262,424 research outputs found
A trace regularity result for thermoelastic equations with application to optimal boundary control
No full textAbstractWe consider a mixed problem for a Kirchoff thermoelastic plate model with clamped boundary conditions. We establish a sharp regularity result for the outer normal derivative of the thermal velocity on the boundary. The proof, based upon interpolation techniques, benefits from the exceptional regularity of traces of solutions to the elastic Kirchoff equation. This result, which complements recent results obtained by the second and third authors, is critical in the study of optimal control problems associated with the thermoelastic system when subject to thermal boundary control. Indeed, the present regularity estimate can be interpreted as a suitable control-theoretic property of the corresponding abstract dynamics, which is crucial to guarantee well-posedness for the associated differential Riccati equations
A theory of the infinite horizon LQ-problem for composite systems of PDEs with boundary control
No full textWe study the infinite horizon linear-quadratic (LQ) problem and the associated algebraic Riccati equations for systems with unbounded control actions. The operator-theoretic context is motivated by composite systems of partial differential equations (PDEs) with boundary or point control. Specific focus is placed on systems of coupled hyperbolic/parabolic PDE with an overall predominant hyperbolic character, such as, e.g., some models for thermoelastic or fluid-structure interactions. While unbounded control actions lead to Riccati equations with unbounded (operator) coefficients, unlike in the parabolic case solvability of these equations becomes a major issue, owing to the lack of sufficient regularity of the solutions to the composite dynamics. In the present case, even the more general theory appealing to estimates of the singularity displayed by the kernel which occurs in the integral representation of the solution to the control system fails. A novel framework which embodies possible hyperbolic components of the dynamics was introduced by the authors in 2005, and a full theory of the LQ-problem on a finite time horizon has been developed. The present paper provides the infinite time horizon theory, culminating in well-posedness of the corresponding (algebraic) Riccati equations. New technical challenges are encountered and new tools are needed, especially in order to pinpoint the differentiability of the optimal solution. The theory is illustrated by means of a boundary control problem arising in thermoelasticity. © 2013 Society for Industrial and Applied Mathematics
Optimal boundary control and Riccati theory for abstract dynamics motivated by hybrid systems of PDEs
No full textWe study the quadratic optimal control problem over a finite time horizon for a class of abstract systems with non analytic underlying semigroup etA and unbounded control operator B. It is assumed that a suitable decomposition of the operator B*etA* is valid, where only one component satisfies a \u27singular estimate\u27, whereas for the other component specific regularity properties hold. Under these conditions, we prove well posedness of the associated differential Riccati equation, and in particular that the gain operator is bounded on a dense set. In spite of the unifying abstract framework used, the prime motivation (and application) of the resulting theory of linear-quadratic problems comes from optimal boundary control of a thermoelastic system with clamped boundary conditions. The non-trivial trace regularity estimate showing that this PDE mixed problem fits into the distinct class of models under examination-for which we have developed the present, novel optimal control theory-is established, as well
Uniform energy decay rates of hyperbolic equations with nonlinear boundary and interior dissipation
No full textWe consider the problem of uniform stabilization of nonlinear hyperbolic equations, epitomized by the following three canonical dynamics: (1) the wave equation in the natural state space L2(ω) x H-1 (ω), under nonlinear (and non-local) boundary dissipation in the Dirichlet B.C., as well as nonlinear internal damping; (2) a corresponding Kirchhoff equation in the natural state space [H2(ω) H01(ω)] x H10(ω), under nonlinear boundary dissipation in the moment B.C. as well as nonlinear internal damping; (3) the system of dynamic elasticity corresponding to (1). All three dynamics possess a strong, hard-to-show boundary → boundary regularity property, which was proved, also by invoking a micro-local argument, in Lasiecka and Triggiani (2004, 2008). This is by no means a general property of hyperbolic or hyperbolic-like dynamics (Lasiecka and Triggiani, 2003, 2008). The present paper, as a continuation of Lasiecka and Triggiani (2008), seeks to take advantage of this strong regularity property in the case of those PDE dynamics where it holds true. Thus, under the above boundary → boundary regularity, as well as exact controllability of the corresponding linear model, uniform stabilization of nonlinear models is obtained under minimal nonlinear assumptions, provided that a corresponding unique continuation property holds true. The treatment of the present paper is cast in the abstract setting (Lasiecka, 1989, 2001; Lasiecka and Triggiani, 2000, Ch. 7, 2003, 2008), which is proper for these hyperbolic dynamics and recovers the results of Lasiecka and Triggiani (2003, 2008) in the absence of the nonlinear interior damping, in particular in the linear case
