Time-energy suboptimal control of nonlinear input-affine systems

In this paper, we derive the necessary conditions for time-energy optimal control of input-affine nonlinear systems using Hamilton–Jacobi–Bellman equation. The objective function is chosen as a weighted combination of control effort and time-horizon minimisation. The necessary conditions of optimality are transformed such that the resultant conditions are structurally similar to a differential matrix Riccati equation. A closed-form approximate solution to these necessary conditions is derived such that the resulting closed-loop dynamics is locally exponentially stable around the origin. Furthermore, it is also shown that the states converge within a ball of an arbitrarily small radius around the origin within a finite time. Finally, the performance of the proposed controller is tested using numerical simulations for Vander Pol's oscillator as well as spacecraft detumbling problem. Furthermore, the performance of the proposed controller is also tested in comparison to the infinite-horizon state-dependent Riccati equation controller.