State and parameter estimation in 1-D hyperbolic PDEs based on an adjoint method

International audienceAn optimal estimation method for state and distributed parameters in1-D hyperbolic system based on adjoint method is proposed in thispaper. A general form of the partial differential equations governingthe dynamics of system is first introduced. In this equation, theinitial condition or state variable as well as some empiricalparameters are supposed to be unknown and need to be estimated. TheLagrangian multiplier method is used to connect the dynamics of thesystem and the cost function defined as the least square error betweenthe simulation values and the measurements. The adjoint state method isapplied to the objective functional in order to get the adjoint systemand the gradients with respect to parameters and initial state. Theobjective functional is minimized by Broyden–Fletcher–Goldfarb–Shanno(BFGS) method. Due to the non-linearity of both direct and adjointsystem, the nonlinear explicit Lax–Wendroff scheme is used to solvethem numerically. The presented optimal estimation approach isvalidated by two illustrative examples, the first one about state andparameter estimation in a traffic flow, and the second one in anoverland flow system