On a state space approach to nonlinear Hl control

Arjan van der Schaft

    Research output: Contribution to journalArticleAcademicpeer-review

    344 Citations (Scopus)
    203 Downloads (Pure)


    We study the standard H∞ optimal control problem using state feedback for smooth nonlinear control systems. The main theorem obtained roughly states that the L2-induced norm (from disturbances to inputs and outputs) can be made smaller than a constant γ > 0 if the corresponding H∞ norm for the system linearized at the equilibrium can be made smaller than γ by linear state feedback. Necessary and sufficient conditions for the latter problem are by now well-known, e.g. from the state space approach to linear H∞ optimal control. Our approach to the nonlinear H∞ optimal control problem generalizes the state space approach to the linear H∞ problem by replacing the Hamiltonian matrix and corresponding Riccati equation as used in the linear context by a Hamiltonian vector field together with a Hamiltonian-Jacobi equation corresponding to its stable invariant manifold.
    Original languageUndefined
    Pages (from-to)1-8
    Number of pages8
    JournalSystems and control letters
    Issue number1
    Publication statusPublished - 1991


    • Nonlinear H∞ control
    • State feedback
    • Lagrangian submanifolds
    • L2-induced norm
    • METIS-140587
    • stable manifolds
    • hyperbolic Hamiltonian vector fields
    • IR-72984
    • Hamilton-Jacobi equation
    • linearization

    Cite this