Extending nonlinear H2, H∞ optimisation to W1,2, W1,∞ spaces - part I: optimal control

In this article, we introduce, formulate and solve the W-1,W-2, W-1,W-infinity nonlinear optimal control problems as extensions of H-2, H-infinity optimal control problems, respectively. As these spaces contain less smooth functions, a larger number of problems could be solved in this framework, and by a suitable choice of weighting functions, additional design objectives could be achieved using the present formulation. Moreover, any solution of the W-1,W-p, p = 2, infinity problem, is automatically a solution of the corresponding H-p-problem. Sufficient conditions for the solvability of the problems are given in terms of new Hamilton-Jacobi equations (HJEs). These new HJEs may also be easier to solve because of the additional degrees of freedom offered by the current norms. Both the state-feedback and output-feedback problems are discussed. The results are then specialised to linear systems, in which case the solutions are characterised in terms of new algebraic-Riccati equations.