A singular control approach to highly damped second-order abstract equations and applications

In this paper we restudy, by a radically different approach, the optimal quadratic cost problem for an abstract dynamics, which models a special class of second-order partial differential equations subject to high internal damping and acted upon by boundary control. A theory for this problem was recently derived in [LLP] and [T1] (see also [T2]) by a change of variable method and by a direct approach, respectively. Unlike [LLP] and [T1], the approach of the present paper is based on singular control theory, combined with regularity theory of the optimal pair from [T1]. This way, not only do we rederive the basic control-theoretic results of [LLP] and [T1]—the (first) synthesis of the optimal pair, and the (first) nonstandard algebraic Riccati equation for the (unique) Riccati operator which enters into the gain operator of the synthesis—but in addition, this method also yields new results—a second form of the synthesis of the optimal pair, and a second (still nonstandard) algebraic Riccati equation for the (still unique) Riccati operator of the synthesis. These results, which show new pathologies in the solution of the problem, are new even in the finite-dimensional case. This research was made possible by NATO Collaborative Research Grant SA.5-2-05 (CRG.940161) 274/94/JARC-501, whose support is gratefully acknowledged. The research of I. Lasiecka and R. Triggiani was supported also by the National Science Foundation under Grant NSF-DMS-92-04338. The research of L. Pandolfi was written with the programs of GNAFA-CNR. The main results of the present paper were announced in [LPT].     

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.Research partially supported by National Science Foundation Grant DMS #9504822 and Army Research Office Grant #35170-MA.     

A trace regularity result for thermoelastic equations with application to optimal boundary control

We 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.     

Galerkin projections for some wave and plate boundary control problems

In this paper we describe a method for projecting the solutions of a wide class of wave and plate control problems with one active boundary control. These projections are reduced to solving completely controllable finite dimensional linear control problems with a scalar control where the control is introduced through a fixed function defined on the boundary of the wave or plate.     

On exact boundary controllability for linearly coupled wave equations

In this paper we study exact boundary controllability for a system of two linear wave equations coupled by lower order terms. We obtain square integrable control of Neuman type for initial state with finite energy, in nonsmooth domains of the plane.     

Min-max game theory and algebraic Riccati equations for boundary control problems with continuous input-solution map. Part II: The general case

We consider the abstract dynamical framework of [LT3, class (H.2)] which models a variety of mixed partial differential equation (PDE) problems in a smooth bounded domain ⁿ , arbitraryn, with boundaryL ₂-control functions. We then set and solve a min-max game theory problem in terms of an algebraic Riccati operator, to express the optimal quantities in pointwise feedback form. The theory obtained is sharp. It requires the usual Finite Cost Condition and Detectability Condition, the first for existence of the Riccati operator, the second for its uniqueness and for exponential decay of the optimal trajectory. It produces an intrinsically defined sharp value of the parameter, here called _c (critical), _c0, such that a complete theory is available for > _c, while the maximization problem does not have a finite solution if 0 < < _c. Mixed PDE problems, all on arbitrary dimensions, except where noted, where all the assumptions are satisfied, and to which, therefore, the theory is automatically applicable include: second-order hyperbolic equations with Dirichlet control, as well as with Neumann control, the latter in the one-dimensional case; Euler-Bernoulli and Kirchhoff equations under a variety of boundary controls involving boundary operators of order zero, one, and two; Schroedinger equations with Dirichlet control; first-order hyperbolic systems, etc., all on explicitly defined (optimal) spaces [LT3, Section 7]. Solution of the min-max problem implies solution of theH -robust stabilization problem with partial observation.The research of C. McMillan was partially supported by an IBM Graduate Student Fellowship and that of R. Triggiani was partially supported by the National Science Foundation under Grant NSF-DMS-8902811-01 and by the Air Force Office of Scientific Research under Grant AFOSR-87-0321.     

Boundary regularity for Maxwell's equations with applications to shape optimization

The dynamic Maxwell equations with a strictly dissipative boundary condition is considered. Sharp trace regularity for the electric and the magnetic field are established for both: weak and differentiable solutions. As an application a shape optimization problem for Maxwell's equations is considered. In order to characterize the shape derivative as a solution to a boundary value problem, the aforementioned sharp regularity of the boundary traces is critical.     

Stability of boundary layer and rarefaction wave to an outflow problem for compressible Navier-Stokes equations under large perturbation

In this paper, we investigate the large-time behavior of solutions to an outflow problem for compressible Navier-Stokes equations. In 2003, Kawashima, Nishibata and Zhu [S. Kawashima, S. Nishibata, P. Zhu, Asymptotic stability of the stationary solution to the compressible Navier-Stokes equations in the half space, Comm. Math. Phys. 240 (2003) 483-500] showed there exists a boundary layer (i.e., stationary solution) to the outflow problem and the boundary layer is nonlinearly stable under small initial perturbation. In the present paper, we show that not only the boundary layer above but also the superposition of a boundary layer and a rarefaction wave are stable under large initial perturbation. The proofs are given by an elementary energy method.     

