Riesz Systems and Controllability of Heat Equations with Memory

Abstract

The main result we derive is the proof that a particular set of functions related to the controllability of the heat equation with memory and finite signal speed, with suitable kernel, is a Riesz system. Riesz systems are important tools in applied mathematics, for example for the solution of inverse problems. In this paper we shows that the Riesz system we identify can be used to give a constructive method for the computation of the control steering a given initial condition to a prescribed target. (2000). Primary 44A60; Secondary 93B05. Mathematics Subject Classification

Controllability of a Nonhomogeneous String and Ring under Time Dependent Tension

“…In other words, we study boundary controllability of this system. The paper represents a continuation of the previous papers by first two authors [1], [2], and [4] and the papers [24]- [26] of the third author. In the papers [1], [4], the density of the string and ring correspondingly is constant but the tension is supposed to be the sum of a constant the method of moments to equations with time dependent coefficients.…”

“…Controllability is proved in these papers, but they do not suggest a constructive method of finding a control which steers a given initial condition to hit a prescribed target. (To be fair, a constructive method of finding control is mostly studied in one-dimensional case, see [1]- [10], [12], [17], [24]- [27], and the literature cited therein.) The paper [24] (as well as [25]- [26]) is especially important to us.…”

“…Controllability problem is reduced to a moment problem with respect to a system of functions. Techniques developed in [24,26] allow proving that the aforementioned system is a Riesz system in spite of the fact that an explicit expression for the elements of the basis is not available. That allows solving controllability problem in a constructive way.…”

“…The main goal of this paper, concerning the wave equation with time varying coefficient P (t) is to remove the restriction on the tension (used in the papers [1,2,3,4]) with the help of an idea suggested in [24]- [26].…”

Exact boundary controllability of coupled hyperbolic equations

“…The latter is defined to be a Riesz basis in the closure of the linear span of the family. Recent investigations into new classes of distributed systems such as hybrid systems, systems with memory, and damped systems as well as problems of simultaneous control have raised a number of new and difficult problems in the theory of exponential families (see, e.g., Hansen and Zuazua, 1995;Avdonin and Moran, 2001b;Pandolfi, 2009;Avdonin and Pandolfi, 2011) including the theory of vector-valued exponentials (Avdonin and Ivanov, 1995) and many other interesting topics.…”