Mathematical Tools of Dynamic Systems in Hilbert Spaces

Abstract

In this chapter, dynamic systems and solution concepts are reviewed in the infinite-dimensional setting. After brief introduction of linear PDEs of parabolic, elliptic, and hyperbolic types, strong, mild, and weak solutions are addressed for these equations. Mathematical tools, presented for their analysis, involve Sturm–Liouville operators and their properties as well as the powerful method of separation of variables. Viscosity and proximal solutions are additionally revisited for nonlinear first-order partial differential inequalities with discontinuous terms. Finally, modern stability paradigms such as ISS and finite time stability among others are recalled with special attention to sliding mode dynamics in Hilbert space and to homogeneous differential inclusions.