Viscosity Solutions and Computation for Dissipative Systems

Recent activity in nonlinear H∞ control has sparked renewed interest in the theory of dissipative nonlinear systems. This theory was initiated byWillems (1972) and developed further by Hill & Moylan (1976). Central to this theory are the so-called storage functions. These functions are generalized energy functions defined in terms of dissipation inequalities. The natural infinitesimal versions of such dissipation inequalities (obtained by dynamic programming) are partial differential inequalities (PDIs), special cases of which relate to nonlinear generalizations of the Positive Real Lemma and the Bounded Real Lemma. Such PDIs are important since one hopes to obtain storage functions by solving them. However, the PDIs involve first order partial derivatives of the storage functions, which unfortunately need not exist in general. The purpose of these notes is to explain (i) how functions which need not be differentiable can solve such PDIs in a generalized sense (viscosity sense), and (ii) how to compute numerical approximations to solutions (i.e. storage functions) of such PDIs. In §2 we consider a simple optimal control problem and present the corresponding PDE (a nonlinear first-order equation) which arises in dynamic programming. In §3 we discuss such nonlinear PDEs, pointing out the difficulty concerning the correct definition of solution and uniqueness. The definition of viscosity solution is given in §4. A simple finite difference numerical approximation scheme is discussed in §5, and an example is given in §6 which illustrates the scheme and a consequence of the lack of smoothness. In §7 the discussion turns to dissipative systems, PDIs, and the viscosity solution interpretation. These results are applied in §8 to the problem of computing the “H∞ norm” of nonlinear systems. Two examples are given in §§9, 10.