Exact Controllability of the Distributed System, Governed by String Equation with Memory

Abstract

We will consider exact controllability of the distributed system governed by the wave equation with memory. It will be proved that this mechanical system can be driven to the null state in finite time, the absolute value of the distributed control function being bounded. In this case the memory kernel is a linear combination of exponentials.

Noncontrollability to Rest of the Two-Dimensional Distributed System Governed by the Integrodifferential Equation

“…Positive results on controllability of a one-dimensional wave equation with memory were obtained in [7]. It was shown that this equation can be driven to rest by applying a bounded distributed control.…”

“…It is a well-known fact that this system is a basis for L 2 (Ω 0 ). We substitute ϕ = ϕ nm in (7). Then using the notation θ nm (t) = θ(t), ϕ nm we obtain…”

Distributed Systems with Persistent Memory

“…In this case, the crochet denotes the duality pairing. Several moment problems, which are practically and historically important are formulated in this more general settings, see [56,58,87]. But, in the cases of our interest H will be a Hilbert space and…”

“…This equivalence of a moment problem and an interpolation problem has been exploited for example in [40,47,48,87]. …”

Null controllability for wave equations with memory

“…In the literature, the controllability problems for evolution equations with memory terms have been studied extensively (See [2,15,17,21,22,25,26,28,29,30,32,33,35] and the references therein). However, in most of the existing works the problem has been addressed analyzing whether the state can be driven to zero at time t = T , without paying attention to the memory term.…”