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

This is the authors' abstract. We don't add key points for this paper.

Nonlinear systems of the formẊ(t)=g[x(t);t]+u(t), wherex(t), u(t), andg[x(t); t]arenvectors, are examined in this paper. It is shown that if(t) = √{x₁²(t) + ... + xₙ²(t)}is constant along trajectories of the homogeneous systemẊ(t)=g[x(t); t]and if the controlu(t)is constrained to lie within a sphere of radiusM, i.e.,(t} ≤ M, for allt, then the controlu^{}(t)= - Mx(t} /(t) any initial state 0 in minimum time and with minimum fuel, where the consumed fuel is measured by∫ {0} {T} u(t) . Moreover, for a given response timeT, the control(t) = -ξ x(t)/T x(t) 0 and minimizes the energy measured by1/2∫ {0} {T}(t)²dt. The theory is applied to the problem of reducing the angular velocities of a tumbling asymmetrical space body to zero.