Control Theory for Partial Differential Equations

Abstract

Originally published in 2000, this is the first volume of a comprehensive two-volume treatment of quadratic optimal control theory for partial differential equations over a finite or infinite time horizon, and related differential (integral) and algebraic Riccati equations. Both continuous theory and numerical approximation theory are included. The authors use an abstract space, operator theoretic approach, which is based on semigroups methods, and which is unifying across a few basic classes of evolution. The various abstract frameworks are motivated by, and ultimately directed to, partial differential equations with boundary/point control. Volume 1 includes the abstract parabolic theory for the finite and infinite cases and corresponding PDE illustrations as well as various abstract hyperbolic settings in the finite case. It presents numerous fascinating results. These volumes will appeal to graduate students and researchers in pure and applied mathematics and theoretical engineering with an interest in optimal control problems.

A Heat–Viscoelastic Structure Interaction Model with Neumann and Dirichlet Boundary Control at the Interface: Optimal Regularity, Control Theoretic Implications

“…Regarding our problem at hands, one key ingredient is precisely discovering the optimal value γ = 1 2 in Corollaries 4.3 and 4.4. Thus, we conclude that the analysis of the boundary (interface) fluid-structure interaction model (1.1a-f)-abstractly, of the pair {A, B} in (1.12)-carried out in the present paper and its predecessor [27], has led to three critical properties: (1) that A −γ B ∈ L(L 2 ( s ); H), γ = 1 2 , as stated in [24] applies, with control space U = L 2 ( s ), as an illustration with γ = 1 2 , and moreover in the special case (which simplifies the theory on the infinite time horizon problems), where the free dynamics is, moreover, exponentially stable. More specifically, such theory includes:…”

“…214-220], next Balakrishnan employed that model to study the Linear Quadratic Regulator Problem for a parabolic equation with Dirichlet boundary control in [7]. This topic was revisited and studied also in the case of a variety of different boundary control problems under an abstract setting in [24] and references therein. Regarding our problem at hands, one key ingredient is precisely discovering the optimal value γ = 1 2 in Corollaries 4.3 and 4.4.…”

Local Exact Controllability of Micropolar Fluids

“…(11) The controllability of linear and nonlinear parabolic partial differential equations and systems has been the subject of many works these last years; see for instance [11], [13], [20] and the references therein. In the context of systems of the Navier-Stokes kind, it has motivated a lot of work; see for instance [3], [12], [13], [15], [17], [8], [9], [16].…”

“…We will need some notation. Thus, let us set Lz = z t − ∆z + (y · ∇)z + (z · ∇)y, and M ρ = ρ t − ∆ρ − ∇(∇ · ρ) + y · ∇ρ (20) and let us introduce the space…”

A Semigroup Approach to Boundary Feedback Systems

“…Many authors have used semigroup theory to study linear initial boundary value or boundary control problems (see, e.g., the monographs by Lasiecka-Triggiani [LT00], the papers by Desch et al [DLS85], [DMS01], the - * approach in Heijmans [Hei87], or the approach via characteristic matrices by Kaashoek and Verduyn Lunel [KVL92].) In this paper we propose an approach in which we convert the given boundary value problem on some domain Ω ⊂ R n into an (inhomogeneous) Abstract Cauchy Problem Ẋ(t) = AX(t) + F (t), t ≥ 0, X(0) = X 0 (iACP)…”