The Continuing Joy of Dissipation Inequalities

Dissipation inequalities play a fundamental role in systems and control theory and dissipativity is a very useful concept in the analysis and design of nonlinear control systems. The idea of dissipativity was introduced in the early 1970s as a generalization of Lyapunov inequalities to systems having inputs and outputs. While Lyapunov functions serve to show the stability of dynamical systems, dissipation inequalities can be applied more widely depending on the choice of the so-called supply rate. Classical special cases being for example the well-known passivity or the L2-norm characterization of nonlinear systems. Like in Lyapunov theory the biggest problem in applications is the construction of a storage function, which is the generalization of the Lyapunov function, that normally requires the solution of a partial differential equation. However for the important class of polynomial systems, i.e. systems with polynomial nonlinearities, recent advances in the area of computational semialgebraic geometry, namely semidefinite programming and the sum of squares decomposition, allow a reliable and efficient solution in many cases.