A note on the Moore–Gibson–Thompson equation with memory of type II
No full textWe consider the Moore–Gibson–Thompson equation with memory of type II ∂tttu(t)+α∂ttu(t)+βA∂tu(t)+γAu(t)-∫0tg(t-s)A∂tu(s)ds=0where A is a strictly positive selfadjoint linear operator (bounded or unbounded) and α, β, γ\u3e 0 satisfy the relation γ≤ αβ. First, we prove well-posedness of finite energy solutions, without requiring any restriction on the total mass ϱ of g. This extends previous results in the literature, where such a restriction was imposed. Second, we address an open question within the context of longtime behavior of solutions. We show that an “overdamping” in the memory term can destabilize the originally stable dynamics. In fact, it is always possible to find memory kernels g, complying with the usual mass restriction ϱ\u3c β, such that the equation admits solutions with energy growing exponentially fast, even in the regime γ\u3c αβ where the corresponding model without memory is exponentially stable. In particular, this provides an answer to a question recently raised in the literature
Regularity theory of hyperbolic equations with non-homogeneous Neumann boundary conditions. II. General boundary data
No full textAbstractThis paper studies the regularity of solutions of general, mixed, second-order, time-dependent, hyperbolic problems of Neumann type. In a previous paper [I. Lasiecka and R. Triggiani, Ann. Mat. Pura Appl. (IV) CLVII (1990), 285–367] using pseudo-differential calculus, we have provided sharp regularity results of the solutions and their traces, when the non-homogeneous data are in L2. Now, we complement this study by providing a regularity when the non-homogeneous data are more regular than, as well as less regular than, L2. In contrast with our previous paper, a functional analytic approach based on the L2-results of our previous paper is used throughout
Exact controllability of the Euler-Bernoulli equation with boundary controls for displacement and moment
No full textAbstractWe consider the Euler-Bernoulli problem (1.1) in the solution w(t, ·) with boundary controls g1 and g2 acting in the Dirichlet traces for w and Δw, respectively. We show two main exact controllability results both in the spaces of maximal regularity of the solution at t = T (see I. Lasiecka and R. Triggiani, Regularity theory for a class of Euler Bernoulli equations: a cosine operator approach, Boll. Unione Mat. Ital. (7), 3-B (1989).) and both without geometrical conditions on the open bounded domain Ω (except for smoothness of its boundary Γ): one with g1 ϵ L2(0, T; L2(Γ)) and g2 ϵ [H1(0, T; L2(Γ))]′ for any T > 0 arbitrarily short; and one with g1 ϵ H01(0, T; L2(Γ)) and g2 ϵ L2(0, T; L2(Γ)) for all T > 0 sufficiently large. An interpolation result between these two cases is also presented. A direct approach is given based on two main steps. First, by means of an operator model (I. Lasiecka and R. Triggiani, Regularity theory for a class of Euler Bernoulli equations: a cosine operator approach, Boll. Unione Mat. Ital. (7), 3-B (1989).) for problem (1.1) and a functional analytic approach, the question of exact controllability is shown to be equivalent to an a-priori inequality for the corresponding homogeneous problem. Next, this a-priori inequality is proved to hold true by means of multiplier techniques. These are inspired by recent progress on the maximal regularity, exact controllability and uniform stabilization questions for second order hyperbolic equations
Uniform stabilization of the quasi-linear Kirchhoff wave equation with a nonlinear boundary feedback
No full textAn n-dimensional quasi-linear wave equation defined on bounded domain Ω with Neumann boundary conditions imposed on the boundary Γ and with a nonlinear boundary feedback acting on a portion of the boundary Γ1⊂ ⊂ Γ is considered. Global existence, uniqueness and uniform decay rates are established for the model, under the assumption that the H1(Ω) × L2(Ω) norms of the initial data are sufficiently small. The result presented in this paper extends these obtained recently in Lasiecka and Ong (1999), where the Dirichlet boundary conditions are imposed on the uncontrolled portion of the boundary Γ0 = Γ \ Γ1̄, and the two portions of the boundary are assumed disjoint, i.e. Γ1̄∩Γ0 = θ. The goal of this paper is to remove this restriction. This is achieved by considering the pure Neumann problem subject to convexity assumption imposed on Γ0
Dirichlet boundary stabilization of the wave equation with damping feedback of finite range
No full textAbstractA “closed loop” system consisting of the wave equation with a feedback acting in the Dirichlet boundary condition in the form of a nonlocal, one-dimensional range operator, defined on the “velocity” vector (damping) is considered. Beside the well-posedness question (generation of a feedback C0-semigroup), the boundary feedback stabilization problem is studied: while asymptotic decay to zero in the uniform operator norm can never occur for the considered class of closed loop systems, however checkable sufficient conditions on the vectors in the boundary conditions are provided that guarantee asymptotic decay to zero in the strong norm of appropriate Sobolev spaces. These conditions include, but are not limited to, the standard case of dissipativity of the feedback system. The starting point of this approach is a functional analytic model for the study of nonsmooth boundary input hyperbolic equations recently developed (I. Lasiecka and R. Triggiani, Appl. Math. Optim. 7 (1) (1981), 35–93)
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