Calculus of variations approach for state and parameter estimation in switched 1-D hyperbolic PDEs
Résumé
This paper proposes the use of calculus of variations to solve the problem of state and parameter estimation
for a class of switched 1-D hyperbolic partial differential equations (PDE) coupled with an ordinary
differential equation (ODE). The term ‘switched’ here refers to a system changing its characteristics
according to a switching rule which may depend on time, parameters of the system and/or state of the
system. The estimation method is based on a smooth approximation of the system dynamics and the use
of variational calculus on an augmented Lagrangian cost functional to get the sensitivity with respect to
initial state and some (possibly distributed) parameters of the system. Those sensitivities – or variations,
together with related adjoint systems, are used as inputs for an optimization algorithm to identify values of
the variables to be estimated. Two examples are provided to demonstrate the effectiveness of the proposed
method: the first one is concerned with a switched overland flow model, developed from Saint-Venant
equations and Green-Ampt law; the second example deals with a switched free traffic flow model based
on Lighthill-Whitham-Richards (LWR) representation, modified by the presence of a relief route.