Examples for the Spectrum

Abstract

The spectrum of a bilinear control system on a vector bundle is given in terms of the solutions of the system. Its computation at the continuity points, where the different spectra agree, is reduced in Appendix D to the solution of certain optimal control problems, i.e., basically to the numerical solution of a class of Hamilton-Jacobi-Bellman equations. This chapter shows how the resulting algorithm can be used to compute the spectral intervals of systems of-currently-dimension up to 4. Our examples include linear oscillators and a model of a stirred tank reactor (compare Sections 2.5 and 9.1). These examples are presented in Section 10.2. Section 10.1 deals with bilinear systems in ℝ2. In this case the projected system on the projective space ℙ1 is one-dimensional, and the results from Chapters 7 and 8 lead to a more explicit characterization of the spectral intervals. Often it is only necessary to determine the sign of the boundary points of the spectral intervals, such as for the computation of stability and stabilization radii (compare Chapter 11) or for the characterization of feedback stabilizability at singular points (compare Section 12.2). In dimension d = 2 the sign of the extremal spectral values can be determined by an explicit control strategy, which is presented at the end of Section 10.1.