Appl Math Optim 39:17–32 (1999)
1999 Springer-Verlag New York Inc.
Equivalence between Nonlinear H
and Existence of Viscosity Solutions of
Dipartimento di Matematica Pura e Applicata,
Universit`a Degli Studi di Padova,
via Belzoni 7, 35131 Padova, Italy
Communicated by A. Bensoussan
Abstract. In this paper we extend to completely general nonlinear systems the re-
sult stating that the H
suboptimal control problem is solved if and only if the corre-
sponding Hamilton–Jacobi–Isaacs (HJI) equation has a nonnegative(super)solution.
This is well known for linear systems, using the Riccati equation instead of the HJI
equation. We do this using the theory of differential games and viscosity solutions.
KeyWords. Nonlinear H
control,Differentialgames,Viscosity solutions, Isaacs
equations, Nonlinear systems.
AMS Classiﬁcation. 93B36, 49L25, 90D25.
In this paper our goal is to extend to nonlinear systems the well-known result in linear
control theory stating that, for a given disturbance attenuation level γ>0, the H
suboptimal problem can be solved if and only if the corresponding Riccati equation has
a positive deﬁnite solution. The classical problem is studied in the frequency domain
which cannot be applied to the nonlinear case, so, as usual, we consider the equivalent
This work was written while the author was visiting the University of California at Santa Barbara. He
was partially supported by Consiglio Nazionale delle Ricerche of Italy.