Remarks on the computation and structure of the sub-optimal solutions of generalized Hamilton-Jacobi inequalities

Hamilton Jacobi inequalities play an important role in the design of robust controllers for nonlinear systems. In this note we address the problem of how to compute their suboptimal solutions and their structure. We identify a very conservative lower bound for the L₂ gain obtainable through the solution of HJ inequality, and formalize conditions under which seeking the optimal solution of the problem, for some general classes of systems yields "very degenerative" controllers. We also introduce a class of neural networks all consisting of global diffeomorphisms. to which can be associated positive definite and proper functions, for the approximation of solutions in HJ inequalities. together with the description of an evolutionary numerical strategy specifically designed for their computation.