abstract

In this paper, we prove a logarithmic convexity that reflects an observability estimate at a single point of time for 1-D heat equation with dynamic boundary conditions. Consequently, we establish the impulse approximate controllability for the impulsive heat equation with dynamic boundary conditions. Moreover, we obtain an explicit upper bound of the cost of impulse control. At the end, we give a constructive algorithm for computing the impulsive control of minimal $L^2$-norm. We also present some numerical tests to validate the theoretical results and show the efficiency of the designed algorithm.