Controllability of Partial Differential Equations Governed by Multiplicative Controls

Abstract

Setting of Controllability ProblemIn this chapter we discuss multiplicative controllability of hyperbolic equations along the approach due to J.M. Ball, J.E. Marsden, and M. Slemrod. We describe the main ideas of this approach and the principal results relevant to the global reachability properties of these equations. For the complete account of all details we refer the reader to the original paper [8].Chapter 9 is organized as follows. In section 9.1 a class of abstract evolution equations, governed in a Banach space X by multiplicative time-dependent controls, is introduced. However, it turns out that the local controllability of these equations (i.e., anywhere within some neighborhood of a drifting trajectory) is out of question when X is infinite dimensional. In section 9.2 we distinguish a more specific subclass within the aforementioned class of equations -the abstract hyperbolic equations, and also remind the reader the concept of Riesz bases. In section 9.3 we show how the global approximate reachability of the latter equations can be established under a number of additional assumptions. In particular, it is assumed that the eigenvalues of the linear part of the equation at hand must all be integers that are multiplied by the same number and that all modes in the initial datum must be present (see Assumptions 1-4 below). In section 9.4 this abstract theory is applied to the 1-D wave and rod equations. In section 9.5 we discuss two results on local controllability for these equations due to K. Beauchard.

The well-posedness of a swimming model in the 3-D incompressible fluid governed by the nonstationary Stokes equation

“…In this paper we deal with the swimming phenomenon in the framework of non-stationary PDEs along the immersed body approach summarized by Khapalov (2010), who was also inspired by the ideas of the above-cited Peskin's method, introduced a 2-D model for "small" flexible swimmers assuming that their bodies are identified with the fluid occupying their shapes (Khapalov, 2005). This approach views such a swimmer as an already discretized, aforementioned immersed boundary supported on the respective grid cells (see, e.g., Figs.…”

“…We established the well-posedness of this model up to the contact either between the swimmer's body parts or with the boundary of the space domain. The need for such a type of models was motivated by the intention to investigate controllability properties of swimming phenomenon (see Khapalov, 2010). Our goal in this paper is to introduce a possible 3-D extension of this model and to investigate its well-posedness.…”

Source localization and sensor placement in environmental monitoring

“…Then, exactly as in Sections 3 and 4 (see, e.g., (19)), we have the following system of equations to find x 0 : Let us note here that the approximate controllability of the system (39) in L 2 (0, 1) was studied by Khapalov (1994(a);1998;2001) (see the work of Khapalov (2010) and the references therein for the case of multiplicative controls), with an algorithm on how one can construct suitable mobile supports S j (t), t ∈ (0, T ), j = 1, . .…”

Addendum to “The well-posedness of a swimming model in the 3-D incompressible fluid governed by the nonstationary Stokes equation”

“…Namely, the forces described in this term are A. Khapalov intended to be internal relative to the swimmer. However, the form of (3) satisfies this condition in terms of forces applied to z i (t) as the centers of mass of S i (z i (t)) only if the sets S i (0)'s have identical measure (for details, see Khapalov, 2010;2013).…”

“…The added extra coefficient (mes (S i (0)) −1 at each characteristic function ξ i (x, t) ensures that all the forces of the swimmer are internal (see also the respective discussion in the end of Chapter 11 of the book by Khapalov (2010) for the 2-D case). In the case of sets S i (z i (t)) of identical measure, the aforementioned (identical) extra coefficients can be viewed as included in k i 's and v i 's.…”