##
**Inverse optimal design of input-to-state stabilizing nonlinear controllers.**
*(English)*
Zbl 0910.93064

The systems considered in this paper are described by
\[
\dot x= f(x)+g_1 (x)d+ g_2(x)u
\]
with the state \(x\), the disturbance \(d\) and the control input \(u\). Input-to-state stabilizability is defined. Conditions for input-to-state stability with respect to the disturbance \(d\) are derived. A differential game problem with a suitable chosen cost functional is defined. An inverse optimal gain assignment problem is said to be solvable if penality functions on the state, the control and the disturbance and a feedback control law exist which minimize the cost functional for worst case disturbances. It is shown that stabilizability is a necessary and sufficient condition for solvability of this problem.

Stability margins are derived by chosing suitable Lyapunov functions. Relations to the nonlinear \(H_\infty\) problem are shown. Stability margins for unmodelled dynamics with passity properties are derived. The design of smooth control laws by integrator backstepping is described.

Stability margins are derived by chosing suitable Lyapunov functions. Relations to the nonlinear \(H_\infty\) problem are shown. Stability margins for unmodelled dynamics with passity properties are derived. The design of smooth control laws by integrator backstepping is described.

Reviewer: R.Tracht (Essen)

### MSC:

93D15 | Stabilization of systems by feedback |

93D30 | Lyapunov and storage functions |

49L99 | Hamilton-Jacobi theories |

93B36 | \(H^\infty\)-control |

93C10 | Nonlinear systems in control theory |