Irena Lasiecka

Long-time behavior of second order evolution equations with nonlinear damping

Introduction Abstract results on global attractors Existence of compact global attractors for evolutions of the second order in time Properties of global attractors for evolutions of the second order in time Semilinear wave equation with a nonlinear dissipation Von Karman evolutions with a nonlinear dissipation Other models from continuum mechanics Bibliography Index

Unified theory for abstract parabolic boundary problems—a semigroup approach

This paper presents and abstract semigroup formulation ofparabolic boundary value problems. Smoothness of solutions, represented by a semigroup formula, is the primary object of discussion. The generality of our approach enables us to treat in a unified manner theregularity of solutions to parabolic equations for a large variety of nonhomogeneous boundary value problems. In particular, the approach presented here allows us to translate known regularity results of the elliptic theory directly into regularity results for the parabolic solutions. On the one hand, our theory recaptures known regularity results of the parabolic solutions over smooth spatial domains. On the other hand, however, our theory also covers the case of conical spatial domains, for which the standard assumption ofC∞-boundaries is violated by suitable application of recent relevant results of elliptic theory for such domains. In the concluding section, an application of our general theory to a boundary control problem with a quadratic performance index is presented.

Exact controllability of semilinear abstract systems with application to waves and plates boundary control problems

This paper studies (global) exact controllability of abstract semilinear equations. Applications include boundary control problems for wave and plate equations on the explicitly identified spaces of exact controllability of the corresponding linear systems.Contents. 1. Motivating examples, corresponding results, literature. 1.1. Motivating examples and corresponding results. 1.2. Literature. 2. Abstract formulation. Statement of main result. Proof. 2.1. Abstract formulation. Exact controllability problem. 2.2. Assumptions and statement of main result. 2.3. Proof of Theorem 2.1. 3. Application: a semilinear wave equation with Dirichlet boundary control. Problem (1.1). 3.1. The caseγ = 1 in Theorem 1.1 for problem (1.1). 3.2. The caseγ = 0 in Theorem 1.1 for problem (1.1). 4. Application: a semilinear Euler—Bernoulli equation with boundary controls. Problem (1.14). 4.1. Verification of assumption (C.1): exact controllability of the linear system. 4.2. Abstract setting for problem (1.14). 4.3. Verification of assumptions (A.1)–(A.5). 4.4. Verification of assumption (C.2). 5. Proof of Theorem 1.2 and of Remark 1.2. Appendix A: Proof of Theorem 3.1. Appendix B: Proof of (4.9) and of (4.10b). References.

Algebraic Riccati equations with non-smoothing observation arising in hyperbolic and Euler-Bernoulli boundary control problems

SummaryThis paper considers the optimal quadratic cost problem (regulator problem) for a class of abstract differential equations with unbounded operators which, under the same unified framework, model in particular «concrete» boundary control problems for partial differential equations defined on a bounded open domain of any dimension, including: second order hyperbolic scalar equations with control in the Dirichlet or in the Neumann boundary conditions; first order hyperbolic systems with boundary control; and Euler-Bernoulli (plate) equations with (for instance) control(s) in the Dirichlet and/or Neumann boundary conditions. The observation operator in the quadratic cost functional is assumed to be non-smoothing (in particular, it may be the identity operator), a case which introduces technical difficulties due to the low regularity of the solutions. The paper studies existence and uniqueness of the resulting algebraic (operator) Riccati equation, as well as the relationship between exact controllability and the property that the Riccati operator be an isomorphism, a distinctive feature of the dynamics in question (emphatically not true for, say, parabolic boundary control problems). This isomorphism allows one to introduce a «dual» Riccati equation, corresponding to a «dual» optimal control problem. Properties between the original and the «dual» problem are also investigated.

Stabilization of wave and plate-like equations with nonlinear dissipation on the boundary

Abstract In this paper we consider second order in time equations of Petrovsky type (wave and plate-like) with nonlinear dissipative boundary conditions. Under certain geometric conditions imposed on the domain Ω the results on asymptotic behavior of the solutions are established.

ON THE ATTRACTOR FOR A SEMILINEAR WAVE EQUATION WITH CRITICAL EXPONENT AND NONLINEAR BOUNDARY DISSIPATION

ABSTRACT Long time behavior of a semilinear wave equation with nonlinear boundary dissipation and critical exponent is considered. It is shown that weak solutions generated by the wave dynamics converge asymptotically to a global and compact attractor. In addition, regularity and structure of the attractor are discussed in the paper. While this type of results are known for wave dynamics with interior dissipation this is, to our best knowledge, first result pertaining to boundary and nonlinear dissipation in the context of global attractors and their properties.

