Logarithmic convexity and impulsive controllability for the one-dimensional heat equation with dynamic boundary conditions

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