Restricted frequency inequality is equivalent to restricted dissipativity

A variety of powerful tools and results in systems and control theory rely on classical Kalman-Yakubovich-Popov-Zames results establishing equivalence between special frequency domain inequalities (FDIs), linear matrix inequalities (LMIs) and time domain inequalities (TDIs). Recent developments have addressed FDIs within (semi)finite frequency ranges to increase flexibility in the system analysis and synthesis. In this paper it is shown that validity of a general FDI within a restricted frequency range is equivalent to validity of the corresponding TDI under rate limitations specified by a matrix-valued integral quadratic constraint. The latter property of a system is termed "restricted dissipativity". Its special cases are "restricted passivity" and "restricted finite gain property". The equivalence between restricted FDI and restricted dissipativity is established for both continuous-time and discrete-time settings. The paper together with the previous developments extends Kalman-Yakubovich-Popov-Zames FDI-LMI-passivity results to FDIs specified within restricted frequency ranges.