4 research outputs found
Ensembles of Hyperbolic PDEs: Stabilization by Backstepping
For the quite extensively developed PDE backstepping methodology for coupled
linear hyperbolic PDEs, we provide a generalization from finite collections of
such PDEs, whose states at each location in space are vector-valued, to
previously unstudied infinite (continuum) ensembles of such hyperbolic PDEs,
whose states are function-valued. The motivation for studying such systems
comes from traffic applications (where driver and vehicle characteristics are
continuously parametrized), fluid and structural applications, and future
applications in population dynamics, including epidemiology. Our design is of
an exponentially stabilizing scalar-valued control law for a PDE system in two
independent dimensions, one spatial dimension and one ensemble dimension. In
the process of generalizing PDE backstepping from finite to infinite
collections of PDE systems, we generalize the results for PDE backstepping
kernels to the continuously parametrized Goursat-form PDEs that govern such
continuously parametrized kernels. The theory is illustrated with a simulation
example, which is selected so that the kernels are explicitly solvable, to lend
clarity and interpretability to the simulation results.Comment: 16 pages, 4 figures, to be publishe
Control Problems for Conservation Laws with Traffic Applications
Conservation and balance laws on networks have been the subject of much research interest given their wide range of applications to real-world processes, particularly traffic flow. This open access monograph is the first to investigate different types of control problems for conservation laws that arise in the modeling of vehicular traffic. Four types of control problems are discussed - boundary, decentralized, distributed, and Lagrangian control - corresponding to, respectively, entrance points and tolls, traffic signals at junctions, variable speed limits, and the use of autonomy and communication. Because conservation laws are strictly connected to Hamilton-Jacobi equations, control of the latter is also considered. An appendix reviewing the general theory of initial-boundary value problems for balance laws is included, as well as an appendix illustrating the main concepts in the theory of conservation laws on networks
Control Problems for Conservation Laws with Traffic Applications
Conservation and balance laws on networks have been the subject of much research interest given their wide range of applications to real-world processes, particularly traffic flow. This open access monograph is the first to investigate different types of control problems for conservation laws that arise in the modeling of vehicular traffic. Four types of control problems are discussed - boundary, decentralized, distributed, and Lagrangian control - corresponding to, respectively, entrance points and tolls, traffic signals at junctions, variable speed limits, and the use of autonomy and communication. Because conservation laws are strictly connected to Hamilton-Jacobi equations, control of the latter is also considered. An appendix reviewing the general theory of initial-boundary value problems for balance laws is included, as well as an appendix illustrating the main concepts in the theory of conservation laws on networks