1 Analytical approximation methods for the stabilizing solution of the Hamilton-Jacobi equation Noboru Sakamoto and Arjan J. van der Schaft Abstract In this paper, two methods for approximating the stabilizing solution of the Hamilton-Jacobi equation are proposed using symplectic geometry and a Hamiltonian perturbation technique as well as stable manifold theory. The first method uses the fact that the Hamiltonian lifted system of an integrable system is also integrable and regards the corresponding Hamiltonian system of the Hamilton-Jacobi equation as an integrable Hamiltonian system with a perturbation caused by control. The second method directly approximates the stable flow of the Hamiltonian systems using a modification of stable manifold theory. Both methods provide analytical approximations of the stable Lagrangian submanifold from which the stabilizing solution is derived. Two examples illustrate the effectiveness of the methods. I. Introduction When analyzing a control system or designing a feedback control, one often encounters certain types of equations that describe fundamental properties of the control problem at hand. It is the Riccati equation for linear systems and the Hamilton-Jacobi equation plays the same role in nonlinear systems. For example, an optimal feedback control can be derived from a solution of a Hamilton-Jacobi equation [25] and H ∞ feedback controls are obtained by solving one or two Hamilton-Jacobi equations [5], [21], [38], [39]. Closely related to optimal control and H ∞ control is the notion of dissipativity, which is characterized by a Hamilton-Jacobi inequality (see, e.g., [19], [42]). Some active areas of research in recent years are the factorization problem [6], [7] and the balanced realization problem [36], [15] and the solutions of these problems are again represented by Hamilton-Jacobi equations (or, inequalities). Contrary to the well-developed theory and computational tools for the Riccati equation, which are widely applied, the Hamilton-Jacobi equation is still an impediment to practical applications of nonlinear control theory. In [27], [16], [30], [17] various series expansion techniques are proposed to obtain approximate solu- tions of the Hamilton-Jacobi equation. With these methods, one can calculate sub-optimal solutions using a few terms for simple nonlinearities. Although higher order approximations are possible to obtain for more complicated nonlinearities, their computations are often time-consuming and there is no guarantee that resulting controllers show better performance. Another approach is through succes- sive approximation, where the Hamilton-Jacobi equation is reduced to a sequence of first order linear partial differential equations. The convergence of the algorithm is proven in [24]. In [9] an explicit technique to find approximate solutions to the sequence of partial differential equation is proposed using the Galerkin spectral method and in [41] the authors propose a modification of the successive approximation method and apply the convex optimization technique. The advantage of the Galerkin method is that it is applicable to a larger class of systems, while the disadvantages are that it is depen- dent on how well initial iterate is chosen and requires the calculation of L 2 inner products which can be significantly time-intensive for higher dimensional systems. The state-dependent Riccati equation approach is proposed in [20], [29] where a nonlinear function is rewritten in a linear-like representation. In this method, feedback control is given in a power series form and has a similar disadvantage to the N. Sakamoto is with the Department of Aerospace Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, 464-8603, Japan sakamoto@nuae.nagoya-u.ac.jp A. J. van der Schaft is with the Institute for Mathematics and Computing Science, University of Groningen, 9700 AV Groningen, The Netherlands A.J.van.der.Schaft@math.rug.nl