Time-, fuel-, and energy-optimal control of nonlinear norm-invariant systems

Nonlinear systems of the form
\dot{X}(t)=g[x(t);t]+u(t)
, where
x(t), u(t)
, and
g[x(t); t]
are
n
vectors, are examined in this paper. It is shown that if
\parellelx(t)\parellel = \sqrt{x_{1}^{2}(t) + ... + x_{n}^{2}(t)}
is constant along trajectories of the homogeneous system
\dot{X}(t)=g[x(t); t]
and if the control
u(t)
is constrained to lie within a sphere of radius
M
, i.e.,
\parellelu(t}\parellel \leq M
, for all
t
, then the control
u^{\ast}(t)= - Mx(t} /\parellelx(t)\parellel
drives any initial state
\xi
to 0 in minimum time and with minimum fuel, where the consumed fuel is measured by
\int \liminf{0} \limsup{T}\parellel u(t) \parelleldt
. Moreover, for a given response time
T
, the control
\utilde(t) = -\parellel\xi\parellel x(t)/T \parellel x(t) \parellel
drives
\xi
to 0 and minimizes the energy measured by
\frac{1}{2}\int \liminf{0} \limsup{T}\parellelu(t)\parellel^{2}dt
. The theory is applied to the problem of reducing the angular velocities of a tumbling asymmetrical space body to zero.