On the attainable set for scalar balance laws with distributed control

Abstract

The paper deals with the set of attainable profiles of a solution u to a scalar balance law in one space dimension with strictly convex flux function ∂tu + ∂xf (u) = z(t, x).Here the function z is regarded as a bounded measurable control. We are interested in studying the set of attainable profiles at a fixed time T > 0, both in case z(t, ·) is supported in the all real line, and in case z(t, ·) is supported in a compact interval [a, b] independent on the time variable t. Mathematics Subject Classification. 35L65, 35Q93. and the source term z ∈ L ∞ (]0, T [×R) is regarded as a control. We are interested in studying the set of attainable profiles at time T , i.e. A(T, Z).= v ∈ L ∞ (R) : ∃z ∈ Z and a solution u to (1.1)−(1.2) : u(T, x) = v(x) a.e. , (1.4)

On the global controllability of scalar conservation laws with boundary and source controls

“…In the second setting and for general strictly convex flux f , Perrollaz [38] provided sufficient conditions for the reachability (in arbitrarly small time) of a state ψ ∈ BV ([a, b]) with boundary and source controls, through entropy weak solutions of (1.4). In a related result Corghi and Marson [16] established a characterization of the attainable set for scalar strictly convex balance laws evolving on the whole real line, with the source term (depending on both space and time) regarded as a control.…”

On the Attainable Set for a Scalar Nonconvex Conservation Law

“…so that 8) holds for any v = 0. Symmetrically, we denote by v ♭ the solution to 10) holds for any v = 0.…”

“…Under similar hypothesis, Adimurthi, Ghoshal and Gowda [1,2] exploit the explicit representation of solutions given by the Lax-Oleinik formula to construct an explicit backward solver and give a coincise characterization of the set of attainable profiles for the initial value problem and the boundary value problem in the half-space and in a strip with two boundaries. Using again the method of generalized characteristics, Corghi and the third author characterize in [8] the attainable set for a scalar balance law with strictly convex flux u t + f (u) x = z(t, x), for t ∈ [0, T ], and x ∈ R , (1.2) where the right hand side z acts as distributed control.…”

Exact controllability to trajectories for entropy solutions to scalar conservation laws in several space dimensions

“…A separate issue is related to the irreversibility of entropy solutions : the set of admissible target states in time T is reduced and its description, often involving a number of highly technical conditions, is in itself an open problem in most cases, see [2,3,6,7,14].…”

“…Starting from the pioneering work by Ancona and Marson, [3], several results have been obtained using the theory of generalized characteristics introduced by Dafermos in [20], as [3,7,14,24,32] or the explicit Lax-Oleinik representation formula, as [1,6]. The latter technique is applicable only when the flux function f is strictly convex/concave, while the theory of generalized characteristics covers also the (slightly) more general case of a flux function f with one inflection point.…”