Nonlinear H-Infinity Control, Hamiltonian Systems and Hamilton-Jacobi Equations

13

Solving the Hamilton-Jacobi Equation

In this chapter, we discuss some approaches for solving the Hamilton-Jacobi-equations (HJE) associated with the optimal control problems for affine nonlinear systems discussed in this book. This has been the biggest bottle-neck in the practical application of nonlinear ${\mathcal{H}}_{\infty }$-control theory. There is no systematic numerical approach for solving the HJIEs. Various attempts have however been made in this direction in the past three decades. Starting with the work of Lukes [193], Glad [112], who proposed a polynomial approximation approach, Van der Schaft [264] applied the approach to the HJIEs and developed a recursive approach which was also refined by Isidori [145]. Since then, many other authors have proposed ...