Global uniqueness for a coefficient inverse problem for the non-stationary transport equation via Carleman estimate

Abstract

A coefficient inverse problem for the non-stationary single-speed transport equation for t ∈ (0, T ) with the lateral boundary data and initial condition at t = 0 is considered. Global uniqueness result is obtained by the method of Carleman estimates.

Stability for Some Inverse Problems for Transport Equations

“…(1.4) physical backgrounds such as neutron transport and medical tomography, see e.g., Case and Zweifel [8], Ren, Bal and Hielscher [18]. Our inverse problem is formulated with a single measurement, and Gaitan and Ouzzane [9], Klibanov and Pamyatnykh [15], Machida and Yamamoto [17] discuss the uniqueness and the stability for inverse problems for initial/boundary value problems for transport equations by Carleman estimates. The papers [15] and [17] discuss transport equations with integral terms where solutions y and u depend also on the velocity as well as the location x and the time t.…”

Observability Inequalities for Transport Equations through Carleman Estimates

“…Carleman estimates for transport equations are proved in Gaitan and Ouzzane [4], Gölgeleyen and Yamamoto [5], Klibanov and Pamyatnykh [6], Machida and Yamamoto [7] to be applied to inverse problems of determining spatially varying coefficients, where coefficients of the first-order terms in x are assumed not to depend on t. In order to improve results for inverse problems by the application of Carleman estimates, we need a better choice of the weight function in the Carleman estimate. The works [4] and [6] use one weight function which is very conventional for a second-order hyperbolic equation but seems less useful to derive analogous results for a time-dependent function H(t). Our choice is more similar to the one in [7] and [5], but even these papers allow no time dependence for H. Although it is very difficult to choose the best possible weight function for the partial differential equation under consideration, our choice (8) of the weight function seems more adapted for the nature of the transport equation (1).…”

Identifiability analysis of an epidemiological model in a structured population

“…It can be easily checked that when starting from a positive infected population at time zero, I remains positive on [0, T ], so X(a) = y(t, a) =X(a). Using(18) we similarly obtain X (a) =X (a), so(20) and(21)also hold.…”

“…Results on various classes of linear models with pointwise observation where obtained using Carleman estimates, for instance for the Schrödinger equation [18] or for a non-stationary particle transport equation (see [19] and references therein). In the nonlinear case, we only found results dealing with parabolic equations using Carleman estimates [20][21][22].…”

“…Then either M ∂ a y (t, a) = 0 or M ∂ a y (t, a) = 0.In the first case,(15) implies that M ∂ a y (t, a) and (Y (a) − M y (t, a)) are collinear, and so are M ∂ aȳ (t, a) and (Ȳ (a) − Mȳ(t, a)). It follows that (Y (a) −Ȳ (a)) and M ∂ a y (t, a) are collinear, which yields(20). Moreover, Y (a) − M y (t, a) andȲ (a) − Mȳ(t, a) are also collinear and consequently(21) holds.In the second case, (16) yields βI(t − a)( X(a) − y(t, a)) = 0.…”