Existence and asymptotic behaviour of solutions to weakly damped wave equations of Kirchhoff type with nonlinear damping and source terms

In this paper we consider the existence of a local solution in time to a weakly damped wave equation of Kirchhoff type with the damping term and the source term:

     

Integrodifferential equations with applications to a plate equation with memory

In this paper we study local existence, uniqueness, and continuous dependence of an abstract integrodifferential equation. We also present a result on unique continuation and a blow‐up alternative for mild solutions of the integrodifferential equation. Finally, we apply our results to an interesting strongly damped plate equation with memory.     

On the existence of Nash strategies and solutions to coupled riccati equations in linear-quadratic games

The existence of linear Nash strategies for the linear-quadratic game is considered. The solvability of the coupled Riccati matrix equations and the stability of the closed-loop matrix are investigated by using Brower's fixed-point theorem. The conditions derived state that the linear closed-loop Nash strategies exist, if the open loop matrixA has a sufficient degree of stability which is determined in terms of the norms of the weighting matrices. WhenA is not necessarily stable, sufficient conditions for existence are given in terms of the solutions of auxiliary problems using the same procedure.This work was supported in part by the Joint Services Electronics Program (US Army, US Navy, and US Air Force) under Contract No. DAAG-29-78-C-0016, in part by the National Science Foundation under Grant No. ENG-74-20091, and in part by the Department of Energy, Electric Energy Systems Division, under Contract No. US-ERDA-EX-76-C-01-2088.     

Global solutions and blow-up for a class of strongly damped wave equations systems

The initial-boundary value problem for semilinear wave equation systems with a strong dissipative term in bounded domain is studied. The existence of global solutions for this problem is proved by using potential well method, and the exponential decay of global solutions is given through introducing an appropriate Lyapunov function. Meanwhile, blow-up of solutions in the unstable set is also obtained.     

Inverse problem for a class of one-dimensional wave equations with piecewise constant coefficients

We consider the problem of reconstructing the piecewise constant coefficient of a one-dimensional wave equation on the halfline from the knowledge of the displacement on the boundary caused by an impulse at time zero. This problem is formulated as a nonlinear optimization problem. The objective function of this optimization problem has several special features that have been exploited in building an ad hoc optimization method. The optimization method is based on the solution of a nonlinear system of equations by an algorithm consisting of the evaluation of the unknowns one by one.The research of the third author has been made possible through the support and sponsorship of the Italian Government through the Ministero Pubblica Istruzione under Contract M.P.I. 60% 1987 at the Università di Roma—La Sapienza.     

Bifurcation and nonsmooth dynamics of solitary waves in the generalized long-short wave equations

Generalized wave equations, which model the resonant interaction between the long wave and the short wave, are considered. To understand the underlying complex dynamics, the bifurcations and nonsmooth behaviors of solitary waves for this system are investigated by qualitative techniques in dynamical systems. These complex behaviors may serve as mechanisms for fascinating physical phenomena such as solitons, chaos and turbulence.     

Numerical approximations and regularizations of Riccati equations arising in hyperbolic dynamics with unbounded control operators

This paper provides an approximation theory for numerical computations of the solutions to algebraic Riccati equations arising in hyperbolic, boundary control problems. One of the difficulties in the approximation theory for Riccati equations is that many attractive numerical methods (such as standard finite elements) do not satisfy a uniform stabilizability condition, which is necessary for the stability of the approximate Riccati solutions. To deal with these problems, a regularizationapproximation technique, based on the introduction of special artificial terms to the dynamics of the original model, is proposed. The need for this regularization appears to be a distinct feature of hyperbolic (hyperbolic-like) equations, rather than parabolic (parabolic-like) problems where the smoothing effect of the dynamics is beneficial for the convergence and stability properties of approximate solutions to the associated Riccati equations (see [14]). The ultimate result demonstrates that the regularized, finite-dimensional feedback control yields near optimal performance and that the corresponding Riccati solution satisfies all the desired convergence properties. The general theory is illustrated by an example of a boundary control problem associated with the Kirchoff plate model. Some numerical results are provided for the given example.     

Asymptotic stability and blow up for a semilinear damped wave equation with dynamic boundary conditions

In this paper we consider a multi-dimensional wave equation with dynamic boundary conditions, related to the Kelvin–Voigt damping. Global existence and asymptotic stability of solutions starting in a stable set are proved. Blow up for solutions of the problem with linear dynamic boundary conditions with initial data in the unstable set is also obtained.     

具时间依赖系数和耗散边界的非线性波方程的能量指数衰减性

利用能量法证明了具耗散边界条件和时间依赖系数的非线性波方程的能量指数衰减性.