Optimal control of MIMO input-quadratic nonlinear systems

We study the infinite-horizon optimal control problem for nonlinear, multi-input, input-quadratic systems. It is shown that optimality of the input-quadratic closed-loop system is intimately related to the property that an auxiliary input-affine system possesses a L-2-gain smaller than one. Such equivalence is established, or approximated, by relying on (a combination of) three alternative sets of technical conditions based (i) on the inclusion of the gradient of the underlying storage function in a certain co-distribution, (ii) on verifying specific algebraic inequalities, (iii) or achieved dynamically by considering the immersion of the original nonlinear plant into a system defined on an augmented state-space.