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This article deals with the parabolic equation ∂tw − c(t)∂2x w = f in D, D = { (t, x) ∈ R2 : t > 0, φ1 (t) < x < φ2(t) } with φi : [0,+∞[→ R, i = 1, 2 and c : [0,+∞[→ R satisfying some conditions and the problem is supplemented with boundary conditions of Dirichlet-Robin type. We study the global regularity problem in a suitable parabolic Sobolev s...
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Citations
This paper is devoted to the analysis of the following linear parabolic equation ?tu-?2xu = f, subject to Robin type conditions ?x u + ?u = 0, on the lateral boundary, where coefficient ? satisfies suitable non-degeneracy assumptions and possibly depends on the time variable. The right-hand side ? of the equation is taken in Lp, 1 < p < ?. The problem is set in a domain of the form ? = {(t, x) ? R2 : 0 < t < 1, 0 < x < t?}, ? > 1/2. We use Labbas-Terreni results [23] on the operator?s sum method in the non-commutative case. This work is an extension of the Hilbertian case studied in [15].
This paper addresses linear quadratic control optimal problems for non-autonomous linear control systems using strongly continuous quasi semigroups. Riccati equations are implemented to investigate the control optimal problems of cost functional with finite and infinite horizons. The unique optimal pair for the cost functional is determined by the mild solution of the associated closed-loop problem and the feedback control of the solution of the corresponding Riccati equation. In addition for the infinite horizon, the stabilizability of the system is a sufficiency for the solvability to the Riccati equation. An application in a parabolic system is proposed.