A cosine operator approach to modelingL2(0,T; L2 (Γ))--Boundary input hyperbolic equations

This paper investigates the regularity properties of the solution of a second-order hyperbolic equation defined over a bounded domain Ω with boundary Γ, under the action of a boundary forcing term inL2(0,T; L2(Γ)). Both Dirichlet and Neumann nonhomogeneous cases are considered. A functional analytic model based on cosine operator functions is presented, which provides an input-solution formula to be interpreted in appropriate topologies. With the help of this model, it is shown, for example, that the solution of the nonhomogeneous Dirichlet problem is inL2(0,T; L2(Ω)), when Ω is either a parallelepiped or a sphere, while the solution of the nonhomogeneous Neumann problem is inL2(0,T; H3/4-e(Ω)) when Ω is a parallelepiped and inL2(0,T; H2/3(Ω) when Ω is a sphere. The Dirichlet case for general domains is studied by means of pseudodifferential operator techniques.

Global uniqueness, observability and stabilization of nonconservative Schrödinger equations via pointwise Carleman estimates. Part II: L 2(Ω)-estimates

We consider a general non-conservative Schrödinger equation defined on an open bounded domain Ω in , with C 2-boundary subject to (Dirichlet and, as a main focus, to) Neumann boundary conditions on the entire boundary Γ. Here, Γ0 and Γ1 are the unobserved (or uncontrolled) and observed (or controlled) parts of the boundary, respectively, both being relatively open in Γ. The Schrödinger equation includes energy-level (H 1(Ω)-level) terms, which accordingly may be viewed as unbounded perturbations. The first goal of the paper is to provide Carleman-type inequalities at the H 1-level, which do not contain lower-order terms; this is a distinguishing feature over most of the literature. This goal is accomplished in a few steps: the paper obtains first pointwise Carleman estimates for C 2-solutions; and next, it turns these pointwise estimates into integral-type Carleman estimates with no lower-order terms, originally for H 2-solutions, and ultimately for H 1-solutions. The passage from H 2- to H 1-solutions is readily accomplished in the case of Dirichlet B.C., but it requires a delicate regularization argument in the case of Neumann B.C. This is so since finite energy solutions are known to have L 2-normal traces in the case of Dirichlet B.C., but by contrast do not produce H 1-traces in the case of Neumann B.C. From Carleman-type inequalities with no lower-order terms, one then obtains the sought-after benefits. These consist of deducing, in one shot, as a part of the same flow of arguments, two important implications: (i) global uniqueness results for H 1-solutions satisfying over-determined boundary conditions, and—above all—(ii) continuous observability (or stabilization) inequalities with an explicit constant. The more demanding purely Neumann boundary conditions requires the same geometrical conditions on the triple {Ω,Γ0,Γ1} that arise in the corresponding problems for second-order hyperbolic equations. The most general result, with weakest geometrical conditions, is, in fact, deferred to Section 9. Sections 1 through 8 provide the main body of our treatment with one vector field under a preliminary working geometrical condition, which is then removed in Section 9, by use of two suitable vector fields. The second and final goal of this paper is to shift the Carleman estimates (Hence, the continuous observability/stabilization inequalities) by one unit downward to the lower L 2(Ω)-level. This is accomplished in Section 10 by means of pseudo-differential analysis, and accordingly, it contains lower-order terms. Applications of these L 2(Ω)-Carleman estimate includes a new uniform stabilization of the conservative Schrödinger equation in the state space L 2(Ω), by an attractive boundary feedback.

Differential Riccati equation for the active control of a problem in structural acoustics

In this paper, we provide results concerning the optimal feedback control of a system of partial differential equations which arises within the context of modeling a particular fluid/structure interaction seen in structural acoustics, this application being the primary motivation for our work. This system consists of two coupled PDEs exhibiting hyperbolic and parabolic characteristics, respectively, with the control action being modeled by a highly unbounded operator. We rigorously justify an optimal control theory for this class of problems and further characterize the optimal control through a suitable Riccati equation. This is achieved in part by exploiting recent techniques in the area of optimization of analytic systems with unbounded inputs, along with a local microanalysis of the hyperbolic part of the dynamics, an analysis which considers the propagation of singularities and optimal trace behavior of the solutions.

Ritz–Galerkin Approximation of the Time Optimal Boundary Control Problem for Parabolic Systems with Dirichlet Boundary Conditions

This paper deals with Ritz–Galerkin approximations of the following two problems: (i) boundary-value problems with