We consider the abstract dynamical framework of Lasiecka and Triggiani (2000)
1, Chapter 9, which models a large variety of mixed PDE problems (see specific classes in the Introduction) with ...boundary or point control, all defined on a smooth, bounded domain
Ω
⊂
R
n
,
n
arbitrary. This means that the input
→
solution map is bounded on natural function spaces. We then study min–max game theory problem over a finite time horizon. The solution is expressed in terms of a (positive, self-adjoint) time-dependent Riccati operator, solution of a non-standard differential Riccati equation, which expresses the optimal qualities in pointwise feedback form. In concrete PDE problems, both control and deterministic disturbance may be applied on the boundary, or as a Dirac measure at a point. The observation operator has some smoothing properties.
We provide sharp regularity results for thermoelastic plate-like systems under the action of an interior point control exercised in the Kirchhoff-type mechanical equation, in the case of ...hinged/Dirichlet boundary conditions (B.C.).
Satellite experiments for gamma-ray and cosmic-ray detection employ plastic scintillators to discriminate charged from neutral particles in order to correctly identify gamma-rays and charged nuclei. ...The High Energy Cosmic Radiation Detection (HERD) facility will be among these experiments, to be installed onboard the future Chinese Space Station (CSS), to detect cosmic-rays and gamma-rays up to TeV energies. The plastic scintillator detector (PSD) will consist of scintillator tiles or bars coupled to Silicon Photomultipliers (SiPMs). To discriminate gamma-rays from charged particles and measure the ion charge up to iron nuclei a wide dynamic range is required, from few tens up to thousands of photoelectrons. We have equipped a plastic scintillator tile prototype with SiPMs produced by Hamamatsu and AdvanSiD and coupled their analog signals to the DT5550W board based on the CITIROC ASIC, produced by CAEN SpA. The CITIROC ASIC allows both the formation of a fast trigger with a configurable threshold and the digitization of analog waveforms after a preamplification and shaping stage along two paths with different gain settings. The performance of our prototype will be shown.
<Abstract. < This paper considers a fully general (Riemann) wave equation on a finite-dimensional Riemannian manifold, with energy level <(H<<<1<<< × L<<<2<<<)< -terms, under essentially minimal ...smoothness assumptions on the variable (in time and space) coefficients. The paper provides Carleman-type inequalities: first pointwise, for <C<<<2<< -solutions, then in integral form for <H<<<1,1<<<(Q)< -solutions. The aim of the present approach is to provide Carleman inequalities which do <not< contain lower-order terms, a distinguishing feature over most of the literature. Accordingly, global uniqueness results for overdetermined problems as well as Continuous Observability</< Uniform Stabilization inequalities follow in one shot, as a part of the same stream of arguments. Constants in the estimates are, therefore, generally explicit. The paper emphasizes the more challenging pure Neumann B.C. case. The paper is a generalization from the Euclidean to the Riemannian setting of LTZ in the more difficult case of purely Neumann B.C., and of KK1 in the case of Dirichlet B.C. The approach is Riemann geometric, but different from--indeed, more flexible than--the one in LTY1.
In the abstract hyperbolic-like case, under a mild exact controllability assumption, the Riccati operator is known to be an isomorphism F. Flandoli, I. Lasiecka, R. Triggiani, Algebraic Riccati ...equations with non-smoothing observation arising in hyperbolic and Euler–Bernoulli boundary control problems, Annali di Matematica Pura e Applicata (iv)CLII (1988) 307–382 (Section 6). This property then plays a crucial role in establishing a Dual Algebraic Riccati Theory. Here we strengthen this theory by providing additional results (which we had announced in V. Barbu, I. Lasiecka, R. Triggiani, Extended algebraic Riccati equations in the abstract hyperbolic case, Non-linear Analysis 40 (2000) 105–129 and R. Triggiani, The algebraic Riccati equation with unbounded control operator: the abstract hyperbolic case revisited, AMS, Contemporary Mathematics 209 (1997) 315–338): in particular that
P
D
(
A
F
)
=
D
(
A
∗
)
and that
P
D
(
A
)
=
D
(
A
F
∗
)
.
We consider a general second-order hyperbolic equation defined on an open bounded domain Ω⊂Rn with variable coefficients in both the elliptic principal part and in the first-order terms as well. At ...first, no boundary conditions (B.C.) are imposed. Our main result (Theorem 3.5) is a reconstruction, or inverse, estimate for solutions w: under checkable conditions on the coefficients of the principal part, the H1(Ω)×L2(Ω)-energy at time t=T, or at time t=0, is dominated by the L2(Σ)-norms of the boundary traces ∂w/∂νA and wt, modulo an interior lower-order term. Once homogeneous B.C. are imposed, our results yield—under a uniqueness theorem, needed to absorb the lower-order term—continuous observability estimates for both the Dirichlet and Neumann case, with an explicit, sharp observability time; hence, by duality, exact controllability results. Moreover, no artificial geometrical conditions are imposed on the controlled part of the boundary in the Neumann case. In contrast with existing literature, the first step of our method employs a Riemann geometry approach to reduce the original variable coefficient principal part problem in Ω⊂Rn to a problem on an appropriate Riemann manifold (determined by the coefficients of the principal part), where the principal part is the Laplacian. In our second step, we employ explicit Carleman estimates at the differential level to take care of the variable first-order (energy level) terms. In our third step, we employ micro-local analysis yielding a sharp trace estimate, to remove artificial geometrical conditions on the controlled part of the boundary, in the Neumann case.
Originally published in 2000, this is the second 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 unifies 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 2 is focused on the optimal control problem over a finite time interval for hyperbolic dynamical systems. A few abstract models are considered, each motivated by a particular canonical hyperbolic dynamics. 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.
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