Control of nonlinear distributed parameter systems using generalized invariants (Q1974000)

The authors present a paradigm for the control of a broad class of nonlinear distributed parameter systems, and generalize the notion of invariants. A system of nonlinear partial differential equations together with their initial and boundary conditions determine the tangent field and the symmetry group in a prolonged space [cf. \textit{E. G. Kalnins} and \textit{W. Miller jun.}, SIAM J. Math. Anal. 16, 221-232 (1985; Zbl 0566.58009)]. This is inspired by Lie group theory that has been widely applied in many fields of mathematical science and engineering. The authors' intent is to combine this approach with a strong stability result. Typically, the stability analysis of nonlinear systems stems from three directions: the Lagrangian approach, the Lyapunov theory, and the energy-based arguments. As a stability criterion, the authors assert that the relation between the observed and the input variables must be an invariant function in the prolonged or so-called jet-space. The major contribution of the paper lies in the establishment of this quantitative stability result, and the subsequent control law computation. The paper is organized as follows: first the authors present preliminaries to provide the basics of differential algebra, leading to the generalized invariants. Then, they state Lyapunov's first theorem to motivate the control problem for nonlinear distributed parameter systems. The formulation is presented for the continuous and discontinuous control cases to underscore the generality of the method.