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.

Let A (the elastic operator) be a positive, self-adjoint operator with domain D(A) in the Hubert space X, and let B (the dissipation operator) be another positive, self-adjoint operator satisfying: p\A a < B < p 2 A a for some constants 0 < pi < pi < oo and 0 < a < 1. Consider the operator (corresponding to the elastic model x + Bx + Ax = 0 written as a first order system), which (once closed) is plainly the generator of a strongly continuous semigroup of contractions on the space E = D(A ι/2 ) x X. We prove that if 1/2 < a < 1, then such semigroup is also analytic (holomorphic) on a triangular sector of C containing the positive real axis. This established a fortiori two conjectures of Goong Chen and David L. Russell on structural damping for elastic systems, which referred to the case a = 1/2. Actually, in the special case a = 1/2 we prove a result stronger than the two conjectures, which yields analyticity of the semigroup over an explicitly identified range of spaces which includes E. This latter result was already proved in our previous effort on this problem. Here we provide a technically different and simplified proof of it. We also provide two conceptually and technically different proofs of our main result for 1/2 < a < 1. Finally, we show that for 0 < a < 1/2 the semigroup is not analytic.

This paper considers an abstract third‐order equation in a Hilbert space that is motivated by, and ultimately directed to, the “concrete” Moore–Gibson–Thompson Equation arising in high‐intensity ultrasound. In its simplest form, with certain specific values of the parameters, this third‐order abstract equation (with unbounded free dynamical operator) is not well‐posed. In general, however, in the present physical model, a suitable change of variable permits one to show that it has a special structural decomposition, with a precise, hyperbolic‐dominated driving part. From this, various attractive dynamical properties follow: s.c. group generation; a refined spectral analysis to include a specifically identified point in the continuous spectrum of the generator (so that it does not have compact resolvent) as an accumulation point of eigenvalues; and a consequent theoretically precise exponential decay with the same decay rate in various function spaces. In particular, the latter is explicit and sharp up to a finite number of (stable) eigenvalues of finite multiplicity. A computer‐based analysis confirms the theoretical spectral analysis findings. Moreover, it shows that the dynamic behavior of these unaccounted for finite‐dimensional eigenvalues are the ones that ultimately may dictate the rate of exponential decay, and which can be estimated with arbitrarily preassigned accuracy. Copyright © 2012 John Wiley & Sons, Ltd.

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