The local solvability of a Hamilton-Jacobi-Bellman PDE around a nonhyperbolic critical point (Q2706160)

The author considers the local solvability around an equilibrium point of a Hamilton-Jacobi-Bellman equation related to the minimization of a quadratic functional w.r.t. output and control of a nonlinear system. NEWLINENEWLINENEWLINEWhile a local solution is known to exist if the equilibrium point (taken to be the origin) is exponentially stabilizable for the linearized system, nonlinear regulation theory motivates the search for solutions also in the case when exponential stabilizability does not hold. NEWLINENEWLINENEWLINEThe main theorem shows that a sufficient condition for the local solvability is the existence of a sufficiently smooth solution of the Francis-Isidori-Byrnes (FBI) PDE related to the nonlinear regulator problem. In addition, an approximation method for the local solution is presented (the author's MATLAB routines for this purpose are available via ftp). Furthermore, a variation of the result for \(H_\infty\) control problems is given. NEWLINENEWLINENEWLINEThe core of the proof (and of the paper) is a stable and partial center manifold theorem due to Aulbach, Flockerzi and Knobloch [see \textit{B. Aulbach, D. Flockerzi} and \textit{H. W. Knobloch}, ``Invariant manifolds and the concept of asymptotic phase'', Čas. Pěstování Mat. 111, 156--176 (1986; Zbl 0621.34037) and \textit{B. Aulbach} and \textit{D. Flockerzi}, ``An existence theorem for invariant manifolds'', Z. Angew. Math. Phys. 38, 151--171 (1987; Zbl 0618.58031)], for which an alternative proof is given.