Let A (the elastic operator) be a positive, self-adjoint operator with domain D(A) in the Hubert space X, and let B dissipation another satisfying: p\Aa < p 2Aa for some constants 0 pi oo 1. Consider (corresponding to model x + Bx Ax = written as first order system), which (once closed) is plainly generator of strongly continuous semigroup contractions on E D(Aι/2) X. We prove that if 1/2 1, then such also analytic (holomorphic) triangular sector C containing positive real axis. This...

It is shown that exact controllability in finite time for linear control systems given on an infinite dimensional Banach space integral form (mild solution) can never arise using locally $L_1 $-controls, if the associated $C_0 $ semigroup compact all $t > 0$. This includes, particular, class of parabolic partial differential equations defined bounded spatial domains.

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 high‐intensity ultrasound. In its simplest form, with certain specific values of parameters, this (with unbounded free dynamical operator) not well‐posed. general, however, present physical model, suitable change variable permits one to show it has special structural decomposition, precise, hyperbolic‐dominated...

The classical theory of (state and output) controllability observability in finite-dimensional spaces is extended to linear abstract systems defined on infinite-dimensional Banach spaces, under the basic assumption that operator acting state be bounded. Tests for approximate as well observability, expressed only terms coefficients system, are proved via a consequence Hahn–Banach theorem, new phenomena arising infinite dimensions studied : instance, by using Baire category arguments, it shown...

Controllability of linear retarded systems is investigated by using the abstract representation such given $\dot x = \tilde Ax + Bu$, where belongs to Hilbert space $R^n \times L_2 ([ - h,0],R^n )$ denoted as $M_2 $, and $\tilde A$ generates a $C_0 $-semigroup. It shown that useful, practically verifiable conditions can be obtained this approach. The following problems are investigated: approximate controllability in $ its subspace, $L_2 exact Euclidean $(R^n controllability, spectral...

The steady-state solutions to Navier-Stokes equations on Ω ⊂ R d , = 2, 3, with no-slip boundary conditions, are locally exponentially stabilizable by a finite-dimensional feedback controller support in an arbitrary open subset w of positive measure. (finite) dimension the is related largest algebraic multiplicity unstable eigenvalues linearized equation.

Introduction Main results Proof of Theorems 2.1 and 2.2 on the linearized system (2.4): $d=3$ Boundary feedback uniform stabilization (3.1.4) via an optimal control problem corresponding Riccati theory. Case Theorem 2.3(i): Well-posedness Navier-Stokes equations with Riccati-based boundary control. 2.3(ii): Local stability A PDE-interpretation abstract in Sections 5 6 Appendix A. Technical material complementing Section 3.1 B. arbitrarily small support (3.1.4a) at...

Generalizations of the familiar rank conditions for controllability and observability linear autonomous finite-dimensional systems to general case when both state space control are infinite-dimensional Banach spaces operator A acting on is only assumed generate a strongly continuous semigroup (group) sought. It shown that suitable version condition, although generally sufficient approximate (observability), however "essentially" necessary in two important cases: (i) generates an analytic...