Optimal control of the inhomogeneous relativistic Maxwell-Newton-Lorentz equations (Q2821809)

The physical model is introduced first. It is based on the classical system of inhomogeneous Maxwell's equations subject to boundary and initial conditions and the relativistic Newton-Lorentz equations. The system models the relativistic motion of charged particles in an electromagnetic field. The external electric and magnetic fields are generated by exterior capacitors and magnets for steering the particle beam. The external magnetic field in the overall system serves as control variable and the aim of the considered optimization problem is to find that control that steers the particle beam to a desired track. By means of some slightly modifications, for instance use of Abraham model, the complete rigorous mathematical optimal control problem including state equations, performance functional and state constrains is formulated. Additional assumptions, function spaces, further transformations on the formulation of the state system, definition of solutions, optimal control and optimal control problem as well as an existence and uniqueness result for the reduces state system are largely discussed. Finally, the existence of at least one globally optimal control, first order necessary optimality conditions and Karush-Kuhn-Tucker conditions are proved. Numerical investigations are performed by means of a special example and numerical results are shown.