Qualitative properties for the 1 − D impulsive wave equation: controllability and observability

Abstract

In this paper, we establish some important results for the impulsive wave equation. We begin by proving the existence of a solution. Then, we study the impulse approximate controllability where the control function acts on a subdomain ω and at one instant of time τ ∈ (0, T ). Afterward, we study impulse observability.

Impulse null approximate controllability for heat equation with dynamic boundary conditions

“…They manifest as sudden boundary conditions in switched control inputs or shock waves in compressible flows. Controlling impulsive systems via impulse controls can be challenging (due to the presence of delta functions) requiring specific techniques such as logarithmic convexity, Carleman commutator approach, and optimal impulse control theory, see e.g., [1,2,11].…”

“…This type of control is very weak since it only acts in a subdomain at one instant of time, which makes the problem of controllablity for the heat equation with impulse control very challenging. Unlike the interior controllability for impulsive systems that have been extensively studied in the literature, see for instance [8-10, 24, 25] and the references therein, the problem of controllability with impulse controls has attracted less attention and not as many works are available in this area, we mention [5,19,30].…”

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

“…Unlike the distributed controllability for impulsive systems that has been extensively studied in the literature, see for instance [12,14,21,24,25,27] and the references therein, the problem of controllability with impulse controls has attracted less attention and not as many works are available in this area, we mention [7,18,32,35]. In the later type, the control is a function acting only at one instant of time τ .…”

Impulse null approximate controllability for heat equation with dynamic boundary conditions

“…This type of control is very weak since it only acts in a subdomain at one instant of time, which makes the problem of controllablity for the heat equation with impulse control very challenging. Unlike the interior controllability for impulsive systems that have been extensively studied in the literature, see for instance [4,9,23,24,27] and the references therein, the problem of controllability with impulse controls has attracted less attention and not as many works are available in this area, we mention [6,18,29].…”